| /* |
| * Tiny arbitrary precision floating point library |
| * |
| * Copyright (c) 2017-2020 Fabrice Bellard |
| * |
| * Permission is hereby granted, free of charge, to any person obtaining a copy |
| * of this software and associated documentation files (the "Software"), to deal |
| * in the Software without restriction, including without limitation the rights |
| * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell |
| * copies of the Software, and to permit persons to whom the Software is |
| * furnished to do so, subject to the following conditions: |
| * |
| * The above copyright notice and this permission notice shall be included in |
| * all copies or substantial portions of the Software. |
| * |
| * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR |
| * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, |
| * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL |
| * THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER |
| * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, |
| * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN |
| * THE SOFTWARE. |
| */ |
| #include <stdlib.h> |
| #include <stdio.h> |
| #include <inttypes.h> |
| #include <math.h> |
| #include <string.h> |
| #include <assert.h> |
| |
| #ifdef __AVX2__ |
| #include <immintrin.h> |
| #endif |
| |
| #include "cutils.h" |
| #include "libbf.h" |
| |
| /* enable it to check the multiplication result */ |
| //#define USE_MUL_CHECK |
| /* enable it to use FFT/NTT multiplication */ |
| #define USE_FFT_MUL |
| /* enable decimal floating point support */ |
| #define USE_BF_DEC |
| |
| //#define inline __attribute__((always_inline)) |
| |
| #ifdef __AVX2__ |
| #define FFT_MUL_THRESHOLD 100 /* in limbs of the smallest factor */ |
| #else |
| #define FFT_MUL_THRESHOLD 100 /* in limbs of the smallest factor */ |
| #endif |
| |
| /* XXX: adjust */ |
| #define DIVNORM_LARGE_THRESHOLD 50 |
| #define UDIV1NORM_THRESHOLD 3 |
| |
| #if LIMB_BITS == 64 |
| #define FMT_LIMB1 "%" PRIx64 |
| #define FMT_LIMB "%016" PRIx64 |
| #define PRId_LIMB PRId64 |
| #define PRIu_LIMB PRIu64 |
| |
| #else |
| |
| #define FMT_LIMB1 "%x" |
| #define FMT_LIMB "%08x" |
| #define PRId_LIMB "d" |
| #define PRIu_LIMB "u" |
| |
| #endif |
| |
| typedef intptr_t mp_size_t; |
| |
| typedef int bf_op2_func_t(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec, |
| bf_flags_t flags); |
| |
| #ifdef USE_FFT_MUL |
| |
| #define FFT_MUL_R_OVERLAP_A (1 << 0) |
| #define FFT_MUL_R_OVERLAP_B (1 << 1) |
| #define FFT_MUL_R_NORESIZE (1 << 2) |
| |
| static no_inline int fft_mul(bf_context_t *s, |
| bf_t *res, limb_t *a_tab, limb_t a_len, |
| limb_t *b_tab, limb_t b_len, int mul_flags); |
| static void fft_clear_cache(bf_context_t *s); |
| #endif |
| #ifdef USE_BF_DEC |
| static limb_t get_digit(const limb_t *tab, limb_t len, slimb_t pos); |
| #endif |
| |
| |
| /* could leading zeros */ |
| static inline int clz(limb_t a) |
| { |
| if (a == 0) { |
| return LIMB_BITS; |
| } else { |
| #if LIMB_BITS == 64 |
| return clz64(a); |
| #else |
| return clz32(a); |
| #endif |
| } |
| } |
| |
| static inline int ctz(limb_t a) |
| { |
| if (a == 0) { |
| return LIMB_BITS; |
| } else { |
| #if LIMB_BITS == 64 |
| return ctz64(a); |
| #else |
| return ctz32(a); |
| #endif |
| } |
| } |
| |
| static inline int ceil_log2(limb_t a) |
| { |
| if (a <= 1) |
| return 0; |
| else |
| return LIMB_BITS - clz(a - 1); |
| } |
| |
| /* b must be >= 1 */ |
| static inline slimb_t ceil_div(slimb_t a, slimb_t b) |
| { |
| if (a >= 0) |
| return (a + b - 1) / b; |
| else |
| return a / b; |
| } |
| |
| /* b must be >= 1 */ |
| static inline slimb_t floor_div(slimb_t a, slimb_t b) |
| { |
| if (a >= 0) { |
| return a / b; |
| } else { |
| return (a - b + 1) / b; |
| } |
| } |
| |
| /* return r = a modulo b (0 <= r <= b - 1. b must be >= 1 */ |
| static inline limb_t smod(slimb_t a, slimb_t b) |
| { |
| a = a % (slimb_t)b; |
| if (a < 0) |
| a += b; |
| return a; |
| } |
| |
| /* signed addition with saturation */ |
| static inline slimb_t sat_add(slimb_t a, slimb_t b) |
| { |
| slimb_t r; |
| r = a + b; |
| /* overflow ? */ |
| if (((a ^ r) & (b ^ r)) < 0) |
| r = (a >> (LIMB_BITS - 1)) ^ (((limb_t)1 << (LIMB_BITS - 1)) - 1); |
| return r; |
| } |
| |
| #define malloc(s) malloc_is_forbidden(s) |
| #define free(p) free_is_forbidden(p) |
| #define realloc(p, s) realloc_is_forbidden(p, s) |
| |
| void bf_context_init(bf_context_t *s, bf_realloc_func_t *realloc_func, |
| void *realloc_opaque) |
| { |
| memset(s, 0, sizeof(*s)); |
| s->realloc_func = realloc_func; |
| s->realloc_opaque = realloc_opaque; |
| } |
| |
| void bf_context_end(bf_context_t *s) |
| { |
| bf_clear_cache(s); |
| } |
| |
| void bf_init(bf_context_t *s, bf_t *r) |
| { |
| r->ctx = s; |
| r->sign = 0; |
| r->expn = BF_EXP_ZERO; |
| r->len = 0; |
| r->tab = NULL; |
| } |
| |
| /* return 0 if OK, -1 if alloc error */ |
| int bf_resize(bf_t *r, limb_t len) |
| { |
| limb_t *tab; |
| |
| if (len != r->len) { |
| tab = bf_realloc(r->ctx, r->tab, len * sizeof(limb_t)); |
| if (!tab && len != 0) |
| return -1; |
| r->tab = tab; |
| r->len = len; |
| } |
| return 0; |
| } |
| |
| /* return 0 or BF_ST_MEM_ERROR */ |
| int bf_set_ui(bf_t *r, uint64_t a) |
| { |
| r->sign = 0; |
| if (a == 0) { |
| r->expn = BF_EXP_ZERO; |
| bf_resize(r, 0); /* cannot fail */ |
| } |
| #if LIMB_BITS == 32 |
| else if (a <= 0xffffffff) |
| #else |
| else |
| #endif |
| { |
| int shift; |
| if (bf_resize(r, 1)) |
| goto fail; |
| shift = clz(a); |
| r->tab[0] = a << shift; |
| r->expn = LIMB_BITS - shift; |
| } |
| #if LIMB_BITS == 32 |
| else { |
| uint32_t a1, a0; |
| int shift; |
| if (bf_resize(r, 2)) |
| goto fail; |
| a0 = a; |
| a1 = a >> 32; |
| shift = clz(a1); |
| r->tab[0] = a0 << shift; |
| r->tab[1] = (a1 << shift) | (a0 >> (LIMB_BITS - shift)); |
| r->expn = 2 * LIMB_BITS - shift; |
| } |
| #endif |
| return 0; |
| fail: |
| bf_set_nan(r); |
| return BF_ST_MEM_ERROR; |
| } |
| |
| /* return 0 or BF_ST_MEM_ERROR */ |
| int bf_set_si(bf_t *r, int64_t a) |
| { |
| int ret; |
| |
| if (a < 0) { |
| ret = bf_set_ui(r, -a); |
| r->sign = 1; |
| } else { |
| ret = bf_set_ui(r, a); |
| } |
| return ret; |
| } |
| |
| void bf_set_nan(bf_t *r) |
| { |
| bf_resize(r, 0); /* cannot fail */ |
| r->expn = BF_EXP_NAN; |
| r->sign = 0; |
| } |
| |
| void bf_set_zero(bf_t *r, int is_neg) |
| { |
| bf_resize(r, 0); /* cannot fail */ |
| r->expn = BF_EXP_ZERO; |
| r->sign = is_neg; |
| } |
| |
| void bf_set_inf(bf_t *r, int is_neg) |
| { |
| bf_resize(r, 0); /* cannot fail */ |
| r->expn = BF_EXP_INF; |
| r->sign = is_neg; |
| } |
| |
| /* return 0 or BF_ST_MEM_ERROR */ |
| int bf_set(bf_t *r, const bf_t *a) |
| { |
| if (r == a) |
| return 0; |
| if (bf_resize(r, a->len)) { |
| bf_set_nan(r); |
| return BF_ST_MEM_ERROR; |
| } |
| r->sign = a->sign; |
| r->expn = a->expn; |
| memcpy(r->tab, a->tab, a->len * sizeof(limb_t)); |
| return 0; |
| } |
| |
| /* equivalent to bf_set(r, a); bf_delete(a) */ |
| void bf_move(bf_t *r, bf_t *a) |
| { |
| bf_context_t *s = r->ctx; |
| if (r == a) |
| return; |
| bf_free(s, r->tab); |
| *r = *a; |
| } |
| |
| static limb_t get_limbz(const bf_t *a, limb_t idx) |
| { |
| if (idx >= a->len) |
| return 0; |
| else |
| return a->tab[idx]; |
| } |
| |
| /* get LIMB_BITS at bit position 'pos' in tab */ |
| static inline limb_t get_bits(const limb_t *tab, limb_t len, slimb_t pos) |
| { |
| limb_t i, a0, a1; |
| int p; |
| |
| i = pos >> LIMB_LOG2_BITS; |
| p = pos & (LIMB_BITS - 1); |
| if (i < len) |
| a0 = tab[i]; |
| else |
| a0 = 0; |
| if (p == 0) { |
| return a0; |
| } else { |
| i++; |
| if (i < len) |
| a1 = tab[i]; |
| else |
| a1 = 0; |
| return (a0 >> p) | (a1 << (LIMB_BITS - p)); |
| } |
| } |
| |
| static inline limb_t get_bit(const limb_t *tab, limb_t len, slimb_t pos) |
| { |
| slimb_t i; |
| i = pos >> LIMB_LOG2_BITS; |
| if (i < 0 || i >= len) |
| return 0; |
| return (tab[i] >> (pos & (LIMB_BITS - 1))) & 1; |
| } |
| |
| static inline limb_t limb_mask(int start, int last) |
| { |
| limb_t v; |
| int n; |
| n = last - start + 1; |
| if (n == LIMB_BITS) |
| v = -1; |
| else |
| v = (((limb_t)1 << n) - 1) << start; |
| return v; |
| } |
| |
| static limb_t mp_scan_nz(const limb_t *tab, mp_size_t n) |
| { |
| mp_size_t i; |
| for(i = 0; i < n; i++) { |
| if (tab[i] != 0) |
| return 1; |
| } |
| return 0; |
| } |
| |
| /* return != 0 if one bit between 0 and bit_pos inclusive is not zero. */ |
| static inline limb_t scan_bit_nz(const bf_t *r, slimb_t bit_pos) |
| { |
| slimb_t pos; |
| limb_t v; |
| |
| pos = bit_pos >> LIMB_LOG2_BITS; |
| if (pos < 0) |
| return 0; |
| v = r->tab[pos] & limb_mask(0, bit_pos & (LIMB_BITS - 1)); |
| if (v != 0) |
| return 1; |
| pos--; |
| while (pos >= 0) { |
| if (r->tab[pos] != 0) |
| return 1; |
| pos--; |
| } |
| return 0; |
| } |
| |
| /* return the addend for rounding. Note that prec can be <= 0 (for |
| BF_FLAG_RADPNT_PREC) */ |
| static int bf_get_rnd_add(int *pret, const bf_t *r, limb_t l, |
| slimb_t prec, int rnd_mode) |
| { |
| int add_one, inexact; |
| limb_t bit1, bit0; |
| |
| if (rnd_mode == BF_RNDF) { |
| bit0 = 1; /* faithful rounding does not honor the INEXACT flag */ |
| } else { |
| /* starting limb for bit 'prec + 1' */ |
| bit0 = scan_bit_nz(r, l * LIMB_BITS - 1 - bf_max(0, prec + 1)); |
| } |
| |
| /* get the bit at 'prec' */ |
| bit1 = get_bit(r->tab, l, l * LIMB_BITS - 1 - prec); |
| inexact = (bit1 | bit0) != 0; |
| |
| add_one = 0; |
| switch(rnd_mode) { |
| case BF_RNDZ: |
| break; |
| case BF_RNDN: |
| if (bit1) { |
| if (bit0) { |
| add_one = 1; |
| } else { |
| /* round to even */ |
| add_one = |
| get_bit(r->tab, l, l * LIMB_BITS - 1 - (prec - 1)); |
| } |
| } |
| break; |
| case BF_RNDD: |
| case BF_RNDU: |
| if (r->sign == (rnd_mode == BF_RNDD)) |
| add_one = inexact; |
| break; |
| case BF_RNDA: |
| add_one = inexact; |
| break; |
| case BF_RNDNA: |
| case BF_RNDF: |
| add_one = bit1; |
| break; |
| default: |
| abort(); |
| } |
| |
| if (inexact) |
| *pret |= BF_ST_INEXACT; |
| return add_one; |
| } |
| |
| static int bf_set_overflow(bf_t *r, int sign, limb_t prec, bf_flags_t flags) |
| { |
| slimb_t i, l, e_max; |
| int rnd_mode; |
| |
| rnd_mode = flags & BF_RND_MASK; |
| if (prec == BF_PREC_INF || |
| rnd_mode == BF_RNDN || |
| rnd_mode == BF_RNDNA || |
| rnd_mode == BF_RNDA || |
| (rnd_mode == BF_RNDD && sign == 1) || |
| (rnd_mode == BF_RNDU && sign == 0)) { |
| bf_set_inf(r, sign); |
| } else { |
| /* set to maximum finite number */ |
| l = (prec + LIMB_BITS - 1) / LIMB_BITS; |
| if (bf_resize(r, l)) { |
| bf_set_nan(r); |
| return BF_ST_MEM_ERROR; |
| } |
| r->tab[0] = limb_mask((-prec) & (LIMB_BITS - 1), |
| LIMB_BITS - 1); |
| for(i = 1; i < l; i++) |
| r->tab[i] = (limb_t)-1; |
| e_max = (limb_t)1 << (bf_get_exp_bits(flags) - 1); |
| r->expn = e_max; |
| r->sign = sign; |
| } |
| return BF_ST_OVERFLOW | BF_ST_INEXACT; |
| } |
| |
| /* round to prec1 bits assuming 'r' is non zero and finite. 'r' is |
| assumed to have length 'l' (1 <= l <= r->len). Note: 'prec1' can be |
| infinite (BF_PREC_INF). 'ret' is 0 or BF_ST_INEXACT if the result |
| is known to be inexact. Can fail with BF_ST_MEM_ERROR in case of |
| overflow not returning infinity. */ |
| static int __bf_round(bf_t *r, limb_t prec1, bf_flags_t flags, limb_t l, |
| int ret) |
| { |
| limb_t v, a; |
| int shift, add_one, rnd_mode; |
| slimb_t i, bit_pos, pos, e_min, e_max, e_range, prec; |
| |
| /* e_min and e_max are computed to match the IEEE 754 conventions */ |
| e_range = (limb_t)1 << (bf_get_exp_bits(flags) - 1); |
| e_min = -e_range + 3; |
| e_max = e_range; |
| |
| if (flags & BF_FLAG_RADPNT_PREC) { |
| /* 'prec' is the precision after the radix point */ |
| if (prec1 != BF_PREC_INF) |
| prec = r->expn + prec1; |
| else |
| prec = prec1; |
| } else if (unlikely(r->expn < e_min) && (flags & BF_FLAG_SUBNORMAL)) { |
| /* restrict the precision in case of potentially subnormal |
| result */ |
| assert(prec1 != BF_PREC_INF); |
| prec = prec1 - (e_min - r->expn); |
| } else { |
| prec = prec1; |
| } |
| |
| /* round to prec bits */ |
| rnd_mode = flags & BF_RND_MASK; |
| add_one = bf_get_rnd_add(&ret, r, l, prec, rnd_mode); |
| |
| if (prec <= 0) { |
| if (add_one) { |
| bf_resize(r, 1); /* cannot fail */ |
| r->tab[0] = (limb_t)1 << (LIMB_BITS - 1); |
| r->expn += 1 - prec; |
| ret |= BF_ST_UNDERFLOW | BF_ST_INEXACT; |
| return ret; |
| } else { |
| goto underflow; |
| } |
| } else if (add_one) { |
| limb_t carry; |
| |
| /* add one starting at digit 'prec - 1' */ |
| bit_pos = l * LIMB_BITS - 1 - (prec - 1); |
| pos = bit_pos >> LIMB_LOG2_BITS; |
| carry = (limb_t)1 << (bit_pos & (LIMB_BITS - 1)); |
| |
| for(i = pos; i < l; i++) { |
| v = r->tab[i] + carry; |
| carry = (v < carry); |
| r->tab[i] = v; |
| if (carry == 0) |
| break; |
| } |
| if (carry) { |
| /* shift right by one digit */ |
| v = 1; |
| for(i = l - 1; i >= pos; i--) { |
| a = r->tab[i]; |
| r->tab[i] = (a >> 1) | (v << (LIMB_BITS - 1)); |
| v = a; |
| } |
| r->expn++; |
| } |
| } |
| |
| /* check underflow */ |
| if (unlikely(r->expn < e_min)) { |
| if (flags & BF_FLAG_SUBNORMAL) { |
| /* if inexact, also set the underflow flag */ |
| if (ret & BF_ST_INEXACT) |
| ret |= BF_ST_UNDERFLOW; |
| } else { |
| underflow: |
| ret |= BF_ST_UNDERFLOW | BF_ST_INEXACT; |
| bf_set_zero(r, r->sign); |
| return ret; |
| } |
| } |
| |
| /* check overflow */ |
| if (unlikely(r->expn > e_max)) |
| return bf_set_overflow(r, r->sign, prec1, flags); |
| |
| /* keep the bits starting at 'prec - 1' */ |
| bit_pos = l * LIMB_BITS - 1 - (prec - 1); |
| i = bit_pos >> LIMB_LOG2_BITS; |
| if (i >= 0) { |
| shift = bit_pos & (LIMB_BITS - 1); |
| if (shift != 0) |
| r->tab[i] &= limb_mask(shift, LIMB_BITS - 1); |
| } else { |
| i = 0; |
| } |
| /* remove trailing zeros */ |
| while (r->tab[i] == 0) |
| i++; |
| if (i > 0) { |
| l -= i; |
| memmove(r->tab, r->tab + i, l * sizeof(limb_t)); |
| } |
| bf_resize(r, l); /* cannot fail */ |
| return ret; |
| } |
| |
| /* 'r' must be a finite number. */ |
| int bf_normalize_and_round(bf_t *r, limb_t prec1, bf_flags_t flags) |
| { |
| limb_t l, v, a; |
| int shift, ret; |
| slimb_t i; |
| |
| // bf_print_str("bf_renorm", r); |
| l = r->len; |
| while (l > 0 && r->tab[l - 1] == 0) |
| l--; |
| if (l == 0) { |
| /* zero */ |
| r->expn = BF_EXP_ZERO; |
| bf_resize(r, 0); /* cannot fail */ |
| ret = 0; |
| } else { |
| r->expn -= (r->len - l) * LIMB_BITS; |
| /* shift to have the MSB set to '1' */ |
| v = r->tab[l - 1]; |
| shift = clz(v); |
| if (shift != 0) { |
| v = 0; |
| for(i = 0; i < l; i++) { |
| a = r->tab[i]; |
| r->tab[i] = (a << shift) | (v >> (LIMB_BITS - shift)); |
| v = a; |
| } |
| r->expn -= shift; |
| } |
| ret = __bf_round(r, prec1, flags, l, 0); |
| } |
| // bf_print_str("r_final", r); |
| return ret; |
| } |
| |
| /* return true if rounding can be done at precision 'prec' assuming |
| the exact result r is such that |r-a| <= 2^(EXP(a)-k). */ |
| /* XXX: check the case where the exponent would be incremented by the |
| rounding */ |
| int bf_can_round(const bf_t *a, slimb_t prec, bf_rnd_t rnd_mode, slimb_t k) |
| { |
| BOOL is_rndn; |
| slimb_t bit_pos, n; |
| limb_t bit; |
| |
| if (a->expn == BF_EXP_INF || a->expn == BF_EXP_NAN) |
| return FALSE; |
| if (rnd_mode == BF_RNDF) { |
| return (k >= (prec + 1)); |
| } |
| if (a->expn == BF_EXP_ZERO) |
| return FALSE; |
| is_rndn = (rnd_mode == BF_RNDN || rnd_mode == BF_RNDNA); |
| if (k < (prec + 2)) |
| return FALSE; |
| bit_pos = a->len * LIMB_BITS - 1 - prec; |
| n = k - prec; |
| /* bit pattern for RNDN or RNDNA: 0111.. or 1000... |
| for other rounding modes: 000... or 111... |
| */ |
| bit = get_bit(a->tab, a->len, bit_pos); |
| bit_pos--; |
| n--; |
| bit ^= is_rndn; |
| /* XXX: slow, but a few iterations on average */ |
| while (n != 0) { |
| if (get_bit(a->tab, a->len, bit_pos) != bit) |
| return TRUE; |
| bit_pos--; |
| n--; |
| } |
| return FALSE; |
| } |
| |
| /* Cannot fail with BF_ST_MEM_ERROR. */ |
| int bf_round(bf_t *r, limb_t prec, bf_flags_t flags) |
| { |
| if (r->len == 0) |
| return 0; |
| return __bf_round(r, prec, flags, r->len, 0); |
| } |
| |
| /* for debugging */ |
| static __maybe_unused void dump_limbs(const char *str, const limb_t *tab, limb_t n) |
| { |
| limb_t i; |
| printf("%s: len=%" PRId_LIMB "\n", str, n); |
| for(i = 0; i < n; i++) { |
| printf("%" PRId_LIMB ": " FMT_LIMB "\n", |
| i, tab[i]); |
| } |
| } |
| |
| void mp_print_str(const char *str, const limb_t *tab, limb_t n) |
| { |
| slimb_t i; |
| printf("%s= 0x", str); |
| for(i = n - 1; i >= 0; i--) { |
| if (i != (n - 1)) |
| printf("_"); |
| printf(FMT_LIMB, tab[i]); |
| } |
| printf("\n"); |
| } |
| |
| static __maybe_unused void mp_print_str_h(const char *str, |
| const limb_t *tab, limb_t n, |
| limb_t high) |
| { |
| slimb_t i; |
| printf("%s= 0x", str); |
| printf(FMT_LIMB, high); |
| for(i = n - 1; i >= 0; i--) { |
| printf("_"); |
| printf(FMT_LIMB, tab[i]); |
| } |
| printf("\n"); |
| } |
| |
| /* for debugging */ |
| void bf_print_str(const char *str, const bf_t *a) |
| { |
| slimb_t i; |
| printf("%s=", str); |
| |
| if (a->expn == BF_EXP_NAN) { |
| printf("NaN"); |
| } else { |
| if (a->sign) |
| putchar('-'); |
| if (a->expn == BF_EXP_ZERO) { |
| putchar('0'); |
| } else if (a->expn == BF_EXP_INF) { |
| printf("Inf"); |
| } else { |
| printf("0x0."); |
| for(i = a->len - 1; i >= 0; i--) |
| printf(FMT_LIMB, a->tab[i]); |
| printf("p%" PRId_LIMB, a->expn); |
| } |
| } |
| printf("\n"); |
| } |
| |
| /* compare the absolute value of 'a' and 'b'. Return < 0 if a < b, 0 |
| if a = b and > 0 otherwise. */ |
| int bf_cmpu(const bf_t *a, const bf_t *b) |
| { |
| slimb_t i; |
| limb_t len, v1, v2; |
| |
| if (a->expn != b->expn) { |
| if (a->expn < b->expn) |
| return -1; |
| else |
| return 1; |
| } |
| len = bf_max(a->len, b->len); |
| for(i = len - 1; i >= 0; i--) { |
| v1 = get_limbz(a, a->len - len + i); |
| v2 = get_limbz(b, b->len - len + i); |
| if (v1 != v2) { |
| if (v1 < v2) |
| return -1; |
| else |
| return 1; |
| } |
| } |
| return 0; |
| } |
| |
| /* Full order: -0 < 0, NaN == NaN and NaN is larger than all other numbers */ |
| int bf_cmp_full(const bf_t *a, const bf_t *b) |
| { |
| int res; |
| |
| if (a->expn == BF_EXP_NAN || b->expn == BF_EXP_NAN) { |
| if (a->expn == b->expn) |
| res = 0; |
| else if (a->expn == BF_EXP_NAN) |
| res = 1; |
| else |
| res = -1; |
| } else if (a->sign != b->sign) { |
| res = 1 - 2 * a->sign; |
| } else { |
| res = bf_cmpu(a, b); |
| if (a->sign) |
| res = -res; |
| } |
| return res; |
| } |
| |
| /* Standard floating point comparison: return 2 if one of the operands |
| is NaN (unordered) or -1, 0, 1 depending on the ordering assuming |
| -0 == +0 */ |
| int bf_cmp(const bf_t *a, const bf_t *b) |
| { |
| int res; |
| |
| if (a->expn == BF_EXP_NAN || b->expn == BF_EXP_NAN) { |
| res = 2; |
| } else if (a->sign != b->sign) { |
| if (a->expn == BF_EXP_ZERO && b->expn == BF_EXP_ZERO) |
| res = 0; |
| else |
| res = 1 - 2 * a->sign; |
| } else { |
| res = bf_cmpu(a, b); |
| if (a->sign) |
| res = -res; |
| } |
| return res; |
| } |
| |
| /* Compute the number of bits 'n' matching the pattern: |
| a= X1000..0 |
| b= X0111..1 |
| |
| When computing a-b, the result will have at least n leading zero |
| bits. |
| |
| Precondition: a > b and a.expn - b.expn = 0 or 1 |
| */ |
| static limb_t count_cancelled_bits(const bf_t *a, const bf_t *b) |
| { |
| slimb_t bit_offset, b_offset, n; |
| int p, p1; |
| limb_t v1, v2, mask; |
| |
| bit_offset = a->len * LIMB_BITS - 1; |
| b_offset = (b->len - a->len) * LIMB_BITS - (LIMB_BITS - 1) + |
| a->expn - b->expn; |
| n = 0; |
| |
| /* first search the equals bits */ |
| for(;;) { |
| v1 = get_limbz(a, bit_offset >> LIMB_LOG2_BITS); |
| v2 = get_bits(b->tab, b->len, bit_offset + b_offset); |
| // printf("v1=" FMT_LIMB " v2=" FMT_LIMB "\n", v1, v2); |
| if (v1 != v2) |
| break; |
| n += LIMB_BITS; |
| bit_offset -= LIMB_BITS; |
| } |
| /* find the position of the first different bit */ |
| p = clz(v1 ^ v2) + 1; |
| n += p; |
| /* then search for '0' in a and '1' in b */ |
| p = LIMB_BITS - p; |
| if (p > 0) { |
| /* search in the trailing p bits of v1 and v2 */ |
| mask = limb_mask(0, p - 1); |
| p1 = bf_min(clz(v1 & mask), clz((~v2) & mask)) - (LIMB_BITS - p); |
| n += p1; |
| if (p1 != p) |
| goto done; |
| } |
| bit_offset -= LIMB_BITS; |
| for(;;) { |
| v1 = get_limbz(a, bit_offset >> LIMB_LOG2_BITS); |
| v2 = get_bits(b->tab, b->len, bit_offset + b_offset); |
| // printf("v1=" FMT_LIMB " v2=" FMT_LIMB "\n", v1, v2); |
| if (v1 != 0 || v2 != -1) { |
| /* different: count the matching bits */ |
| p1 = bf_min(clz(v1), clz(~v2)); |
| n += p1; |
| break; |
| } |
| n += LIMB_BITS; |
| bit_offset -= LIMB_BITS; |
| } |
| done: |
| return n; |
| } |
| |
| static int bf_add_internal(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec, |
| bf_flags_t flags, int b_neg) |
| { |
| const bf_t *tmp; |
| int is_sub, ret, cmp_res, a_sign, b_sign; |
| |
| a_sign = a->sign; |
| b_sign = b->sign ^ b_neg; |
| is_sub = a_sign ^ b_sign; |
| cmp_res = bf_cmpu(a, b); |
| if (cmp_res < 0) { |
| tmp = a; |
| a = b; |
| b = tmp; |
| a_sign = b_sign; /* b_sign is never used later */ |
| } |
| /* abs(a) >= abs(b) */ |
| if (cmp_res == 0 && is_sub && a->expn < BF_EXP_INF) { |
| /* zero result */ |
| bf_set_zero(r, (flags & BF_RND_MASK) == BF_RNDD); |
| ret = 0; |
| } else if (a->len == 0 || b->len == 0) { |
| ret = 0; |
| if (a->expn >= BF_EXP_INF) { |
| if (a->expn == BF_EXP_NAN) { |
| /* at least one operand is NaN */ |
| bf_set_nan(r); |
| } else if (b->expn == BF_EXP_INF && is_sub) { |
| /* infinities with different signs */ |
| bf_set_nan(r); |
| ret = BF_ST_INVALID_OP; |
| } else { |
| bf_set_inf(r, a_sign); |
| } |
| } else { |
| /* at least one zero and not subtract */ |
| bf_set(r, a); |
| r->sign = a_sign; |
| goto renorm; |
| } |
| } else { |
| slimb_t d, a_offset, b_bit_offset, i, cancelled_bits; |
| limb_t carry, v1, v2, u, r_len, carry1, precl, tot_len, z, sub_mask; |
| |
| r->sign = a_sign; |
| r->expn = a->expn; |
| d = a->expn - b->expn; |
| /* must add more precision for the leading cancelled bits in |
| subtraction */ |
| if (is_sub) { |
| if (d <= 1) |
| cancelled_bits = count_cancelled_bits(a, b); |
| else |
| cancelled_bits = 1; |
| } else { |
| cancelled_bits = 0; |
| } |
| |
| /* add two extra bits for rounding */ |
| precl = (cancelled_bits + prec + 2 + LIMB_BITS - 1) / LIMB_BITS; |
| tot_len = bf_max(a->len, b->len + (d + LIMB_BITS - 1) / LIMB_BITS); |
| r_len = bf_min(precl, tot_len); |
| if (bf_resize(r, r_len)) |
| goto fail; |
| a_offset = a->len - r_len; |
| b_bit_offset = (b->len - r_len) * LIMB_BITS + d; |
| |
| /* compute the bits before for the rounding */ |
| carry = is_sub; |
| z = 0; |
| sub_mask = -is_sub; |
| i = r_len - tot_len; |
| while (i < 0) { |
| slimb_t ap, bp; |
| BOOL inflag; |
| |
| ap = a_offset + i; |
| bp = b_bit_offset + i * LIMB_BITS; |
| inflag = FALSE; |
| if (ap >= 0 && ap < a->len) { |
| v1 = a->tab[ap]; |
| inflag = TRUE; |
| } else { |
| v1 = 0; |
| } |
| if (bp + LIMB_BITS > 0 && bp < (slimb_t)(b->len * LIMB_BITS)) { |
| v2 = get_bits(b->tab, b->len, bp); |
| inflag = TRUE; |
| } else { |
| v2 = 0; |
| } |
| if (!inflag) { |
| /* outside 'a' and 'b': go directly to the next value |
| inside a or b so that the running time does not |
| depend on the exponent difference */ |
| i = 0; |
| if (ap < 0) |
| i = bf_min(i, -a_offset); |
| /* b_bit_offset + i * LIMB_BITS + LIMB_BITS >= 1 |
| equivalent to |
| i >= ceil(-b_bit_offset + 1 - LIMB_BITS) / LIMB_BITS) |
| */ |
| if (bp + LIMB_BITS <= 0) |
| i = bf_min(i, (-b_bit_offset) >> LIMB_LOG2_BITS); |
| } else { |
| i++; |
| } |
| v2 ^= sub_mask; |
| u = v1 + v2; |
| carry1 = u < v1; |
| u += carry; |
| carry = (u < carry) | carry1; |
| z |= u; |
| } |
| /* and the result */ |
| for(i = 0; i < r_len; i++) { |
| v1 = get_limbz(a, a_offset + i); |
| v2 = get_bits(b->tab, b->len, b_bit_offset + i * LIMB_BITS); |
| v2 ^= sub_mask; |
| u = v1 + v2; |
| carry1 = u < v1; |
| u += carry; |
| carry = (u < carry) | carry1; |
| r->tab[i] = u; |
| } |
| /* set the extra bits for the rounding */ |
| r->tab[0] |= (z != 0); |
| |
| /* carry is only possible in add case */ |
| if (!is_sub && carry) { |
| if (bf_resize(r, r_len + 1)) |
| goto fail; |
| r->tab[r_len] = 1; |
| r->expn += LIMB_BITS; |
| } |
| renorm: |
| ret = bf_normalize_and_round(r, prec, flags); |
| } |
| return ret; |
| fail: |
| bf_set_nan(r); |
| return BF_ST_MEM_ERROR; |
| } |
| |
| static int __bf_add(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec, |
| bf_flags_t flags) |
| { |
| return bf_add_internal(r, a, b, prec, flags, 0); |
| } |
| |
| static int __bf_sub(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec, |
| bf_flags_t flags) |
| { |
| return bf_add_internal(r, a, b, prec, flags, 1); |
| } |
| |
| limb_t mp_add(limb_t *res, const limb_t *op1, const limb_t *op2, |
| limb_t n, limb_t carry) |
| { |
| slimb_t i; |
| limb_t k, a, v, k1; |
| |
| k = carry; |
| for(i=0;i<n;i++) { |
| v = op1[i]; |
| a = v + op2[i]; |
| k1 = a < v; |
| a = a + k; |
| k = (a < k) | k1; |
| res[i] = a; |
| } |
| return k; |
| } |
| |
| limb_t mp_add_ui(limb_t *tab, limb_t b, size_t n) |
| { |
| size_t i; |
| limb_t k, a; |
| |
| k=b; |
| for(i=0;i<n;i++) { |
| if (k == 0) |
| break; |
| a = tab[i] + k; |
| k = (a < k); |
| tab[i] = a; |
| } |
| return k; |
| } |
| |
| limb_t mp_sub(limb_t *res, const limb_t *op1, const limb_t *op2, |
| mp_size_t n, limb_t carry) |
| { |
| int i; |
| limb_t k, a, v, k1; |
| |
| k = carry; |
| for(i=0;i<n;i++) { |
| v = op1[i]; |
| a = v - op2[i]; |
| k1 = a > v; |
| v = a - k; |
| k = (v > a) | k1; |
| res[i] = v; |
| } |
| return k; |
| } |
| |
| /* compute 0 - op2 */ |
| static limb_t mp_neg(limb_t *res, const limb_t *op2, mp_size_t n, limb_t carry) |
| { |
| int i; |
| limb_t k, a, v, k1; |
| |
| k = carry; |
| for(i=0;i<n;i++) { |
| v = 0; |
| a = v - op2[i]; |
| k1 = a > v; |
| v = a - k; |
| k = (v > a) | k1; |
| res[i] = v; |
| } |
| return k; |
| } |
| |
| limb_t mp_sub_ui(limb_t *tab, limb_t b, mp_size_t n) |
| { |
| mp_size_t i; |
| limb_t k, a, v; |
| |
| k=b; |
| for(i=0;i<n;i++) { |
| v = tab[i]; |
| a = v - k; |
| k = a > v; |
| tab[i] = a; |
| if (k == 0) |
| break; |
| } |
| return k; |
| } |
| |
| /* r = (a + high*B^n) >> shift. Return the remainder r (0 <= r < 2^shift). |
| 1 <= shift <= LIMB_BITS - 1 */ |
| static limb_t mp_shr(limb_t *tab_r, const limb_t *tab, mp_size_t n, |
| int shift, limb_t high) |
| { |
| mp_size_t i; |
| limb_t l, a; |
| |
| assert(shift >= 1 && shift < LIMB_BITS); |
| l = high; |
| for(i = n - 1; i >= 0; i--) { |
| a = tab[i]; |
| tab_r[i] = (a >> shift) | (l << (LIMB_BITS - shift)); |
| l = a; |
| } |
| return l & (((limb_t)1 << shift) - 1); |
| } |
| |
| /* tabr[] = taba[] * b + l. Return the high carry */ |
| static limb_t mp_mul1(limb_t *tabr, const limb_t *taba, limb_t n, |
| limb_t b, limb_t l) |
| { |
| limb_t i; |
| dlimb_t t; |
| |
| for(i = 0; i < n; i++) { |
| t = (dlimb_t)taba[i] * (dlimb_t)b + l; |
| tabr[i] = t; |
| l = t >> LIMB_BITS; |
| } |
| return l; |
| } |
| |
| /* tabr[] += taba[] * b, return the high word. */ |
| static limb_t mp_add_mul1(limb_t *tabr, const limb_t *taba, limb_t n, |
| limb_t b) |
| { |
| limb_t i, l; |
| dlimb_t t; |
| |
| l = 0; |
| for(i = 0; i < n; i++) { |
| t = (dlimb_t)taba[i] * (dlimb_t)b + l + tabr[i]; |
| tabr[i] = t; |
| l = t >> LIMB_BITS; |
| } |
| return l; |
| } |
| |
| /* size of the result : op1_size + op2_size. */ |
| static void mp_mul_basecase(limb_t *result, |
| const limb_t *op1, limb_t op1_size, |
| const limb_t *op2, limb_t op2_size) |
| { |
| limb_t i, r; |
| |
| result[op1_size] = mp_mul1(result, op1, op1_size, op2[0], 0); |
| for(i=1;i<op2_size;i++) { |
| r = mp_add_mul1(result + i, op1, op1_size, op2[i]); |
| result[i + op1_size] = r; |
| } |
| } |
| |
| /* return 0 if OK, -1 if memory error */ |
| /* XXX: change API so that result can be allocated */ |
| int mp_mul(bf_context_t *s, limb_t *result, |
| const limb_t *op1, limb_t op1_size, |
| const limb_t *op2, limb_t op2_size) |
| { |
| #ifdef USE_FFT_MUL |
| if (unlikely(bf_min(op1_size, op2_size) >= FFT_MUL_THRESHOLD)) { |
| bf_t r_s, *r = &r_s; |
| r->tab = result; |
| /* XXX: optimize memory usage in API */ |
| if (fft_mul(s, r, (limb_t *)op1, op1_size, |
| (limb_t *)op2, op2_size, FFT_MUL_R_NORESIZE)) |
| return -1; |
| } else |
| #endif |
| { |
| mp_mul_basecase(result, op1, op1_size, op2, op2_size); |
| } |
| return 0; |
| } |
| |
| /* tabr[] -= taba[] * b. Return the value to substract to the high |
| word. */ |
| static limb_t mp_sub_mul1(limb_t *tabr, const limb_t *taba, limb_t n, |
| limb_t b) |
| { |
| limb_t i, l; |
| dlimb_t t; |
| |
| l = 0; |
| for(i = 0; i < n; i++) { |
| t = tabr[i] - (dlimb_t)taba[i] * (dlimb_t)b - l; |
| tabr[i] = t; |
| l = -(t >> LIMB_BITS); |
| } |
| return l; |
| } |
| |
| /* WARNING: d must be >= 2^(LIMB_BITS-1) */ |
| static inline limb_t udiv1norm_init(limb_t d) |
| { |
| limb_t a0, a1; |
| a1 = -d - 1; |
| a0 = -1; |
| return (((dlimb_t)a1 << LIMB_BITS) | a0) / d; |
| } |
| |
| /* return the quotient and the remainder in '*pr'of 'a1*2^LIMB_BITS+a0 |
| / d' with 0 <= a1 < d. */ |
| static inline limb_t udiv1norm(limb_t *pr, limb_t a1, limb_t a0, |
| limb_t d, limb_t d_inv) |
| { |
| limb_t n1m, n_adj, q, r, ah; |
| dlimb_t a; |
| n1m = ((slimb_t)a0 >> (LIMB_BITS - 1)); |
| n_adj = a0 + (n1m & d); |
| a = (dlimb_t)d_inv * (a1 - n1m) + n_adj; |
| q = (a >> LIMB_BITS) + a1; |
| /* compute a - q * r and update q so that the remainder is\ |
| between 0 and d - 1 */ |
| a = ((dlimb_t)a1 << LIMB_BITS) | a0; |
| a = a - (dlimb_t)q * d - d; |
| ah = a >> LIMB_BITS; |
| q += 1 + ah; |
| r = (limb_t)a + (ah & d); |
| *pr = r; |
| return q; |
| } |
| |
| /* b must be >= 1 << (LIMB_BITS - 1) */ |
| static limb_t mp_div1norm(limb_t *tabr, const limb_t *taba, limb_t n, |
| limb_t b, limb_t r) |
| { |
| slimb_t i; |
| |
| if (n >= UDIV1NORM_THRESHOLD) { |
| limb_t b_inv; |
| b_inv = udiv1norm_init(b); |
| for(i = n - 1; i >= 0; i--) { |
| tabr[i] = udiv1norm(&r, r, taba[i], b, b_inv); |
| } |
| } else { |
| dlimb_t a1; |
| for(i = n - 1; i >= 0; i--) { |
| a1 = ((dlimb_t)r << LIMB_BITS) | taba[i]; |
| tabr[i] = a1 / b; |
| r = a1 % b; |
| } |
| } |
| return r; |
| } |
| |
| static int mp_divnorm_large(bf_context_t *s, |
| limb_t *tabq, limb_t *taba, limb_t na, |
| const limb_t *tabb, limb_t nb); |
| |
| /* base case division: divides taba[0..na-1] by tabb[0..nb-1]. tabb[nb |
| - 1] must be >= 1 << (LIMB_BITS - 1). na - nb must be >= 0. 'taba' |
| is modified and contains the remainder (nb limbs). tabq[0..na-nb] |
| contains the quotient with tabq[na - nb] <= 1. */ |
| static int mp_divnorm(bf_context_t *s, limb_t *tabq, limb_t *taba, limb_t na, |
| const limb_t *tabb, limb_t nb) |
| { |
| limb_t r, a, c, q, v, b1, b1_inv, n, dummy_r; |
| slimb_t i, j; |
| |
| b1 = tabb[nb - 1]; |
| if (nb == 1) { |
| taba[0] = mp_div1norm(tabq, taba, na, b1, 0); |
| return 0; |
| } |
| n = na - nb; |
| if (bf_min(n, nb) >= DIVNORM_LARGE_THRESHOLD) { |
| return mp_divnorm_large(s, tabq, taba, na, tabb, nb); |
| } |
| |
| if (n >= UDIV1NORM_THRESHOLD) |
| b1_inv = udiv1norm_init(b1); |
| else |
| b1_inv = 0; |
| |
| /* first iteration: the quotient is only 0 or 1 */ |
| q = 1; |
| for(j = nb - 1; j >= 0; j--) { |
| if (taba[n + j] != tabb[j]) { |
| if (taba[n + j] < tabb[j]) |
| q = 0; |
| break; |
| } |
| } |
| tabq[n] = q; |
| if (q) { |
| mp_sub(taba + n, taba + n, tabb, nb, 0); |
| } |
| |
| for(i = n - 1; i >= 0; i--) { |
| if (unlikely(taba[i + nb] >= b1)) { |
| q = -1; |
| } else if (b1_inv) { |
| q = udiv1norm(&dummy_r, taba[i + nb], taba[i + nb - 1], b1, b1_inv); |
| } else { |
| dlimb_t al; |
| al = ((dlimb_t)taba[i + nb] << LIMB_BITS) | taba[i + nb - 1]; |
| q = al / b1; |
| r = al % b1; |
| } |
| r = mp_sub_mul1(taba + i, tabb, nb, q); |
| |
| v = taba[i + nb]; |
| a = v - r; |
| c = (a > v); |
| taba[i + nb] = a; |
| |
| if (c != 0) { |
| /* negative result */ |
| for(;;) { |
| q--; |
| c = mp_add(taba + i, taba + i, tabb, nb, 0); |
| /* propagate carry and test if positive result */ |
| if (c != 0) { |
| if (++taba[i + nb] == 0) { |
| break; |
| } |
| } |
| } |
| } |
| tabq[i] = q; |
| } |
| return 0; |
| } |
| |
| /* compute r=B^(2*n)/a such as a*r < B^(2*n) < a*r + 2 with n >= 1. 'a' |
| has n limbs with a[n-1] >= B/2 and 'r' has n+1 limbs with r[n] = 1. |
| |
| See Modern Computer Arithmetic by Richard P. Brent and Paul |
| Zimmermann, algorithm 3.5 */ |
| int mp_recip(bf_context_t *s, limb_t *tabr, const limb_t *taba, limb_t n) |
| { |
| mp_size_t l, h, k, i; |
| limb_t *tabxh, *tabt, c, *tabu; |
| |
| if (n <= 2) { |
| /* return ceil(B^(2*n)/a) - 1 */ |
| /* XXX: could avoid allocation */ |
| tabu = bf_malloc(s, sizeof(limb_t) * (2 * n + 1)); |
| tabt = bf_malloc(s, sizeof(limb_t) * (n + 2)); |
| if (!tabt || !tabu) |
| goto fail; |
| for(i = 0; i < 2 * n; i++) |
| tabu[i] = 0; |
| tabu[2 * n] = 1; |
| if (mp_divnorm(s, tabt, tabu, 2 * n + 1, taba, n)) |
| goto fail; |
| for(i = 0; i < n + 1; i++) |
| tabr[i] = tabt[i]; |
| if (mp_scan_nz(tabu, n) == 0) { |
| /* only happens for a=B^n/2 */ |
| mp_sub_ui(tabr, 1, n + 1); |
| } |
| } else { |
| l = (n - 1) / 2; |
| h = n - l; |
| /* n=2p -> l=p-1, h = p + 1, k = p + 3 |
| n=2p+1-> l=p, h = p + 1; k = p + 2 |
| */ |
| tabt = bf_malloc(s, sizeof(limb_t) * (n + h + 1)); |
| tabu = bf_malloc(s, sizeof(limb_t) * (n + 2 * h - l + 2)); |
| if (!tabt || !tabu) |
| goto fail; |
| tabxh = tabr + l; |
| if (mp_recip(s, tabxh, taba + l, h)) |
| goto fail; |
| if (mp_mul(s, tabt, taba, n, tabxh, h + 1)) /* n + h + 1 limbs */ |
| goto fail; |
| while (tabt[n + h] != 0) { |
| mp_sub_ui(tabxh, 1, h + 1); |
| c = mp_sub(tabt, tabt, taba, n, 0); |
| mp_sub_ui(tabt + n, c, h + 1); |
| } |
| /* T = B^(n+h) - T */ |
| mp_neg(tabt, tabt, n + h + 1, 0); |
| tabt[n + h]++; |
| if (mp_mul(s, tabu, tabt + l, n + h + 1 - l, tabxh, h + 1)) |
| goto fail; |
| /* n + 2*h - l + 2 limbs */ |
| k = 2 * h - l; |
| for(i = 0; i < l; i++) |
| tabr[i] = tabu[i + k]; |
| mp_add(tabr + l, tabr + l, tabu + 2 * h, h, 0); |
| } |
| bf_free(s, tabt); |
| bf_free(s, tabu); |
| return 0; |
| fail: |
| bf_free(s, tabt); |
| bf_free(s, tabu); |
| return -1; |
| } |
| |
| /* return -1, 0 or 1 */ |
| static int mp_cmp(const limb_t *taba, const limb_t *tabb, mp_size_t n) |
| { |
| mp_size_t i; |
| for(i = n - 1; i >= 0; i--) { |
| if (taba[i] != tabb[i]) { |
| if (taba[i] < tabb[i]) |
| return -1; |
| else |
| return 1; |
| } |
| } |
| return 0; |
| } |
| |
| //#define DEBUG_DIVNORM_LARGE |
| //#define DEBUG_DIVNORM_LARGE2 |
| |
| /* subquadratic divnorm */ |
| static int mp_divnorm_large(bf_context_t *s, |
| limb_t *tabq, limb_t *taba, limb_t na, |
| const limb_t *tabb, limb_t nb) |
| { |
| limb_t *tabb_inv, nq, *tabt, i, n; |
| nq = na - nb; |
| #ifdef DEBUG_DIVNORM_LARGE |
| printf("na=%d nb=%d nq=%d\n", (int)na, (int)nb, (int)nq); |
| mp_print_str("a", taba, na); |
| mp_print_str("b", tabb, nb); |
| #endif |
| assert(nq >= 1); |
| n = nq; |
| if (nq < nb) |
| n++; |
| tabb_inv = bf_malloc(s, sizeof(limb_t) * (n + 1)); |
| tabt = bf_malloc(s, sizeof(limb_t) * 2 * (n + 1)); |
| if (!tabb_inv || !tabt) |
| goto fail; |
| |
| if (n >= nb) { |
| for(i = 0; i < n - nb; i++) |
| tabt[i] = 0; |
| for(i = 0; i < nb; i++) |
| tabt[i + n - nb] = tabb[i]; |
| } else { |
| /* truncate B: need to increment it so that the approximate |
| inverse is smaller that the exact inverse */ |
| for(i = 0; i < n; i++) |
| tabt[i] = tabb[i + nb - n]; |
| if (mp_add_ui(tabt, 1, n)) { |
| /* tabt = B^n : tabb_inv = B^n */ |
| memset(tabb_inv, 0, n * sizeof(limb_t)); |
| tabb_inv[n] = 1; |
| goto recip_done; |
| } |
| } |
| if (mp_recip(s, tabb_inv, tabt, n)) |
| goto fail; |
| recip_done: |
| /* Q=A*B^-1 */ |
| if (mp_mul(s, tabt, tabb_inv, n + 1, taba + na - (n + 1), n + 1)) |
| goto fail; |
| |
| for(i = 0; i < nq + 1; i++) |
| tabq[i] = tabt[i + 2 * (n + 1) - (nq + 1)]; |
| #ifdef DEBUG_DIVNORM_LARGE |
| mp_print_str("q", tabq, nq + 1); |
| #endif |
| |
| bf_free(s, tabt); |
| bf_free(s, tabb_inv); |
| tabb_inv = NULL; |
| |
| /* R=A-B*Q */ |
| tabt = bf_malloc(s, sizeof(limb_t) * (na + 1)); |
| if (!tabt) |
| goto fail; |
| if (mp_mul(s, tabt, tabq, nq + 1, tabb, nb)) |
| goto fail; |
| /* we add one more limb for the result */ |
| mp_sub(taba, taba, tabt, nb + 1, 0); |
| bf_free(s, tabt); |
| /* the approximated quotient is smaller than than the exact one, |
| hence we may have to increment it */ |
| #ifdef DEBUG_DIVNORM_LARGE2 |
| int cnt = 0; |
| static int cnt_max; |
| #endif |
| for(;;) { |
| if (taba[nb] == 0 && mp_cmp(taba, tabb, nb) < 0) |
| break; |
| taba[nb] -= mp_sub(taba, taba, tabb, nb, 0); |
| mp_add_ui(tabq, 1, nq + 1); |
| #ifdef DEBUG_DIVNORM_LARGE2 |
| cnt++; |
| #endif |
| } |
| #ifdef DEBUG_DIVNORM_LARGE2 |
| if (cnt > cnt_max) { |
| cnt_max = cnt; |
| printf("\ncnt=%d nq=%d nb=%d\n", cnt_max, (int)nq, (int)nb); |
| } |
| #endif |
| return 0; |
| fail: |
| bf_free(s, tabb_inv); |
| bf_free(s, tabt); |
| return -1; |
| } |
| |
| int bf_mul(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec, |
| bf_flags_t flags) |
| { |
| int ret, r_sign; |
| |
| if (a->len < b->len) { |
| const bf_t *tmp = a; |
| a = b; |
| b = tmp; |
| } |
| r_sign = a->sign ^ b->sign; |
| /* here b->len <= a->len */ |
| if (b->len == 0) { |
| if (a->expn == BF_EXP_NAN || b->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| ret = 0; |
| } else if (a->expn == BF_EXP_INF || b->expn == BF_EXP_INF) { |
| if ((a->expn == BF_EXP_INF && b->expn == BF_EXP_ZERO) || |
| (a->expn == BF_EXP_ZERO && b->expn == BF_EXP_INF)) { |
| bf_set_nan(r); |
| ret = BF_ST_INVALID_OP; |
| } else { |
| bf_set_inf(r, r_sign); |
| ret = 0; |
| } |
| } else { |
| bf_set_zero(r, r_sign); |
| ret = 0; |
| } |
| } else { |
| bf_t tmp, *r1 = NULL; |
| limb_t a_len, b_len, precl; |
| limb_t *a_tab, *b_tab; |
| |
| a_len = a->len; |
| b_len = b->len; |
| |
| if ((flags & BF_RND_MASK) == BF_RNDF) { |
| /* faithful rounding does not require using the full inputs */ |
| precl = (prec + 2 + LIMB_BITS - 1) / LIMB_BITS; |
| a_len = bf_min(a_len, precl); |
| b_len = bf_min(b_len, precl); |
| } |
| a_tab = a->tab + a->len - a_len; |
| b_tab = b->tab + b->len - b_len; |
| |
| #ifdef USE_FFT_MUL |
| if (b_len >= FFT_MUL_THRESHOLD) { |
| int mul_flags = 0; |
| if (r == a) |
| mul_flags |= FFT_MUL_R_OVERLAP_A; |
| if (r == b) |
| mul_flags |= FFT_MUL_R_OVERLAP_B; |
| if (fft_mul(r->ctx, r, a_tab, a_len, b_tab, b_len, mul_flags)) |
| goto fail; |
| } else |
| #endif |
| { |
| if (r == a || r == b) { |
| bf_init(r->ctx, &tmp); |
| r1 = r; |
| r = &tmp; |
| } |
| if (bf_resize(r, a_len + b_len)) { |
| fail: |
| bf_set_nan(r); |
| ret = BF_ST_MEM_ERROR; |
| goto done; |
| } |
| mp_mul_basecase(r->tab, a_tab, a_len, b_tab, b_len); |
| } |
| r->sign = r_sign; |
| r->expn = a->expn + b->expn; |
| ret = bf_normalize_and_round(r, prec, flags); |
| done: |
| if (r == &tmp) |
| bf_move(r1, &tmp); |
| } |
| return ret; |
| } |
| |
| /* multiply 'r' by 2^e */ |
| int bf_mul_2exp(bf_t *r, slimb_t e, limb_t prec, bf_flags_t flags) |
| { |
| slimb_t e_max; |
| if (r->len == 0) |
| return 0; |
| e_max = ((limb_t)1 << BF_EXT_EXP_BITS_MAX) - 1; |
| e = bf_max(e, -e_max); |
| e = bf_min(e, e_max); |
| r->expn += e; |
| return __bf_round(r, prec, flags, r->len, 0); |
| } |
| |
| /* Return e such as a=m*2^e with m odd integer. return 0 if a is zero, |
| Infinite or Nan. */ |
| slimb_t bf_get_exp_min(const bf_t *a) |
| { |
| slimb_t i; |
| limb_t v; |
| int k; |
| |
| for(i = 0; i < a->len; i++) { |
| v = a->tab[i]; |
| if (v != 0) { |
| k = ctz(v); |
| return a->expn - (a->len - i) * LIMB_BITS + k; |
| } |
| } |
| return 0; |
| } |
| |
| /* a and b must be finite numbers with a >= 0 and b > 0. 'q' is the |
| integer defined as floor(a/b) and r = a - q * b. */ |
| static void bf_tdivremu(bf_t *q, bf_t *r, |
| const bf_t *a, const bf_t *b) |
| { |
| if (bf_cmpu(a, b) < 0) { |
| bf_set_ui(q, 0); |
| bf_set(r, a); |
| } else { |
| bf_div(q, a, b, bf_max(a->expn - b->expn + 1, 2), BF_RNDZ); |
| bf_rint(q, BF_RNDZ); |
| bf_mul(r, q, b, BF_PREC_INF, BF_RNDZ); |
| bf_sub(r, a, r, BF_PREC_INF, BF_RNDZ); |
| } |
| } |
| |
| static int __bf_div(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec, |
| bf_flags_t flags) |
| { |
| bf_context_t *s = r->ctx; |
| int ret, r_sign; |
| limb_t n, nb, precl; |
| |
| r_sign = a->sign ^ b->sign; |
| if (a->expn >= BF_EXP_INF || b->expn >= BF_EXP_INF) { |
| if (a->expn == BF_EXP_NAN || b->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| return 0; |
| } else if (a->expn == BF_EXP_INF && b->expn == BF_EXP_INF) { |
| bf_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else if (a->expn == BF_EXP_INF) { |
| bf_set_inf(r, r_sign); |
| return 0; |
| } else { |
| bf_set_zero(r, r_sign); |
| return 0; |
| } |
| } else if (a->expn == BF_EXP_ZERO) { |
| if (b->expn == BF_EXP_ZERO) { |
| bf_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else { |
| bf_set_zero(r, r_sign); |
| return 0; |
| } |
| } else if (b->expn == BF_EXP_ZERO) { |
| bf_set_inf(r, r_sign); |
| return BF_ST_DIVIDE_ZERO; |
| } |
| |
| /* number of limbs of the quotient (2 extra bits for rounding) */ |
| precl = (prec + 2 + LIMB_BITS - 1) / LIMB_BITS; |
| nb = b->len; |
| n = bf_max(a->len, precl); |
| |
| { |
| limb_t *taba, na; |
| slimb_t d; |
| |
| na = n + nb; |
| taba = bf_malloc(s, (na + 1) * sizeof(limb_t)); |
| if (!taba) |
| goto fail; |
| d = na - a->len; |
| memset(taba, 0, d * sizeof(limb_t)); |
| memcpy(taba + d, a->tab, a->len * sizeof(limb_t)); |
| if (bf_resize(r, n + 1)) |
| goto fail1; |
| if (mp_divnorm(s, r->tab, taba, na, b->tab, nb)) { |
| fail1: |
| bf_free(s, taba); |
| goto fail; |
| } |
| /* see if non zero remainder */ |
| if (mp_scan_nz(taba, nb)) |
| r->tab[0] |= 1; |
| bf_free(r->ctx, taba); |
| r->expn = a->expn - b->expn + LIMB_BITS; |
| r->sign = r_sign; |
| ret = bf_normalize_and_round(r, prec, flags); |
| } |
| return ret; |
| fail: |
| bf_set_nan(r); |
| return BF_ST_MEM_ERROR; |
| } |
| |
| /* division and remainder. |
| |
| rnd_mode is the rounding mode for the quotient. The additional |
| rounding mode BF_RND_EUCLIDIAN is supported. |
| |
| 'q' is an integer. 'r' is rounded with prec and flags (prec can be |
| BF_PREC_INF). |
| */ |
| int bf_divrem(bf_t *q, bf_t *r, const bf_t *a, const bf_t *b, |
| limb_t prec, bf_flags_t flags, int rnd_mode) |
| { |
| bf_t a1_s, *a1 = &a1_s; |
| bf_t b1_s, *b1 = &b1_s; |
| int q_sign, ret; |
| BOOL is_ceil, is_rndn; |
| |
| assert(q != a && q != b); |
| assert(r != a && r != b); |
| assert(q != r); |
| |
| if (a->len == 0 || b->len == 0) { |
| bf_set_zero(q, 0); |
| if (a->expn == BF_EXP_NAN || b->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| return 0; |
| } else if (a->expn == BF_EXP_INF || b->expn == BF_EXP_ZERO) { |
| bf_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else { |
| bf_set(r, a); |
| return bf_round(r, prec, flags); |
| } |
| } |
| |
| q_sign = a->sign ^ b->sign; |
| is_rndn = (rnd_mode == BF_RNDN || rnd_mode == BF_RNDNA); |
| switch(rnd_mode) { |
| default: |
| case BF_RNDZ: |
| case BF_RNDN: |
| case BF_RNDNA: |
| is_ceil = FALSE; |
| break; |
| case BF_RNDD: |
| is_ceil = q_sign; |
| break; |
| case BF_RNDU: |
| is_ceil = q_sign ^ 1; |
| break; |
| case BF_RNDA: |
| is_ceil = TRUE; |
| break; |
| case BF_DIVREM_EUCLIDIAN: |
| is_ceil = a->sign; |
| break; |
| } |
| |
| a1->expn = a->expn; |
| a1->tab = a->tab; |
| a1->len = a->len; |
| a1->sign = 0; |
| |
| b1->expn = b->expn; |
| b1->tab = b->tab; |
| b1->len = b->len; |
| b1->sign = 0; |
| |
| /* XXX: could improve to avoid having a large 'q' */ |
| bf_tdivremu(q, r, a1, b1); |
| if (bf_is_nan(q) || bf_is_nan(r)) |
| goto fail; |
| |
| if (r->len != 0) { |
| if (is_rndn) { |
| int res; |
| b1->expn--; |
| res = bf_cmpu(r, b1); |
| b1->expn++; |
| if (res > 0 || |
| (res == 0 && |
| (rnd_mode == BF_RNDNA || |
| get_bit(q->tab, q->len, q->len * LIMB_BITS - q->expn)))) { |
| goto do_sub_r; |
| } |
| } else if (is_ceil) { |
| do_sub_r: |
| ret = bf_add_si(q, q, 1, BF_PREC_INF, BF_RNDZ); |
| ret |= bf_sub(r, r, b1, BF_PREC_INF, BF_RNDZ); |
| if (ret & BF_ST_MEM_ERROR) |
| goto fail; |
| } |
| } |
| |
| r->sign ^= a->sign; |
| q->sign = q_sign; |
| return bf_round(r, prec, flags); |
| fail: |
| bf_set_nan(q); |
| bf_set_nan(r); |
| return BF_ST_MEM_ERROR; |
| } |
| |
| int bf_rem(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec, |
| bf_flags_t flags, int rnd_mode) |
| { |
| bf_t q_s, *q = &q_s; |
| int ret; |
| |
| bf_init(r->ctx, q); |
| ret = bf_divrem(q, r, a, b, prec, flags, rnd_mode); |
| bf_delete(q); |
| return ret; |
| } |
| |
| static inline int bf_get_limb(slimb_t *pres, const bf_t *a, int flags) |
| { |
| #if LIMB_BITS == 32 |
| return bf_get_int32(pres, a, flags); |
| #else |
| return bf_get_int64(pres, a, flags); |
| #endif |
| } |
| |
| int bf_remquo(slimb_t *pq, bf_t *r, const bf_t *a, const bf_t *b, limb_t prec, |
| bf_flags_t flags, int rnd_mode) |
| { |
| bf_t q_s, *q = &q_s; |
| int ret; |
| |
| bf_init(r->ctx, q); |
| ret = bf_divrem(q, r, a, b, prec, flags, rnd_mode); |
| bf_get_limb(pq, q, BF_GET_INT_MOD); |
| bf_delete(q); |
| return ret; |
| } |
| |
| static __maybe_unused inline limb_t mul_mod(limb_t a, limb_t b, limb_t m) |
| { |
| dlimb_t t; |
| t = (dlimb_t)a * (dlimb_t)b; |
| return t % m; |
| } |
| |
| #if defined(USE_MUL_CHECK) |
| static limb_t mp_mod1(const limb_t *tab, limb_t n, limb_t m, limb_t r) |
| { |
| slimb_t i; |
| dlimb_t t; |
| |
| for(i = n - 1; i >= 0; i--) { |
| t = ((dlimb_t)r << LIMB_BITS) | tab[i]; |
| r = t % m; |
| } |
| return r; |
| } |
| #endif |
| |
| static const uint16_t sqrt_table[192] = { |
| 128,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,144,145,146,147,148,149,150,150,151,152,153,154,155,155,156,157,158,159,160,160,161,162,163,163,164,165,166,167,167,168,169,170,170,171,172,173,173,174,175,176,176,177,178,178,179,180,181,181,182,183,183,184,185,185,186,187,187,188,189,189,190,191,192,192,193,193,194,195,195,196,197,197,198,199,199,200,201,201,202,203,203,204,204,205,206,206,207,208,208,209,209,210,211,211,212,212,213,214,214,215,215,216,217,217,218,218,219,219,220,221,221,222,222,223,224,224,225,225,226,226,227,227,228,229,229,230,230,231,231,232,232,233,234,234,235,235,236,236,237,237,238,238,239,240,240,241,241,242,242,243,243,244,244,245,245,246,246,247,247,248,248,249,249,250,250,251,251,252,252,253,253,254,254,255, |
| }; |
| |
| /* a >= 2^(LIMB_BITS - 2). Return (s, r) with s=floor(sqrt(a)) and |
| r=a-s^2. 0 <= r <= 2 * s */ |
| static limb_t mp_sqrtrem1(limb_t *pr, limb_t a) |
| { |
| limb_t s1, r1, s, r, q, u, num; |
| |
| /* use a table for the 16 -> 8 bit sqrt */ |
| s1 = sqrt_table[(a >> (LIMB_BITS - 8)) - 64]; |
| r1 = (a >> (LIMB_BITS - 16)) - s1 * s1; |
| if (r1 > 2 * s1) { |
| r1 -= 2 * s1 + 1; |
| s1++; |
| } |
| |
| /* one iteration to get a 32 -> 16 bit sqrt */ |
| num = (r1 << 8) | ((a >> (LIMB_BITS - 32 + 8)) & 0xff); |
| q = num / (2 * s1); /* q <= 2^8 */ |
| u = num % (2 * s1); |
| s = (s1 << 8) + q; |
| r = (u << 8) | ((a >> (LIMB_BITS - 32)) & 0xff); |
| r -= q * q; |
| if ((slimb_t)r < 0) { |
| s--; |
| r += 2 * s + 1; |
| } |
| |
| #if LIMB_BITS == 64 |
| s1 = s; |
| r1 = r; |
| /* one more iteration for 64 -> 32 bit sqrt */ |
| num = (r1 << 16) | ((a >> (LIMB_BITS - 64 + 16)) & 0xffff); |
| q = num / (2 * s1); /* q <= 2^16 */ |
| u = num % (2 * s1); |
| s = (s1 << 16) + q; |
| r = (u << 16) | ((a >> (LIMB_BITS - 64)) & 0xffff); |
| r -= q * q; |
| if ((slimb_t)r < 0) { |
| s--; |
| r += 2 * s + 1; |
| } |
| #endif |
| *pr = r; |
| return s; |
| } |
| |
| /* return floor(sqrt(a)) */ |
| limb_t bf_isqrt(limb_t a) |
| { |
| limb_t s, r; |
| int k; |
| |
| if (a == 0) |
| return 0; |
| k = clz(a) & ~1; |
| s = mp_sqrtrem1(&r, a << k); |
| s >>= (k >> 1); |
| return s; |
| } |
| |
| static limb_t mp_sqrtrem2(limb_t *tabs, limb_t *taba) |
| { |
| limb_t s1, r1, s, q, u, a0, a1; |
| dlimb_t r, num; |
| int l; |
| |
| a0 = taba[0]; |
| a1 = taba[1]; |
| s1 = mp_sqrtrem1(&r1, a1); |
| l = LIMB_BITS / 2; |
| num = ((dlimb_t)r1 << l) | (a0 >> l); |
| q = num / (2 * s1); |
| u = num % (2 * s1); |
| s = (s1 << l) + q; |
| r = ((dlimb_t)u << l) | (a0 & (((limb_t)1 << l) - 1)); |
| if (unlikely((q >> l) != 0)) |
| r -= (dlimb_t)1 << LIMB_BITS; /* special case when q=2^l */ |
| else |
| r -= q * q; |
| if ((slimb_t)(r >> LIMB_BITS) < 0) { |
| s--; |
| r += 2 * (dlimb_t)s + 1; |
| } |
| tabs[0] = s; |
| taba[0] = r; |
| return r >> LIMB_BITS; |
| } |
| |
| //#define DEBUG_SQRTREM |
| |
| /* tmp_buf must contain (n / 2 + 1 limbs). *prh contains the highest |
| limb of the remainder. */ |
| static int mp_sqrtrem_rec(bf_context_t *s, limb_t *tabs, limb_t *taba, limb_t n, |
| limb_t *tmp_buf, limb_t *prh) |
| { |
| limb_t l, h, rh, ql, qh, c, i; |
| |
| if (n == 1) { |
| *prh = mp_sqrtrem2(tabs, taba); |
| return 0; |
| } |
| #ifdef DEBUG_SQRTREM |
| mp_print_str("a", taba, 2 * n); |
| #endif |
| l = n / 2; |
| h = n - l; |
| if (mp_sqrtrem_rec(s, tabs + l, taba + 2 * l, h, tmp_buf, &qh)) |
| return -1; |
| #ifdef DEBUG_SQRTREM |
| mp_print_str("s1", tabs + l, h); |
| mp_print_str_h("r1", taba + 2 * l, h, qh); |
| mp_print_str_h("r2", taba + l, n, qh); |
| #endif |
| |
| /* the remainder is in taba + 2 * l. Its high bit is in qh */ |
| if (qh) { |
| mp_sub(taba + 2 * l, taba + 2 * l, tabs + l, h, 0); |
| } |
| /* instead of dividing by 2*s, divide by s (which is normalized) |
| and update q and r */ |
| if (mp_divnorm(s, tmp_buf, taba + l, n, tabs + l, h)) |
| return -1; |
| qh += tmp_buf[l]; |
| for(i = 0; i < l; i++) |
| tabs[i] = tmp_buf[i]; |
| ql = mp_shr(tabs, tabs, l, 1, qh & 1); |
| qh = qh >> 1; /* 0 or 1 */ |
| if (ql) |
| rh = mp_add(taba + l, taba + l, tabs + l, h, 0); |
| else |
| rh = 0; |
| #ifdef DEBUG_SQRTREM |
| mp_print_str_h("q", tabs, l, qh); |
| mp_print_str_h("u", taba + l, h, rh); |
| #endif |
| |
| mp_add_ui(tabs + l, qh, h); |
| #ifdef DEBUG_SQRTREM |
| mp_print_str_h("s2", tabs, n, sh); |
| #endif |
| |
| /* q = qh, tabs[l - 1 ... 0], r = taba[n - 1 ... l] */ |
| /* subtract q^2. if qh = 1 then q = B^l, so we can take shortcuts */ |
| if (qh) { |
| c = qh; |
| } else { |
| if (mp_mul(s, taba + n, tabs, l, tabs, l)) |
| return -1; |
| c = mp_sub(taba, taba, taba + n, 2 * l, 0); |
| } |
| rh -= mp_sub_ui(taba + 2 * l, c, n - 2 * l); |
| if ((slimb_t)rh < 0) { |
| mp_sub_ui(tabs, 1, n); |
| rh += mp_add_mul1(taba, tabs, n, 2); |
| rh += mp_add_ui(taba, 1, n); |
| } |
| *prh = rh; |
| return 0; |
| } |
| |
| /* 'taba' has 2*n limbs with n >= 1 and taba[2*n-1] >= 2 ^ (LIMB_BITS |
| - 2). Return (s, r) with s=floor(sqrt(a)) and r=a-s^2. 0 <= r <= 2 |
| * s. tabs has n limbs. r is returned in the lower n limbs of |
| taba. Its r[n] is the returned value of the function. */ |
| /* Algorithm from the article "Karatsuba Square Root" by Paul Zimmermann and |
| inspirated from its GMP implementation */ |
| int mp_sqrtrem(bf_context_t *s, limb_t *tabs, limb_t *taba, limb_t n) |
| { |
| limb_t tmp_buf1[8]; |
| limb_t *tmp_buf; |
| mp_size_t n2; |
| int ret; |
| n2 = n / 2 + 1; |
| if (n2 <= countof(tmp_buf1)) { |
| tmp_buf = tmp_buf1; |
| } else { |
| tmp_buf = bf_malloc(s, sizeof(limb_t) * n2); |
| if (!tmp_buf) |
| return -1; |
| } |
| ret = mp_sqrtrem_rec(s, tabs, taba, n, tmp_buf, taba + n); |
| if (tmp_buf != tmp_buf1) |
| bf_free(s, tmp_buf); |
| return ret; |
| } |
| |
| /* Integer square root with remainder. 'a' must be an integer. r = |
| floor(sqrt(a)) and rem = a - r^2. BF_ST_INEXACT is set if the result |
| is inexact. 'rem' can be NULL if the remainder is not needed. */ |
| int bf_sqrtrem(bf_t *r, bf_t *rem1, const bf_t *a) |
| { |
| int ret; |
| |
| if (a->len == 0) { |
| if (a->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| } else if (a->expn == BF_EXP_INF && a->sign) { |
| goto invalid_op; |
| } else { |
| bf_set(r, a); |
| } |
| if (rem1) |
| bf_set_ui(rem1, 0); |
| ret = 0; |
| } else if (a->sign) { |
| invalid_op: |
| bf_set_nan(r); |
| if (rem1) |
| bf_set_ui(rem1, 0); |
| ret = BF_ST_INVALID_OP; |
| } else { |
| bf_t rem_s, *rem; |
| |
| bf_sqrt(r, a, (a->expn + 1) / 2, BF_RNDZ); |
| bf_rint(r, BF_RNDZ); |
| /* see if the result is exact by computing the remainder */ |
| if (rem1) { |
| rem = rem1; |
| } else { |
| rem = &rem_s; |
| bf_init(r->ctx, rem); |
| } |
| /* XXX: could avoid recomputing the remainder */ |
| bf_mul(rem, r, r, BF_PREC_INF, BF_RNDZ); |
| bf_neg(rem); |
| bf_add(rem, rem, a, BF_PREC_INF, BF_RNDZ); |
| if (bf_is_nan(rem)) { |
| ret = BF_ST_MEM_ERROR; |
| goto done; |
| } |
| if (rem->len != 0) { |
| ret = BF_ST_INEXACT; |
| } else { |
| ret = 0; |
| } |
| done: |
| if (!rem1) |
| bf_delete(rem); |
| } |
| return ret; |
| } |
| |
| int bf_sqrt(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags) |
| { |
| bf_context_t *s = a->ctx; |
| int ret; |
| |
| assert(r != a); |
| |
| if (a->len == 0) { |
| if (a->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| } else if (a->expn == BF_EXP_INF && a->sign) { |
| goto invalid_op; |
| } else { |
| bf_set(r, a); |
| } |
| ret = 0; |
| } else if (a->sign) { |
| invalid_op: |
| bf_set_nan(r); |
| ret = BF_ST_INVALID_OP; |
| } else { |
| limb_t *a1; |
| slimb_t n, n1; |
| limb_t res; |
| |
| /* convert the mantissa to an integer with at least 2 * |
| prec + 4 bits */ |
| n = (2 * (prec + 2) + 2 * LIMB_BITS - 1) / (2 * LIMB_BITS); |
| if (bf_resize(r, n)) |
| goto fail; |
| a1 = bf_malloc(s, sizeof(limb_t) * 2 * n); |
| if (!a1) |
| goto fail; |
| n1 = bf_min(2 * n, a->len); |
| memset(a1, 0, (2 * n - n1) * sizeof(limb_t)); |
| memcpy(a1 + 2 * n - n1, a->tab + a->len - n1, n1 * sizeof(limb_t)); |
| if (a->expn & 1) { |
| res = mp_shr(a1, a1, 2 * n, 1, 0); |
| } else { |
| res = 0; |
| } |
| if (mp_sqrtrem(s, r->tab, a1, n)) { |
| bf_free(s, a1); |
| goto fail; |
| } |
| if (!res) { |
| res = mp_scan_nz(a1, n + 1); |
| } |
| bf_free(s, a1); |
| if (!res) { |
| res = mp_scan_nz(a->tab, a->len - n1); |
| } |
| if (res != 0) |
| r->tab[0] |= 1; |
| r->sign = 0; |
| r->expn = (a->expn + 1) >> 1; |
| ret = bf_round(r, prec, flags); |
| } |
| return ret; |
| fail: |
| bf_set_nan(r); |
| return BF_ST_MEM_ERROR; |
| } |
| |
| static no_inline int bf_op2(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec, |
| bf_flags_t flags, bf_op2_func_t *func) |
| { |
| bf_t tmp; |
| int ret; |
| |
| if (r == a || r == b) { |
| bf_init(r->ctx, &tmp); |
| ret = func(&tmp, a, b, prec, flags); |
| bf_move(r, &tmp); |
| } else { |
| ret = func(r, a, b, prec, flags); |
| } |
| return ret; |
| } |
| |
| int bf_add(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec, |
| bf_flags_t flags) |
| { |
| return bf_op2(r, a, b, prec, flags, __bf_add); |
| } |
| |
| int bf_sub(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec, |
| bf_flags_t flags) |
| { |
| return bf_op2(r, a, b, prec, flags, __bf_sub); |
| } |
| |
| int bf_div(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec, |
| bf_flags_t flags) |
| { |
| return bf_op2(r, a, b, prec, flags, __bf_div); |
| } |
| |
| int bf_mul_ui(bf_t *r, const bf_t *a, uint64_t b1, limb_t prec, |
| bf_flags_t flags) |
| { |
| bf_t b; |
| int ret; |
| bf_init(r->ctx, &b); |
| ret = bf_set_ui(&b, b1); |
| ret |= bf_mul(r, a, &b, prec, flags); |
| bf_delete(&b); |
| return ret; |
| } |
| |
| int bf_mul_si(bf_t *r, const bf_t *a, int64_t b1, limb_t prec, |
| bf_flags_t flags) |
| { |
| bf_t b; |
| int ret; |
| bf_init(r->ctx, &b); |
| ret = bf_set_si(&b, b1); |
| ret |= bf_mul(r, a, &b, prec, flags); |
| bf_delete(&b); |
| return ret; |
| } |
| |
| int bf_add_si(bf_t *r, const bf_t *a, int64_t b1, limb_t prec, |
| bf_flags_t flags) |
| { |
| bf_t b; |
| int ret; |
| |
| bf_init(r->ctx, &b); |
| ret = bf_set_si(&b, b1); |
| ret |= bf_add(r, a, &b, prec, flags); |
| bf_delete(&b); |
| return ret; |
| } |
| |
| static int bf_pow_ui(bf_t *r, const bf_t *a, limb_t b, limb_t prec, |
| bf_flags_t flags) |
| { |
| int ret, n_bits, i; |
| |
| assert(r != a); |
| if (b == 0) |
| return bf_set_ui(r, 1); |
| ret = bf_set(r, a); |
| n_bits = LIMB_BITS - clz(b); |
| for(i = n_bits - 2; i >= 0; i--) { |
| ret |= bf_mul(r, r, r, prec, flags); |
| if ((b >> i) & 1) |
| ret |= bf_mul(r, r, a, prec, flags); |
| } |
| return ret; |
| } |
| |
| static int bf_pow_ui_ui(bf_t *r, limb_t a1, limb_t b, |
| limb_t prec, bf_flags_t flags) |
| { |
| bf_t a; |
| int ret; |
| |
| if (a1 == 10 && b <= LIMB_DIGITS) { |
| /* use precomputed powers. We do not round at this point |
| because we expect the caller to do it */ |
| ret = bf_set_ui(r, mp_pow_dec[b]); |
| } else { |
| bf_init(r->ctx, &a); |
| ret = bf_set_ui(&a, a1); |
| ret |= bf_pow_ui(r, &a, b, prec, flags); |
| bf_delete(&a); |
| } |
| return ret; |
| } |
| |
| /* convert to integer (infinite precision) */ |
| int bf_rint(bf_t *r, int rnd_mode) |
| { |
| return bf_round(r, 0, rnd_mode | BF_FLAG_RADPNT_PREC); |
| } |
| |
| /* logical operations */ |
| #define BF_LOGIC_OR 0 |
| #define BF_LOGIC_XOR 1 |
| #define BF_LOGIC_AND 2 |
| |
| static inline limb_t bf_logic_op1(limb_t a, limb_t b, int op) |
| { |
| switch(op) { |
| case BF_LOGIC_OR: |
| return a | b; |
| case BF_LOGIC_XOR: |
| return a ^ b; |
| default: |
| case BF_LOGIC_AND: |
| return a & b; |
| } |
| } |
| |
| static int bf_logic_op(bf_t *r, const bf_t *a1, const bf_t *b1, int op) |
| { |
| bf_t b1_s, a1_s, *a, *b; |
| limb_t a_sign, b_sign, r_sign; |
| slimb_t l, i, a_bit_offset, b_bit_offset; |
| limb_t v1, v2, v1_mask, v2_mask, r_mask; |
| int ret; |
| |
| assert(r != a1 && r != b1); |
| |
| if (a1->expn <= 0) |
| a_sign = 0; /* minus zero is considered as positive */ |
| else |
| a_sign = a1->sign; |
| |
| if (b1->expn <= 0) |
| b_sign = 0; /* minus zero is considered as positive */ |
| else |
| b_sign = b1->sign; |
| |
| if (a_sign) { |
| a = &a1_s; |
| bf_init(r->ctx, a); |
| if (bf_add_si(a, a1, 1, BF_PREC_INF, BF_RNDZ)) { |
| b = NULL; |
| goto fail; |
| } |
| } else { |
| a = (bf_t *)a1; |
| } |
| |
| if (b_sign) { |
| b = &b1_s; |
| bf_init(r->ctx, b); |
| if (bf_add_si(b, b1, 1, BF_PREC_INF, BF_RNDZ)) |
| goto fail; |
| } else { |
| b = (bf_t *)b1; |
| } |
| |
| r_sign = bf_logic_op1(a_sign, b_sign, op); |
| if (op == BF_LOGIC_AND && r_sign == 0) { |
| /* no need to compute extra zeros for and */ |
| if (a_sign == 0 && b_sign == 0) |
| l = bf_min(a->expn, b->expn); |
| else if (a_sign == 0) |
| l = a->expn; |
| else |
| l = b->expn; |
| } else { |
| l = bf_max(a->expn, b->expn); |
| } |
| /* Note: a or b can be zero */ |
| l = (bf_max(l, 1) + LIMB_BITS - 1) / LIMB_BITS; |
| if (bf_resize(r, l)) |
| goto fail; |
| a_bit_offset = a->len * LIMB_BITS - a->expn; |
| b_bit_offset = b->len * LIMB_BITS - b->expn; |
| v1_mask = -a_sign; |
| v2_mask = -b_sign; |
| r_mask = -r_sign; |
| for(i = 0; i < l; i++) { |
| v1 = get_bits(a->tab, a->len, a_bit_offset + i * LIMB_BITS) ^ v1_mask; |
| v2 = get_bits(b->tab, b->len, b_bit_offset + i * LIMB_BITS) ^ v2_mask; |
| r->tab[i] = bf_logic_op1(v1, v2, op) ^ r_mask; |
| } |
| r->expn = l * LIMB_BITS; |
| r->sign = r_sign; |
| bf_normalize_and_round(r, BF_PREC_INF, BF_RNDZ); /* cannot fail */ |
| if (r_sign) { |
| if (bf_add_si(r, r, -1, BF_PREC_INF, BF_RNDZ)) |
| goto fail; |
| } |
| ret = 0; |
| done: |
| if (a == &a1_s) |
| bf_delete(a); |
| if (b == &b1_s) |
| bf_delete(b); |
| return ret; |
| fail: |
| bf_set_nan(r); |
| ret = BF_ST_MEM_ERROR; |
| goto done; |
| } |
| |
| /* 'a' and 'b' must be integers. Return 0 or BF_ST_MEM_ERROR. */ |
| int bf_logic_or(bf_t *r, const bf_t *a, const bf_t *b) |
| { |
| return bf_logic_op(r, a, b, BF_LOGIC_OR); |
| } |
| |
| /* 'a' and 'b' must be integers. Return 0 or BF_ST_MEM_ERROR. */ |
| int bf_logic_xor(bf_t *r, const bf_t *a, const bf_t *b) |
| { |
| return bf_logic_op(r, a, b, BF_LOGIC_XOR); |
| } |
| |
| /* 'a' and 'b' must be integers. Return 0 or BF_ST_MEM_ERROR. */ |
| int bf_logic_and(bf_t *r, const bf_t *a, const bf_t *b) |
| { |
| return bf_logic_op(r, a, b, BF_LOGIC_AND); |
| } |
| |
| /* conversion between fixed size types */ |
| |
| typedef union { |
| double d; |
| uint64_t u; |
| } Float64Union; |
| |
| int bf_get_float64(const bf_t *a, double *pres, bf_rnd_t rnd_mode) |
| { |
| Float64Union u; |
| int e, ret; |
| uint64_t m; |
| |
| ret = 0; |
| if (a->expn == BF_EXP_NAN) { |
| u.u = 0x7ff8000000000000; /* quiet nan */ |
| } else { |
| bf_t b_s, *b = &b_s; |
| |
| bf_init(a->ctx, b); |
| bf_set(b, a); |
| if (bf_is_finite(b)) { |
| ret = bf_round(b, 53, rnd_mode | BF_FLAG_SUBNORMAL | bf_set_exp_bits(11)); |
| } |
| if (b->expn == BF_EXP_INF) { |
| e = (1 << 11) - 1; |
| m = 0; |
| } else if (b->expn == BF_EXP_ZERO) { |
| e = 0; |
| m = 0; |
| } else { |
| e = b->expn + 1023 - 1; |
| #if LIMB_BITS == 32 |
| if (b->len == 2) { |
| m = ((uint64_t)b->tab[1] << 32) | b->tab[0]; |
| } else { |
| m = ((uint64_t)b->tab[0] << 32); |
| } |
| #else |
| m = b->tab[0]; |
| #endif |
| if (e <= 0) { |
| /* subnormal */ |
| m = m >> (12 - e); |
| e = 0; |
| } else { |
| m = (m << 1) >> 12; |
| } |
| } |
| u.u = m | ((uint64_t)e << 52) | ((uint64_t)b->sign << 63); |
| bf_delete(b); |
| } |
| *pres = u.d; |
| return ret; |
| } |
| |
| int bf_set_float64(bf_t *a, double d) |
| { |
| Float64Union u; |
| uint64_t m; |
| int shift, e, sgn; |
| |
| u.d = d; |
| sgn = u.u >> 63; |
| e = (u.u >> 52) & ((1 << 11) - 1); |
| m = u.u & (((uint64_t)1 << 52) - 1); |
| if (e == ((1 << 11) - 1)) { |
| if (m != 0) { |
| bf_set_nan(a); |
| } else { |
| bf_set_inf(a, sgn); |
| } |
| } else if (e == 0) { |
| if (m == 0) { |
| bf_set_zero(a, sgn); |
| } else { |
| /* subnormal number */ |
| m <<= 12; |
| shift = clz64(m); |
| m <<= shift; |
| e = -shift; |
| goto norm; |
| } |
| } else { |
| m = (m << 11) | ((uint64_t)1 << 63); |
| norm: |
| a->expn = e - 1023 + 1; |
| #if LIMB_BITS == 32 |
| if (bf_resize(a, 2)) |
| goto fail; |
| a->tab[0] = m; |
| a->tab[1] = m >> 32; |
| #else |
| if (bf_resize(a, 1)) |
| goto fail; |
| a->tab[0] = m; |
| #endif |
| a->sign = sgn; |
| } |
| return 0; |
| fail: |
| bf_set_nan(a); |
| return BF_ST_MEM_ERROR; |
| } |
| |
| /* The rounding mode is always BF_RNDZ. Return BF_ST_INVALID_OP if there |
| is an overflow and 0 otherwise. */ |
| int bf_get_int32(int *pres, const bf_t *a, int flags) |
| { |
| uint32_t v; |
| int ret; |
| if (a->expn >= BF_EXP_INF) { |
| ret = BF_ST_INVALID_OP; |
| if (flags & BF_GET_INT_MOD) { |
| v = 0; |
| } else if (a->expn == BF_EXP_INF) { |
| v = (uint32_t)INT32_MAX + a->sign; |
| } else { |
| v = INT32_MAX; |
| } |
| } else if (a->expn <= 0) { |
| v = 0; |
| ret = 0; |
| } else if (a->expn <= 31) { |
| v = a->tab[a->len - 1] >> (LIMB_BITS - a->expn); |
| if (a->sign) |
| v = -v; |
| ret = 0; |
| } else if (!(flags & BF_GET_INT_MOD)) { |
| ret = BF_ST_INVALID_OP; |
| if (a->sign) { |
| v = (uint32_t)INT32_MAX + 1; |
| if (a->expn == 32 && |
| (a->tab[a->len - 1] >> (LIMB_BITS - 32)) == v) { |
| ret = 0; |
| } |
| } else { |
| v = INT32_MAX; |
| } |
| } else { |
| v = get_bits(a->tab, a->len, a->len * LIMB_BITS - a->expn); |
| if (a->sign) |
| v = -v; |
| ret = 0; |
| } |
| *pres = v; |
| return ret; |
| } |
| |
| /* The rounding mode is always BF_RNDZ. Return BF_ST_INVALID_OP if there |
| is an overflow and 0 otherwise. */ |
| int bf_get_int64(int64_t *pres, const bf_t *a, int flags) |
| { |
| uint64_t v; |
| int ret; |
| if (a->expn >= BF_EXP_INF) { |
| ret = BF_ST_INVALID_OP; |
| if (flags & BF_GET_INT_MOD) { |
| v = 0; |
| } else if (a->expn == BF_EXP_INF) { |
| v = (uint64_t)INT64_MAX + a->sign; |
| } else { |
| v = INT64_MAX; |
| } |
| } else if (a->expn <= 0) { |
| v = 0; |
| ret = 0; |
| } else if (a->expn <= 63) { |
| #if LIMB_BITS == 32 |
| if (a->expn <= 32) |
| v = a->tab[a->len - 1] >> (LIMB_BITS - a->expn); |
| else |
| v = (((uint64_t)a->tab[a->len - 1] << 32) | |
| get_limbz(a, a->len - 2)) >> (64 - a->expn); |
| #else |
| v = a->tab[a->len - 1] >> (LIMB_BITS - a->expn); |
| #endif |
| if (a->sign) |
| v = -v; |
| ret = 0; |
| } else if (!(flags & BF_GET_INT_MOD)) { |
| ret = BF_ST_INVALID_OP; |
| if (a->sign) { |
| uint64_t v1; |
| v = (uint64_t)INT64_MAX + 1; |
| if (a->expn == 64) { |
| v1 = a->tab[a->len - 1]; |
| #if LIMB_BITS == 32 |
| v1 = (v1 << 32) | get_limbz(a, a->len - 2); |
| #endif |
| if (v1 == v) |
| ret = 0; |
| } |
| } else { |
| v = INT64_MAX; |
| } |
| } else { |
| slimb_t bit_pos = a->len * LIMB_BITS - a->expn; |
| v = get_bits(a->tab, a->len, bit_pos); |
| #if LIMB_BITS == 32 |
| v |= (uint64_t)get_bits(a->tab, a->len, bit_pos + 32) << 32; |
| #endif |
| if (a->sign) |
| v = -v; |
| ret = 0; |
| } |
| *pres = v; |
| return ret; |
| } |
| |
| /* The rounding mode is always BF_RNDZ. Return BF_ST_INVALID_OP if there |
| is an overflow and 0 otherwise. */ |
| int bf_get_uint64(uint64_t *pres, const bf_t *a) |
| { |
| uint64_t v; |
| int ret; |
| if (a->expn == BF_EXP_NAN) { |
| goto overflow; |
| } else if (a->expn <= 0) { |
| v = 0; |
| ret = 0; |
| } else if (a->sign) { |
| v = 0; |
| ret = BF_ST_INVALID_OP; |
| } else if (a->expn <= 64) { |
| #if LIMB_BITS == 32 |
| if (a->expn <= 32) |
| v = a->tab[a->len - 1] >> (LIMB_BITS - a->expn); |
| else |
| v = (((uint64_t)a->tab[a->len - 1] << 32) | |
| get_limbz(a, a->len - 2)) >> (64 - a->expn); |
| #else |
| v = a->tab[a->len - 1] >> (LIMB_BITS - a->expn); |
| #endif |
| ret = 0; |
| } else { |
| overflow: |
| v = UINT64_MAX; |
| ret = BF_ST_INVALID_OP; |
| } |
| *pres = v; |
| return ret; |
| } |
| |
| /* base conversion from radix */ |
| |
| static const uint8_t digits_per_limb_table[BF_RADIX_MAX - 1] = { |
| #if LIMB_BITS == 32 |
| 32,20,16,13,12,11,10,10, 9, 9, 8, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, |
| #else |
| 64,40,32,27,24,22,21,20,19,18,17,17,16,16,16,15,15,15,14,14,14,14,13,13,13,13,13,13,13,12,12,12,12,12,12, |
| #endif |
| }; |
| |
| static limb_t get_limb_radix(int radix) |
| { |
| int i, k; |
| limb_t radixl; |
| |
| k = digits_per_limb_table[radix - 2]; |
| radixl = radix; |
| for(i = 1; i < k; i++) |
| radixl *= radix; |
| return radixl; |
| } |
| |
| /* return != 0 if error */ |
| static int bf_integer_from_radix_rec(bf_t *r, const limb_t *tab, |
| limb_t n, int level, limb_t n0, |
| limb_t radix, bf_t *pow_tab) |
| { |
| int ret; |
| if (n == 1) { |
| ret = bf_set_ui(r, tab[0]); |
| } else { |
| bf_t T_s, *T = &T_s, *B; |
| limb_t n1, n2; |
| |
| n2 = (((n0 * 2) >> (level + 1)) + 1) / 2; |
| n1 = n - n2; |
| // printf("level=%d n0=%ld n1=%ld n2=%ld\n", level, n0, n1, n2); |
| B = &pow_tab[level]; |
| if (B->len == 0) { |
| ret = bf_pow_ui_ui(B, radix, n2, BF_PREC_INF, BF_RNDZ); |
| if (ret) |
| return ret; |
| } |
| ret = bf_integer_from_radix_rec(r, tab + n2, n1, level + 1, n0, |
| radix, pow_tab); |
| if (ret) |
| return ret; |
| ret = bf_mul(r, r, B, BF_PREC_INF, BF_RNDZ); |
| if (ret) |
| return ret; |
| bf_init(r->ctx, T); |
| ret = bf_integer_from_radix_rec(T, tab, n2, level + 1, n0, |
| radix, pow_tab); |
| if (!ret) |
| ret = bf_add(r, r, T, BF_PREC_INF, BF_RNDZ); |
| bf_delete(T); |
| } |
| return ret; |
| // bf_print_str(" r=", r); |
| } |
| |
| /* return 0 if OK != 0 if memory error */ |
| static int bf_integer_from_radix(bf_t *r, const limb_t *tab, |
| limb_t n, limb_t radix) |
| { |
| bf_context_t *s = r->ctx; |
| int pow_tab_len, i, ret; |
| limb_t radixl; |
| bf_t *pow_tab; |
| |
| radixl = get_limb_radix(radix); |
| pow_tab_len = ceil_log2(n) + 2; /* XXX: check */ |
| pow_tab = bf_malloc(s, sizeof(pow_tab[0]) * pow_tab_len); |
| if (!pow_tab) |
| return -1; |
| for(i = 0; i < pow_tab_len; i++) |
| bf_init(r->ctx, &pow_tab[i]); |
| ret = bf_integer_from_radix_rec(r, tab, n, 0, n, radixl, pow_tab); |
| for(i = 0; i < pow_tab_len; i++) { |
| bf_delete(&pow_tab[i]); |
| } |
| bf_free(s, pow_tab); |
| return ret; |
| } |
| |
| /* compute and round T * radix^expn. */ |
| int bf_mul_pow_radix(bf_t *r, const bf_t *T, limb_t radix, |
| slimb_t expn, limb_t prec, bf_flags_t flags) |
| { |
| int ret, expn_sign, overflow; |
| slimb_t e, extra_bits, prec1, ziv_extra_bits; |
| bf_t B_s, *B = &B_s; |
| |
| if (T->len == 0) { |
| return bf_set(r, T); |
| } else if (expn == 0) { |
| ret = bf_set(r, T); |
| ret |= bf_round(r, prec, flags); |
| return ret; |
| } |
| |
| e = expn; |
| expn_sign = 0; |
| if (e < 0) { |
| e = -e; |
| expn_sign = 1; |
| } |
| bf_init(r->ctx, B); |
| if (prec == BF_PREC_INF) { |
| /* infinite precision: only used if the result is known to be exact */ |
| ret = bf_pow_ui_ui(B, radix, e, BF_PREC_INF, BF_RNDN); |
| if (expn_sign) { |
| ret |= bf_div(r, T, B, T->len * LIMB_BITS, BF_RNDN); |
| } else { |
| ret |= bf_mul(r, T, B, BF_PREC_INF, BF_RNDN); |
| } |
| } else { |
| ziv_extra_bits = 16; |
| for(;;) { |
| prec1 = prec + ziv_extra_bits; |
| /* XXX: correct overflow/underflow handling */ |
| /* XXX: rigorous error analysis needed */ |
| extra_bits = ceil_log2(e) * 2 + 1; |
| ret = bf_pow_ui_ui(B, radix, e, prec1 + extra_bits, BF_RNDN | BF_FLAG_EXT_EXP); |
| overflow = !bf_is_finite(B); |
| /* XXX: if bf_pow_ui_ui returns an exact result, can stop |
| after the next operation */ |
| if (expn_sign) |
| ret |= bf_div(r, T, B, prec1 + extra_bits, BF_RNDN | BF_FLAG_EXT_EXP); |
| else |
| ret |= bf_mul(r, T, B, prec1 + extra_bits, BF_RNDN | BF_FLAG_EXT_EXP); |
| if (ret & BF_ST_MEM_ERROR) |
| break; |
| if ((ret & BF_ST_INEXACT) && |
| !bf_can_round(r, prec, flags & BF_RND_MASK, prec1) && |
| !overflow) { |
| /* and more precision and retry */ |
| ziv_extra_bits = ziv_extra_bits + (ziv_extra_bits / 2); |
| } else { |
| /* XXX: need to use __bf_round() to pass the inexact |
| flag for the subnormal case */ |
| ret = bf_round(r, prec, flags) | (ret & BF_ST_INEXACT); |
| break; |
| } |
| } |
| } |
| bf_delete(B); |
| return ret; |
| } |
| |
| static inline int to_digit(int c) |
| { |
| if (c >= '0' && c <= '9') |
| return c - '0'; |
| else if (c >= 'A' && c <= 'Z') |
| return c - 'A' + 10; |
| else if (c >= 'a' && c <= 'z') |
| return c - 'a' + 10; |
| else |
| return 36; |
| } |
| |
| /* add a limb at 'pos' and decrement pos. new space is created if |
| needed. Return 0 if OK, -1 if memory error */ |
| static int bf_add_limb(bf_t *a, slimb_t *ppos, limb_t v) |
| { |
| slimb_t pos; |
| pos = *ppos; |
| if (unlikely(pos < 0)) { |
| limb_t new_size, d, *new_tab; |
| new_size = bf_max(a->len + 1, a->len * 3 / 2); |
| new_tab = bf_realloc(a->ctx, a->tab, sizeof(limb_t) * new_size); |
| if (!new_tab) |
| return -1; |
| a->tab = new_tab; |
| d = new_size - a->len; |
| memmove(a->tab + d, a->tab, a->len * sizeof(limb_t)); |
| a->len = new_size; |
| pos += d; |
| } |
| a->tab[pos--] = v; |
| *ppos = pos; |
| return 0; |
| } |
| |
| static int bf_tolower(int c) |
| { |
| if (c >= 'A' && c <= 'Z') |
| c = c - 'A' + 'a'; |
| return c; |
| } |
| |
| static int strcasestart(const char *str, const char *val, const char **ptr) |
| { |
| const char *p, *q; |
| p = str; |
| q = val; |
| while (*q != '\0') { |
| if (bf_tolower(*p) != *q) |
| return 0; |
| p++; |
| q++; |
| } |
| if (ptr) |
| *ptr = p; |
| return 1; |
| } |
| |
| static int bf_atof_internal(bf_t *r, slimb_t *pexponent, |
| const char *str, const char **pnext, int radix, |
| limb_t prec, bf_flags_t flags, BOOL is_dec) |
| { |
| const char *p, *p_start; |
| int is_neg, radix_bits, exp_is_neg, ret, digits_per_limb, shift; |
| limb_t cur_limb; |
| slimb_t pos, expn, int_len, digit_count; |
| BOOL has_decpt, is_bin_exp; |
| bf_t a_s, *a; |
| |
| *pexponent = 0; |
| p = str; |
| if (!(flags & BF_ATOF_NO_NAN_INF) && radix <= 16 && |
| strcasestart(p, "nan", &p)) { |
| bf_set_nan(r); |
| ret = 0; |
| goto done; |
| } |
| is_neg = 0; |
| |
| if (p[0] == '+') { |
| p++; |
| p_start = p; |
| } else if (p[0] == '-') { |
| is_neg = 1; |
| p++; |
| p_start = p; |
| } else { |
| p_start = p; |
| } |
| if (p[0] == '0') { |
| if ((p[1] == 'x' || p[1] == 'X') && |
| (radix == 0 || radix == 16) && |
| !(flags & BF_ATOF_NO_HEX)) { |
| radix = 16; |
| p += 2; |
| } else if ((p[1] == 'o' || p[1] == 'O') && |
| radix == 0 && (flags & BF_ATOF_BIN_OCT)) { |
| p += 2; |
| radix = 8; |
| } else if ((p[1] == 'b' || p[1] == 'B') && |
| radix == 0 && (flags & BF_ATOF_BIN_OCT)) { |
| p += 2; |
| radix = 2; |
| } else { |
| goto no_prefix; |
| } |
| /* there must be a digit after the prefix */ |
| if (to_digit((uint8_t)*p) >= radix) { |
| bf_set_nan(r); |
| ret = 0; |
| goto done; |
| } |
| no_prefix: ; |
| } else { |
| if (!(flags & BF_ATOF_NO_NAN_INF) && radix <= 16 && |
| strcasestart(p, "inf", &p)) { |
| bf_set_inf(r, is_neg); |
| ret = 0; |
| goto done; |
| } |
| } |
| |
| if (radix == 0) |
| radix = 10; |
| if (is_dec) { |
| assert(radix == 10); |
| radix_bits = 0; |
| a = r; |
| } else if ((radix & (radix - 1)) != 0) { |
| radix_bits = 0; /* base is not a power of two */ |
| a = &a_s; |
| bf_init(r->ctx, a); |
| } else { |
| radix_bits = ceil_log2(radix); |
| a = r; |
| } |
| |
| /* skip leading zeros */ |
| /* XXX: could also skip zeros after the decimal point */ |
| while (*p == '0') |
| p++; |
| |
| if (radix_bits) { |
| shift = digits_per_limb = LIMB_BITS; |
| } else { |
| radix_bits = 0; |
| shift = digits_per_limb = digits_per_limb_table[radix - 2]; |
| } |
| cur_limb = 0; |
| bf_resize(a, 1); |
| pos = 0; |
| has_decpt = FALSE; |
| int_len = digit_count = 0; |
| for(;;) { |
| limb_t c; |
| if (*p == '.' && (p > p_start || to_digit(p[1]) < radix)) { |
| if (has_decpt) |
| break; |
| has_decpt = TRUE; |
| int_len = digit_count; |
| p++; |
| } |
| c = to_digit(*p); |
| if (c >= radix) |
| break; |
| digit_count++; |
| p++; |
| if (radix_bits) { |
| shift -= radix_bits; |
| if (shift <= 0) { |
| cur_limb |= c >> (-shift); |
| if (bf_add_limb(a, &pos, cur_limb)) |
| goto mem_error; |
| if (shift < 0) |
| cur_limb = c << (LIMB_BITS + shift); |
| else |
| cur_limb = 0; |
| shift += LIMB_BITS; |
| } else { |
| cur_limb |= c << shift; |
| } |
| } else { |
| cur_limb = cur_limb * radix + c; |
| shift--; |
| if (shift == 0) { |
| if (bf_add_limb(a, &pos, cur_limb)) |
| goto mem_error; |
| shift = digits_per_limb; |
| cur_limb = 0; |
| } |
| } |
| } |
| if (!has_decpt) |
| int_len = digit_count; |
| |
| /* add the last limb and pad with zeros */ |
| if (shift != digits_per_limb) { |
| if (radix_bits == 0) { |
| while (shift != 0) { |
| cur_limb *= radix; |
| shift--; |
| } |
| } |
| if (bf_add_limb(a, &pos, cur_limb)) { |
| mem_error: |
| ret = BF_ST_MEM_ERROR; |
| if (!radix_bits) |
| bf_delete(a); |
| bf_set_nan(r); |
| goto done; |
| } |
| } |
| |
| /* reset the next limbs to zero (we prefer to reallocate in the |
| renormalization) */ |
| memset(a->tab, 0, (pos + 1) * sizeof(limb_t)); |
| |
| if (p == p_start) { |
| ret = 0; |
| if (!radix_bits) |
| bf_delete(a); |
| bf_set_nan(r); |
| goto done; |
| } |
| |
| /* parse the exponent, if any */ |
| expn = 0; |
| is_bin_exp = FALSE; |
| if (((radix == 10 && (*p == 'e' || *p == 'E')) || |
| (radix != 10 && (*p == '@' || |
| (radix_bits && (*p == 'p' || *p == 'P'))))) && |
| p > p_start) { |
| is_bin_exp = (*p == 'p' || *p == 'P'); |
| p++; |
| exp_is_neg = 0; |
| if (*p == '+') { |
| p++; |
| } else if (*p == '-') { |
| exp_is_neg = 1; |
| p++; |
| } |
| for(;;) { |
| int c; |
| c = to_digit(*p); |
| if (c >= 10) |
| break; |
| if (unlikely(expn > ((BF_RAW_EXP_MAX - 2 - 9) / 10))) { |
| /* exponent overflow */ |
| if (exp_is_neg) { |
| bf_set_zero(r, is_neg); |
| ret = BF_ST_UNDERFLOW | BF_ST_INEXACT; |
| } else { |
| bf_set_inf(r, is_neg); |
| ret = BF_ST_OVERFLOW | BF_ST_INEXACT; |
| } |
| goto done; |
| } |
| p++; |
| expn = expn * 10 + c; |
| } |
| if (exp_is_neg) |
| expn = -expn; |
| } |
| if (is_dec) { |
| a->expn = expn + int_len; |
| a->sign = is_neg; |
| ret = bfdec_normalize_and_round((bfdec_t *)a, prec, flags); |
| } else if (radix_bits) { |
| /* XXX: may overflow */ |
| if (!is_bin_exp) |
| expn *= radix_bits; |
| a->expn = expn + (int_len * radix_bits); |
| a->sign = is_neg; |
| ret = bf_normalize_and_round(a, prec, flags); |
| } else { |
| limb_t l; |
| pos++; |
| l = a->len - pos; /* number of limbs */ |
| if (l == 0) { |
| bf_set_zero(r, is_neg); |
| ret = 0; |
| } else { |
| bf_t T_s, *T = &T_s; |
| |
| expn -= l * digits_per_limb - int_len; |
| bf_init(r->ctx, T); |
| if (bf_integer_from_radix(T, a->tab + pos, l, radix)) { |
| bf_set_nan(r); |
| ret = BF_ST_MEM_ERROR; |
| } else { |
| T->sign = is_neg; |
| if (flags & BF_ATOF_EXPONENT) { |
| /* return the exponent */ |
| *pexponent = expn; |
| ret = bf_set(r, T); |
| } else { |
| ret = bf_mul_pow_radix(r, T, radix, expn, prec, flags); |
| } |
| } |
| bf_delete(T); |
| } |
| bf_delete(a); |
| } |
| done: |
| if (pnext) |
| *pnext = p; |
| return ret; |
| } |
| |
| /* |
| Return (status, n, exp). 'status' is the floating point status. 'n' |
| is the parsed number. |
| |
| If (flags & BF_ATOF_EXPONENT) and if the radix is not a power of |
| two, the parsed number is equal to r * |
| (*pexponent)^radix. Otherwise *pexponent = 0. |
| */ |
| int bf_atof2(bf_t *r, slimb_t *pexponent, |
| const char *str, const char **pnext, int radix, |
| limb_t prec, bf_flags_t flags) |
| { |
| return bf_atof_internal(r, pexponent, str, pnext, radix, prec, flags, |
| FALSE); |
| } |
| |
| int bf_atof(bf_t *r, const char *str, const char **pnext, int radix, |
| limb_t prec, bf_flags_t flags) |
| { |
| slimb_t dummy_exp; |
| return bf_atof_internal(r, &dummy_exp, str, pnext, radix, prec, flags, FALSE); |
| } |
| |
| /* base conversion to radix */ |
| |
| #if LIMB_BITS == 64 |
| #define RADIXL_10 UINT64_C(10000000000000000000) |
| #else |
| #define RADIXL_10 UINT64_C(1000000000) |
| #endif |
| |
| static const uint32_t inv_log2_radix[BF_RADIX_MAX - 1][LIMB_BITS / 32 + 1] = { |
| #if LIMB_BITS == 32 |
| { 0x80000000, 0x00000000,}, |
| { 0x50c24e60, 0xd4d4f4a7,}, |
| { 0x40000000, 0x00000000,}, |
| { 0x372068d2, 0x0a1ee5ca,}, |
| { 0x3184648d, 0xb8153e7a,}, |
| { 0x2d983275, 0x9d5369c4,}, |
| { 0x2aaaaaaa, 0xaaaaaaab,}, |
| { 0x28612730, 0x6a6a7a54,}, |
| { 0x268826a1, 0x3ef3fde6,}, |
| { 0x25001383, 0xbac8a744,}, |
| { 0x23b46706, 0x82c0c709,}, |
| { 0x229729f1, 0xb2c83ded,}, |
| { 0x219e7ffd, 0xa5ad572b,}, |
| { 0x20c33b88, 0xda7c29ab,}, |
| { 0x20000000, 0x00000000,}, |
| { 0x1f50b57e, 0xac5884b3,}, |
| { 0x1eb22cc6, 0x8aa6e26f,}, |
| { 0x1e21e118, 0x0c5daab2,}, |
| { 0x1d9dcd21, 0x439834e4,}, |
| { 0x1d244c78, 0x367a0d65,}, |
| { 0x1cb40589, 0xac173e0c,}, |
| { 0x1c4bd95b, 0xa8d72b0d,}, |
| { 0x1bead768, 0x98f8ce4c,}, |
| { 0x1b903469, 0x050f72e5,}, |
| { 0x1b3b433f, 0x2eb06f15,}, |
| { 0x1aeb6f75, 0x9c46fc38,}, |
| { 0x1aa038eb, 0x0e3bfd17,}, |
| { 0x1a593062, 0xb38d8c56,}, |
| { 0x1a15f4c3, 0x2b95a2e6,}, |
| { 0x19d630dc, 0xcc7ddef9,}, |
| { 0x19999999, 0x9999999a,}, |
| { 0x195fec80, 0x8a609431,}, |
| { 0x1928ee7b, 0x0b4f22f9,}, |
| { 0x18f46acf, 0x8c06e318,}, |
| { 0x18c23246, 0xdc0a9f3d,}, |
| #else |
| { 0x80000000, 0x00000000, 0x00000000,}, |
| { 0x50c24e60, 0xd4d4f4a7, 0x021f57bc,}, |
| { 0x40000000, 0x00000000, 0x00000000,}, |
| { 0x372068d2, 0x0a1ee5ca, 0x19ea911b,}, |
| { 0x3184648d, 0xb8153e7a, 0x7fc2d2e1,}, |
| { 0x2d983275, 0x9d5369c4, 0x4dec1661,}, |
| { 0x2aaaaaaa, 0xaaaaaaaa, 0xaaaaaaab,}, |
| { 0x28612730, 0x6a6a7a53, 0x810fabde,}, |
| { 0x268826a1, 0x3ef3fde6, 0x23e2566b,}, |
| { 0x25001383, 0xbac8a744, 0x385a3349,}, |
| { 0x23b46706, 0x82c0c709, 0x3f891718,}, |
| { 0x229729f1, 0xb2c83ded, 0x15fba800,}, |
| { 0x219e7ffd, 0xa5ad572a, 0xe169744b,}, |
| { 0x20c33b88, 0xda7c29aa, 0x9bddee52,}, |
| { 0x20000000, 0x00000000, 0x00000000,}, |
| { 0x1f50b57e, 0xac5884b3, 0x70e28eee,}, |
| { 0x1eb22cc6, 0x8aa6e26f, 0x06d1a2a2,}, |
| { 0x1e21e118, 0x0c5daab1, 0x81b4f4bf,}, |
| { 0x1d9dcd21, 0x439834e3, 0x81667575,}, |
| { 0x1d244c78, 0x367a0d64, 0xc8204d6d,}, |
| { 0x1cb40589, 0xac173e0c, 0x3b7b16ba,}, |
| { 0x1c4bd95b, 0xa8d72b0d, 0x5879f25a,}, |
| { 0x1bead768, 0x98f8ce4c, 0x66cc2858,}, |
| { 0x1b903469, 0x050f72e5, 0x0cf5488e,}, |
| { 0x1b3b433f, 0x2eb06f14, 0x8c89719c,}, |
| { 0x1aeb6f75, 0x9c46fc37, 0xab5fc7e9,}, |
| { 0x1aa038eb, 0x0e3bfd17, 0x1bd62080,}, |
| { 0x1a593062, 0xb38d8c56, 0x7998ab45,}, |
| { 0x1a15f4c3, 0x2b95a2e6, 0x46aed6a0,}, |
| { 0x19d630dc, 0xcc7ddef9, 0x5aadd61b,}, |
| { 0x19999999, 0x99999999, 0x9999999a,}, |
| { 0x195fec80, 0x8a609430, 0xe1106014,}, |
| { 0x1928ee7b, 0x0b4f22f9, 0x5f69791d,}, |
| { 0x18f46acf, 0x8c06e318, 0x4d2aeb2c,}, |
| { 0x18c23246, 0xdc0a9f3d, 0x3fe16970,}, |
| #endif |
| }; |
| |
| static const limb_t log2_radix[BF_RADIX_MAX - 1] = { |
| #if LIMB_BITS == 32 |
| 0x20000000, |
| 0x32b80347, |
| 0x40000000, |
| 0x4a4d3c26, |
| 0x52b80347, |
| 0x59d5d9fd, |
| 0x60000000, |
| 0x6570068e, |
| 0x6a4d3c26, |
| 0x6eb3a9f0, |
| 0x72b80347, |
| 0x766a008e, |
| 0x79d5d9fd, |
| 0x7d053f6d, |
| 0x80000000, |
| 0x82cc7edf, |
| 0x8570068e, |
| 0x87ef05ae, |
| 0x8a4d3c26, |
| 0x8c8ddd45, |
| 0x8eb3a9f0, |
| 0x90c10501, |
| 0x92b80347, |
| 0x949a784c, |
| 0x966a008e, |
| 0x982809d6, |
| 0x99d5d9fd, |
| 0x9b74948f, |
| 0x9d053f6d, |
| 0x9e88c6b3, |
| 0xa0000000, |
| 0xa16bad37, |
| 0xa2cc7edf, |
| 0xa4231623, |
| 0xa570068e, |
| #else |
| 0x2000000000000000, |
| 0x32b803473f7ad0f4, |
| 0x4000000000000000, |
| 0x4a4d3c25e68dc57f, |
| 0x52b803473f7ad0f4, |
| 0x59d5d9fd5010b366, |
| 0x6000000000000000, |
| 0x6570068e7ef5a1e8, |
| 0x6a4d3c25e68dc57f, |
| 0x6eb3a9f01975077f, |
| 0x72b803473f7ad0f4, |
| 0x766a008e4788cbcd, |
| 0x79d5d9fd5010b366, |
| 0x7d053f6d26089673, |
| 0x8000000000000000, |
| 0x82cc7edf592262d0, |
| 0x8570068e7ef5a1e8, |
| 0x87ef05ae409a0289, |
| 0x8a4d3c25e68dc57f, |
| 0x8c8ddd448f8b845a, |
| 0x8eb3a9f01975077f, |
| 0x90c10500d63aa659, |
| 0x92b803473f7ad0f4, |
| 0x949a784bcd1b8afe, |
| 0x966a008e4788cbcd, |
| 0x982809d5be7072dc, |
| 0x99d5d9fd5010b366, |
| 0x9b74948f5532da4b, |
| 0x9d053f6d26089673, |
| 0x9e88c6b3626a72aa, |
| 0xa000000000000000, |
| 0xa16bad3758efd873, |
| 0xa2cc7edf592262d0, |
| 0xa4231623369e78e6, |
| 0xa570068e7ef5a1e8, |
| #endif |
| }; |
| |
| /* compute floor(a*b) or ceil(a*b) with b = log2(radix) or |
| b=1/log2(radix). For is_inv = 0, strict accuracy is not guaranteed |
| when radix is not a power of two. */ |
| slimb_t bf_mul_log2_radix(slimb_t a1, unsigned int radix, int is_inv, |
| int is_ceil1) |
| { |
| int is_neg; |
| limb_t a; |
| BOOL is_ceil; |
| |
| is_ceil = is_ceil1; |
| a = a1; |
| if (a1 < 0) { |
| a = -a; |
| is_neg = 1; |
| } else { |
| is_neg = 0; |
| } |
| is_ceil ^= is_neg; |
| if ((radix & (radix - 1)) == 0) { |
| int radix_bits; |
| /* radix is a power of two */ |
| radix_bits = ceil_log2(radix); |
| if (is_inv) { |
| if (is_ceil) |
| a += radix_bits - 1; |
| a = a / radix_bits; |
| } else { |
| a = a * radix_bits; |
| } |
| } else { |
| const uint32_t *tab; |
| limb_t b0, b1; |
| dlimb_t t; |
| |
| if (is_inv) { |
| tab = inv_log2_radix[radix - 2]; |
| #if LIMB_BITS == 32 |
| b1 = tab[0]; |
| b0 = tab[1]; |
| #else |
| b1 = ((limb_t)tab[0] << 32) | tab[1]; |
| b0 = (limb_t)tab[2] << 32; |
| #endif |
| t = (dlimb_t)b0 * (dlimb_t)a; |
| t = (dlimb_t)b1 * (dlimb_t)a + (t >> LIMB_BITS); |
| a = t >> (LIMB_BITS - 1); |
| } else { |
| b0 = log2_radix[radix - 2]; |
| t = (dlimb_t)b0 * (dlimb_t)a; |
| a = t >> (LIMB_BITS - 3); |
| } |
| /* a = floor(result) and 'result' cannot be an integer */ |
| a += is_ceil; |
| } |
| if (is_neg) |
| a = -a; |
| return a; |
| } |
| |
| /* 'n' is the number of output limbs */ |
| static int bf_integer_to_radix_rec(bf_t *pow_tab, |
| limb_t *out, const bf_t *a, limb_t n, |
| int level, limb_t n0, limb_t radixl, |
| unsigned int radixl_bits) |
| { |
| limb_t n1, n2, q_prec; |
| int ret; |
| |
| assert(n >= 1); |
| if (n == 1) { |
| out[0] = get_bits(a->tab, a->len, a->len * LIMB_BITS - a->expn); |
| } else if (n == 2) { |
| dlimb_t t; |
| slimb_t pos; |
| pos = a->len * LIMB_BITS - a->expn; |
| t = ((dlimb_t)get_bits(a->tab, a->len, pos + LIMB_BITS) << LIMB_BITS) | |
| get_bits(a->tab, a->len, pos); |
| if (likely(radixl == RADIXL_10)) { |
| /* use division by a constant when possible */ |
| out[0] = t % RADIXL_10; |
| out[1] = t / RADIXL_10; |
| } else { |
| out[0] = t % radixl; |
| out[1] = t / radixl; |
| } |
| } else { |
| bf_t Q, R, *B, *B_inv; |
| int q_add; |
| bf_init(a->ctx, &Q); |
| bf_init(a->ctx, &R); |
| n2 = (((n0 * 2) >> (level + 1)) + 1) / 2; |
| n1 = n - n2; |
| B = &pow_tab[2 * level]; |
| B_inv = &pow_tab[2 * level + 1]; |
| ret = 0; |
| if (B->len == 0) { |
| /* compute BASE^n2 */ |
| ret |= bf_pow_ui_ui(B, radixl, n2, BF_PREC_INF, BF_RNDZ); |
| /* we use enough bits for the maximum possible 'n1' value, |
| i.e. n2 + 1 */ |
| ret |= bf_set_ui(&R, 1); |
| ret |= bf_div(B_inv, &R, B, (n2 + 1) * radixl_bits + 2, BF_RNDN); |
| } |
| // printf("%d: n1=% " PRId64 " n2=%" PRId64 "\n", level, n1, n2); |
| q_prec = n1 * radixl_bits; |
| ret |= bf_mul(&Q, a, B_inv, q_prec, BF_RNDN); |
| ret |= bf_rint(&Q, BF_RNDZ); |
| |
| ret |= bf_mul(&R, &Q, B, BF_PREC_INF, BF_RNDZ); |
| ret |= bf_sub(&R, a, &R, BF_PREC_INF, BF_RNDZ); |
| |
| if (ret & BF_ST_MEM_ERROR) |
| goto fail; |
| /* adjust if necessary */ |
| q_add = 0; |
| while (R.sign && R.len != 0) { |
| if (bf_add(&R, &R, B, BF_PREC_INF, BF_RNDZ)) |
| goto fail; |
| q_add--; |
| } |
| while (bf_cmpu(&R, B) >= 0) { |
| if (bf_sub(&R, &R, B, BF_PREC_INF, BF_RNDZ)) |
| goto fail; |
| q_add++; |
| } |
| if (q_add != 0) { |
| if (bf_add_si(&Q, &Q, q_add, BF_PREC_INF, BF_RNDZ)) |
| goto fail; |
| } |
| if (bf_integer_to_radix_rec(pow_tab, out + n2, &Q, n1, level + 1, n0, |
| radixl, radixl_bits)) |
| goto fail; |
| if (bf_integer_to_radix_rec(pow_tab, out, &R, n2, level + 1, n0, |
| radixl, radixl_bits)) { |
| fail: |
| bf_delete(&Q); |
| bf_delete(&R); |
| return -1; |
| } |
| bf_delete(&Q); |
| bf_delete(&R); |
| } |
| return 0; |
| } |
| |
| /* return 0 if OK != 0 if memory error */ |
| static int bf_integer_to_radix(bf_t *r, const bf_t *a, limb_t radixl) |
| { |
| bf_context_t *s = r->ctx; |
| limb_t r_len; |
| bf_t *pow_tab; |
| int i, pow_tab_len, ret; |
| |
| r_len = r->len; |
| pow_tab_len = (ceil_log2(r_len) + 2) * 2; /* XXX: check */ |
| pow_tab = bf_malloc(s, sizeof(pow_tab[0]) * pow_tab_len); |
| if (!pow_tab) |
| return -1; |
| for(i = 0; i < pow_tab_len; i++) |
| bf_init(r->ctx, &pow_tab[i]); |
| |
| ret = bf_integer_to_radix_rec(pow_tab, r->tab, a, r_len, 0, r_len, radixl, |
| ceil_log2(radixl)); |
| |
| for(i = 0; i < pow_tab_len; i++) { |
| bf_delete(&pow_tab[i]); |
| } |
| bf_free(s, pow_tab); |
| return ret; |
| } |
| |
| /* a must be >= 0. 'P' is the wanted number of digits in radix |
| 'radix'. 'r' is the mantissa represented as an integer. *pE |
| contains the exponent. Return != 0 if memory error. */ |
| static int bf_convert_to_radix(bf_t *r, slimb_t *pE, |
| const bf_t *a, int radix, |
| limb_t P, bf_rnd_t rnd_mode, |
| BOOL is_fixed_exponent) |
| { |
| slimb_t E, e, prec, extra_bits, ziv_extra_bits, prec0; |
| bf_t B_s, *B = &B_s; |
| int e_sign, ret, res; |
| |
| if (a->len == 0) { |
| /* zero case */ |
| *pE = 0; |
| return bf_set(r, a); |
| } |
| |
| if (is_fixed_exponent) { |
| E = *pE; |
| } else { |
| /* compute the new exponent */ |
| E = 1 + bf_mul_log2_radix(a->expn - 1, radix, TRUE, FALSE); |
| } |
| // bf_print_str("a", a); |
| // printf("E=%ld P=%ld radix=%d\n", E, P, radix); |
| |
| for(;;) { |
| e = P - E; |
| e_sign = 0; |
| if (e < 0) { |
| e = -e; |
| e_sign = 1; |
| } |
| /* Note: precision for log2(radix) is not critical here */ |
| prec0 = bf_mul_log2_radix(P, radix, FALSE, TRUE); |
| ziv_extra_bits = 16; |
| for(;;) { |
| prec = prec0 + ziv_extra_bits; |
| /* XXX: rigorous error analysis needed */ |
| extra_bits = ceil_log2(e) * 2 + 1; |
| ret = bf_pow_ui_ui(r, radix, e, prec + extra_bits, |
| BF_RNDN | BF_FLAG_EXT_EXP); |
| if (!e_sign) |
| ret |= bf_mul(r, r, a, prec + extra_bits, |
| BF_RNDN | BF_FLAG_EXT_EXP); |
| else |
| ret |= bf_div(r, a, r, prec + extra_bits, |
| BF_RNDN | BF_FLAG_EXT_EXP); |
| if (ret & BF_ST_MEM_ERROR) |
| return BF_ST_MEM_ERROR; |
| /* if the result is not exact, check that it can be safely |
| rounded to an integer */ |
| if ((ret & BF_ST_INEXACT) && |
| !bf_can_round(r, r->expn, rnd_mode, prec)) { |
| /* and more precision and retry */ |
| ziv_extra_bits = ziv_extra_bits + (ziv_extra_bits / 2); |
| continue; |
| } else { |
| ret = bf_rint(r, rnd_mode); |
| if (ret & BF_ST_MEM_ERROR) |
| return BF_ST_MEM_ERROR; |
| break; |
| } |
| } |
| if (is_fixed_exponent) |
| break; |
| /* check that the result is < B^P */ |
| /* XXX: do a fast approximate test first ? */ |
| bf_init(r->ctx, B); |
| ret = bf_pow_ui_ui(B, radix, P, BF_PREC_INF, BF_RNDZ); |
| if (ret) { |
| bf_delete(B); |
| return ret; |
| } |
| res = bf_cmpu(r, B); |
| bf_delete(B); |
| if (res < 0) |
| break; |
| /* try a larger exponent */ |
| E++; |
| } |
| *pE = E; |
| return 0; |
| } |
| |
| static void limb_to_a(char *buf, limb_t n, unsigned int radix, int len) |
| { |
| int digit, i; |
| |
| if (radix == 10) { |
| /* specific case with constant divisor */ |
| for(i = len - 1; i >= 0; i--) { |
| digit = (limb_t)n % 10; |
| n = (limb_t)n / 10; |
| buf[i] = digit + '0'; |
| } |
| } else { |
| for(i = len - 1; i >= 0; i--) { |
| digit = (limb_t)n % radix; |
| n = (limb_t)n / radix; |
| if (digit < 10) |
| digit += '0'; |
| else |
| digit += 'a' - 10; |
| buf[i] = digit; |
| } |
| } |
| } |
| |
| /* for power of 2 radixes */ |
| static void limb_to_a2(char *buf, limb_t n, unsigned int radix_bits, int len) |
| { |
| int digit, i; |
| unsigned int mask; |
| |
| mask = (1 << radix_bits) - 1; |
| for(i = len - 1; i >= 0; i--) { |
| digit = n & mask; |
| n >>= radix_bits; |
| if (digit < 10) |
| digit += '0'; |
| else |
| digit += 'a' - 10; |
| buf[i] = digit; |
| } |
| } |
| |
| /* 'a' must be an integer if the is_dec = FALSE or if the radix is not |
| a power of two. A dot is added before the 'dot_pos' digit. dot_pos |
| = n_digits does not display the dot. 0 <= dot_pos <= |
| n_digits. n_digits >= 1. */ |
| static void output_digits(DynBuf *s, const bf_t *a1, int radix, limb_t n_digits, |
| limb_t dot_pos, BOOL is_dec) |
| { |
| limb_t i, v, l; |
| slimb_t pos, pos_incr; |
| int digits_per_limb, buf_pos, radix_bits, first_buf_pos; |
| char buf[65]; |
| bf_t a_s, *a; |
| |
| if (is_dec) { |
| digits_per_limb = LIMB_DIGITS; |
| a = (bf_t *)a1; |
| radix_bits = 0; |
| pos = a->len; |
| pos_incr = 1; |
| first_buf_pos = 0; |
| } else if ((radix & (radix - 1)) == 0) { |
| a = (bf_t *)a1; |
| radix_bits = ceil_log2(radix); |
| digits_per_limb = LIMB_BITS / radix_bits; |
| pos_incr = digits_per_limb * radix_bits; |
| /* digits are aligned relative to the radix point */ |
| pos = a->len * LIMB_BITS + smod(-a->expn, radix_bits); |
| first_buf_pos = 0; |
| } else { |
| limb_t n, radixl; |
| |
| digits_per_limb = digits_per_limb_table[radix - 2]; |
| radixl = get_limb_radix(radix); |
| a = &a_s; |
| bf_init(a1->ctx, a); |
| n = (n_digits + digits_per_limb - 1) / digits_per_limb; |
| if (bf_resize(a, n)) { |
| dbuf_set_error(s); |
| goto done; |
| } |
| if (bf_integer_to_radix(a, a1, radixl)) { |
| dbuf_set_error(s); |
| goto done; |
| } |
| radix_bits = 0; |
| pos = n; |
| pos_incr = 1; |
| first_buf_pos = pos * digits_per_limb - n_digits; |
| } |
| buf_pos = digits_per_limb; |
| i = 0; |
| while (i < n_digits) { |
| if (buf_pos == digits_per_limb) { |
| pos -= pos_incr; |
| if (radix_bits == 0) { |
| v = get_limbz(a, pos); |
| limb_to_a(buf, v, radix, digits_per_limb); |
| } else { |
| v = get_bits(a->tab, a->len, pos); |
| limb_to_a2(buf, v, radix_bits, digits_per_limb); |
| } |
| buf_pos = first_buf_pos; |
| first_buf_pos = 0; |
| } |
| if (i < dot_pos) { |
| l = dot_pos; |
| } else { |
| if (i == dot_pos) |
| dbuf_putc(s, '.'); |
| l = n_digits; |
| } |
| l = bf_min(digits_per_limb - buf_pos, l - i); |
| dbuf_put(s, (uint8_t *)(buf + buf_pos), l); |
| buf_pos += l; |
| i += l; |
| } |
| done: |
| if (a != a1) |
| bf_delete(a); |
| } |
| |
| static void *bf_dbuf_realloc(void *opaque, void *ptr, size_t size) |
| { |
| bf_context_t *s = opaque; |
| return bf_realloc(s, ptr, size); |
| } |
| |
| /* return the length in bytes. A trailing '\0' is added */ |
| static char *bf_ftoa_internal(size_t *plen, const bf_t *a2, int radix, |
| limb_t prec, bf_flags_t flags, BOOL is_dec) |
| { |
| bf_context_t *ctx = a2->ctx; |
| DynBuf s_s, *s = &s_s; |
| int radix_bits; |
| |
| // bf_print_str("ftoa", a2); |
| // printf("radix=%d\n", radix); |
| dbuf_init2(s, ctx, bf_dbuf_realloc); |
| if (a2->expn == BF_EXP_NAN) { |
| dbuf_putstr(s, "NaN"); |
| } else { |
| if (a2->sign) |
| dbuf_putc(s, '-'); |
| if (a2->expn == BF_EXP_INF) { |
| if (flags & BF_FTOA_JS_QUIRKS) |
| dbuf_putstr(s, "Infinity"); |
| else |
| dbuf_putstr(s, "Inf"); |
| } else { |
| int fmt, ret; |
| slimb_t n_digits, n, i, n_max, n1; |
| bf_t a1_s, *a1 = &a1_s; |
| |
| if ((radix & (radix - 1)) != 0) |
| radix_bits = 0; |
| else |
| radix_bits = ceil_log2(radix); |
| |
| fmt = flags & BF_FTOA_FORMAT_MASK; |
| bf_init(ctx, a1); |
| if (fmt == BF_FTOA_FORMAT_FRAC) { |
| if (is_dec || radix_bits != 0) { |
| if (bf_set(a1, a2)) |
| goto fail1; |
| #ifdef USE_BF_DEC |
| if (is_dec) { |
| if (bfdec_round((bfdec_t *)a1, prec, (flags & BF_RND_MASK) | BF_FLAG_RADPNT_PREC) & BF_ST_MEM_ERROR) |
| goto fail1; |
| n = a1->expn; |
| } else |
| #endif |
| { |
| if (bf_round(a1, prec * radix_bits, (flags & BF_RND_MASK) | BF_FLAG_RADPNT_PREC) & BF_ST_MEM_ERROR) |
| goto fail1; |
| n = ceil_div(a1->expn, radix_bits); |
| } |
| if (flags & BF_FTOA_ADD_PREFIX) { |
| if (radix == 16) |
| dbuf_putstr(s, "0x"); |
| else if (radix == 8) |
| dbuf_putstr(s, "0o"); |
| else if (radix == 2) |
| dbuf_putstr(s, "0b"); |
| } |
| if (a1->expn == BF_EXP_ZERO) { |
| dbuf_putstr(s, "0"); |
| if (prec > 0) { |
| dbuf_putstr(s, "."); |
| for(i = 0; i < prec; i++) { |
| dbuf_putc(s, '0'); |
| } |
| } |
| } else { |
| n_digits = prec + n; |
| if (n <= 0) { |
| /* 0.x */ |
| dbuf_putstr(s, "0."); |
| for(i = 0; i < -n; i++) { |
| dbuf_putc(s, '0'); |
| } |
| if (n_digits > 0) { |
| output_digits(s, a1, radix, n_digits, n_digits, is_dec); |
| } |
| } else { |
| output_digits(s, a1, radix, n_digits, n, is_dec); |
| } |
| } |
| } else { |
| size_t pos, start; |
| bf_t a_s, *a = &a_s; |
| |
| /* make a positive number */ |
| a->tab = a2->tab; |
| a->len = a2->len; |
| a->expn = a2->expn; |
| a->sign = 0; |
| |
| /* one more digit for the rounding */ |
| n = 1 + bf_mul_log2_radix(bf_max(a->expn, 0), radix, TRUE, TRUE); |
| n_digits = n + prec; |
| n1 = n; |
| if (bf_convert_to_radix(a1, &n1, a, radix, n_digits, |
| flags & BF_RND_MASK, TRUE)) |
| goto fail1; |
| start = s->size; |
| output_digits(s, a1, radix, n_digits, n, is_dec); |
| /* remove leading zeros because we allocated one more digit */ |
| pos = start; |
| while ((pos + 1) < s->size && s->buf[pos] == '0' && |
| s->buf[pos + 1] != '.') |
| pos++; |
| if (pos > start) { |
| memmove(s->buf + start, s->buf + pos, s->size - pos); |
| s->size -= (pos - start); |
| } |
| } |
| } else { |
| #ifdef USE_BF_DEC |
| if (is_dec) { |
| if (bf_set(a1, a2)) |
| goto fail1; |
| if (fmt == BF_FTOA_FORMAT_FIXED) { |
| n_digits = prec; |
| n_max = n_digits; |
| if (bfdec_round((bfdec_t *)a1, prec, (flags & BF_RND_MASK)) & BF_ST_MEM_ERROR) |
| goto fail1; |
| } else { |
| /* prec is ignored */ |
| prec = n_digits = a1->len * LIMB_DIGITS; |
| /* remove the trailing zero digits */ |
| while (n_digits > 1 && |
| get_digit(a1->tab, a1->len, prec - n_digits) == 0) { |
| n_digits--; |
| } |
| n_max = n_digits + 4; |
| } |
| n = a1->expn; |
| } else |
| #endif |
| if (radix_bits != 0) { |
| if (bf_set(a1, a2)) |
| goto fail1; |
| if (fmt == BF_FTOA_FORMAT_FIXED) { |
| slimb_t prec_bits; |
| n_digits = prec; |
| n_max = n_digits; |
| /* align to the radix point */ |
| prec_bits = prec * radix_bits - |
| smod(-a1->expn, radix_bits); |
| if (bf_round(a1, prec_bits, |
| (flags & BF_RND_MASK)) & BF_ST_MEM_ERROR) |
| goto fail1; |
| } else { |
| limb_t digit_mask; |
| slimb_t pos; |
| /* position of the digit before the most |
| significant digit in bits */ |
| pos = a1->len * LIMB_BITS + |
| smod(-a1->expn, radix_bits); |
| n_digits = ceil_div(pos, radix_bits); |
| /* remove the trailing zero digits */ |
| digit_mask = ((limb_t)1 << radix_bits) - 1; |
| while (n_digits > 1 && |
| (get_bits(a1->tab, a1->len, pos - n_digits * radix_bits) & digit_mask) == 0) { |
| n_digits--; |
| } |
| n_max = n_digits + 4; |
| } |
| n = ceil_div(a1->expn, radix_bits); |
| } else { |
| bf_t a_s, *a = &a_s; |
| |
| /* make a positive number */ |
| a->tab = a2->tab; |
| a->len = a2->len; |
| a->expn = a2->expn; |
| a->sign = 0; |
| |
| if (fmt == BF_FTOA_FORMAT_FIXED) { |
| n_digits = prec; |
| n_max = n_digits; |
| } else { |
| slimb_t n_digits_max, n_digits_min; |
| |
| assert(prec != BF_PREC_INF); |
| n_digits = 1 + bf_mul_log2_radix(prec, radix, TRUE, TRUE); |
| /* max number of digits for non exponential |
| notation. The rational is to have the same rule |
| as JS i.e. n_max = 21 for 64 bit float in base 10. */ |
| n_max = n_digits + 4; |
| if (fmt == BF_FTOA_FORMAT_FREE_MIN) { |
| bf_t b_s, *b = &b_s; |
| |
| /* find the minimum number of digits by |
| dichotomy. */ |
| /* XXX: inefficient */ |
| n_digits_max = n_digits; |
| n_digits_min = 1; |
| bf_init(ctx, b); |
| while (n_digits_min < n_digits_max) { |
| n_digits = (n_digits_min + n_digits_max) / 2; |
| if (bf_convert_to_radix(a1, &n, a, radix, n_digits, |
| flags & BF_RND_MASK, FALSE)) { |
| bf_delete(b); |
| goto fail1; |
| } |
| /* convert back to a number and compare */ |
| ret = bf_mul_pow_radix(b, a1, radix, n - n_digits, |
| prec, |
| (flags & ~BF_RND_MASK) | |
| BF_RNDN); |
| if (ret & BF_ST_MEM_ERROR) { |
| bf_delete(b); |
| goto fail1; |
| } |
| if (bf_cmpu(b, a) == 0) { |
| n_digits_max = n_digits; |
| } else { |
| n_digits_min = n_digits + 1; |
| } |
| } |
| bf_delete(b); |
| n_digits = n_digits_max; |
| } |
| } |
| if (bf_convert_to_radix(a1, &n, a, radix, n_digits, |
| flags & BF_RND_MASK, FALSE)) { |
| fail1: |
| bf_delete(a1); |
| goto fail; |
| } |
| } |
| if (a1->expn == BF_EXP_ZERO && |
| fmt != BF_FTOA_FORMAT_FIXED && |
| !(flags & BF_FTOA_FORCE_EXP)) { |
| /* just output zero */ |
| dbuf_putstr(s, "0"); |
| } else { |
| if (flags & BF_FTOA_ADD_PREFIX) { |
| if (radix == 16) |
| dbuf_putstr(s, "0x"); |
| else if (radix == 8) |
| dbuf_putstr(s, "0o"); |
| else if (radix == 2) |
| dbuf_putstr(s, "0b"); |
| } |
| if (a1->expn == BF_EXP_ZERO) |
| n = 1; |
| if ((flags & BF_FTOA_FORCE_EXP) || |
| n <= -6 || n > n_max) { |
| const char *fmt; |
| /* exponential notation */ |
| output_digits(s, a1, radix, n_digits, 1, is_dec); |
| if (radix_bits != 0 && radix <= 16) { |
| if (flags & BF_FTOA_JS_QUIRKS) |
| fmt = "p%+" PRId_LIMB; |
| else |
| fmt = "p%" PRId_LIMB; |
| dbuf_printf(s, fmt, (n - 1) * radix_bits); |
| } else { |
| if (flags & BF_FTOA_JS_QUIRKS) |
| fmt = "%c%+" PRId_LIMB; |
| else |
| fmt = "%c%" PRId_LIMB; |
| dbuf_printf(s, fmt, |
| radix <= 10 ? 'e' : '@', n - 1); |
| } |
| } else if (n <= 0) { |
| /* 0.x */ |
| dbuf_putstr(s, "0."); |
| for(i = 0; i < -n; i++) { |
| dbuf_putc(s, '0'); |
| } |
| output_digits(s, a1, radix, n_digits, n_digits, is_dec); |
| } else { |
| if (n_digits <= n) { |
| /* no dot */ |
| output_digits(s, a1, radix, n_digits, n_digits, is_dec); |
| for(i = 0; i < (n - n_digits); i++) |
| dbuf_putc(s, '0'); |
| } else { |
| output_digits(s, a1, radix, n_digits, n, is_dec); |
| } |
| } |
| } |
| } |
| bf_delete(a1); |
| } |
| } |
| dbuf_putc(s, '\0'); |
| if (dbuf_error(s)) |
| goto fail; |
| if (plen) |
| *plen = s->size - 1; |
| return (char *)s->buf; |
| fail: |
| bf_free(ctx, s->buf); |
| if (plen) |
| *plen = 0; |
| return NULL; |
| } |
| |
| char *bf_ftoa(size_t *plen, const bf_t *a, int radix, limb_t prec, |
| bf_flags_t flags) |
| { |
| return bf_ftoa_internal(plen, a, radix, prec, flags, FALSE); |
| } |
| |
| /***************************************************************/ |
| /* transcendental functions */ |
| |
| /* Note: the algorithm is from MPFR */ |
| static void bf_const_log2_rec(bf_t *T, bf_t *P, bf_t *Q, limb_t n1, |
| limb_t n2, BOOL need_P) |
| { |
| bf_context_t *s = T->ctx; |
| if ((n2 - n1) == 1) { |
| if (n1 == 0) { |
| bf_set_ui(P, 3); |
| } else { |
| bf_set_ui(P, n1); |
| P->sign = 1; |
| } |
| bf_set_ui(Q, 2 * n1 + 1); |
| Q->expn += 2; |
| bf_set(T, P); |
| } else { |
| limb_t m; |
| bf_t T1_s, *T1 = &T1_s; |
| bf_t P1_s, *P1 = &P1_s; |
| bf_t Q1_s, *Q1 = &Q1_s; |
| |
| m = n1 + ((n2 - n1) >> 1); |
| bf_const_log2_rec(T, P, Q, n1, m, TRUE); |
| bf_init(s, T1); |
| bf_init(s, P1); |
| bf_init(s, Q1); |
| bf_const_log2_rec(T1, P1, Q1, m, n2, need_P); |
| bf_mul(T, T, Q1, BF_PREC_INF, BF_RNDZ); |
| bf_mul(T1, T1, P, BF_PREC_INF, BF_RNDZ); |
| bf_add(T, T, T1, BF_PREC_INF, BF_RNDZ); |
| if (need_P) |
| bf_mul(P, P, P1, BF_PREC_INF, BF_RNDZ); |
| bf_mul(Q, Q, Q1, BF_PREC_INF, BF_RNDZ); |
| bf_delete(T1); |
| bf_delete(P1); |
| bf_delete(Q1); |
| } |
| } |
| |
| /* compute log(2) with faithful rounding at precision 'prec' */ |
| static void bf_const_log2_internal(bf_t *T, limb_t prec) |
| { |
| limb_t w, N; |
| bf_t P_s, *P = &P_s; |
| bf_t Q_s, *Q = &Q_s; |
| |
| w = prec + 15; |
| N = w / 3 + 1; |
| bf_init(T->ctx, P); |
| bf_init(T->ctx, Q); |
| bf_const_log2_rec(T, P, Q, 0, N, FALSE); |
| bf_div(T, T, Q, prec, BF_RNDN); |
| bf_delete(P); |
| bf_delete(Q); |
| } |
| |
| /* PI constant */ |
| |
| #define CHUD_A 13591409 |
| #define CHUD_B 545140134 |
| #define CHUD_C 640320 |
| #define CHUD_BITS_PER_TERM 47 |
| |
| static void chud_bs(bf_t *P, bf_t *Q, bf_t *G, int64_t a, int64_t b, int need_g, |
| limb_t prec) |
| { |
| bf_context_t *s = P->ctx; |
| int64_t c; |
| |
| if (a == (b - 1)) { |
| bf_t T0, T1; |
| |
| bf_init(s, &T0); |
| bf_init(s, &T1); |
| bf_set_ui(G, 2 * b - 1); |
| bf_mul_ui(G, G, 6 * b - 1, prec, BF_RNDN); |
| bf_mul_ui(G, G, 6 * b - 5, prec, BF_RNDN); |
| bf_set_ui(&T0, CHUD_B); |
| bf_mul_ui(&T0, &T0, b, prec, BF_RNDN); |
| bf_set_ui(&T1, CHUD_A); |
| bf_add(&T0, &T0, &T1, prec, BF_RNDN); |
| bf_mul(P, G, &T0, prec, BF_RNDN); |
| P->sign = b & 1; |
| |
| bf_set_ui(Q, b); |
| bf_mul_ui(Q, Q, b, prec, BF_RNDN); |
| bf_mul_ui(Q, Q, b, prec, BF_RNDN); |
| bf_mul_ui(Q, Q, (uint64_t)CHUD_C * CHUD_C * CHUD_C / 24, prec, BF_RNDN); |
| bf_delete(&T0); |
| bf_delete(&T1); |
| } else { |
| bf_t P2, Q2, G2; |
| |
| bf_init(s, &P2); |
| bf_init(s, &Q2); |
| bf_init(s, &G2); |
| |
| c = (a + b) / 2; |
| chud_bs(P, Q, G, a, c, 1, prec); |
| chud_bs(&P2, &Q2, &G2, c, b, need_g, prec); |
| |
| /* Q = Q1 * Q2 */ |
| /* G = G1 * G2 */ |
| /* P = P1 * Q2 + P2 * G1 */ |
| bf_mul(&P2, &P2, G, prec, BF_RNDN); |
| if (!need_g) |
| bf_set_ui(G, 0); |
| bf_mul(P, P, &Q2, prec, BF_RNDN); |
| bf_add(P, P, &P2, prec, BF_RNDN); |
| bf_delete(&P2); |
| |
| bf_mul(Q, Q, &Q2, prec, BF_RNDN); |
| bf_delete(&Q2); |
| if (need_g) |
| bf_mul(G, G, &G2, prec, BF_RNDN); |
| bf_delete(&G2); |
| } |
| } |
| |
| /* compute Pi with faithful rounding at precision 'prec' using the |
| Chudnovsky formula */ |
| static void bf_const_pi_internal(bf_t *Q, limb_t prec) |
| { |
| bf_context_t *s = Q->ctx; |
| int64_t n, prec1; |
| bf_t P, G; |
| |
| /* number of serie terms */ |
| n = prec / CHUD_BITS_PER_TERM + 1; |
| /* XXX: precision analysis */ |
| prec1 = prec + 32; |
| |
| bf_init(s, &P); |
| bf_init(s, &G); |
| |
| chud_bs(&P, Q, &G, 0, n, 0, BF_PREC_INF); |
| |
| bf_mul_ui(&G, Q, CHUD_A, prec1, BF_RNDN); |
| bf_add(&P, &G, &P, prec1, BF_RNDN); |
| bf_div(Q, Q, &P, prec1, BF_RNDF); |
| |
| bf_set_ui(&P, CHUD_C); |
| bf_sqrt(&G, &P, prec1, BF_RNDF); |
| bf_mul_ui(&G, &G, (uint64_t)CHUD_C / 12, prec1, BF_RNDF); |
| bf_mul(Q, Q, &G, prec, BF_RNDN); |
| bf_delete(&P); |
| bf_delete(&G); |
| } |
| |
| static int bf_const_get(bf_t *T, limb_t prec, bf_flags_t flags, |
| BFConstCache *c, |
| void (*func)(bf_t *res, limb_t prec), int sign) |
| { |
| limb_t ziv_extra_bits, prec1; |
| |
| ziv_extra_bits = 32; |
| for(;;) { |
| prec1 = prec + ziv_extra_bits; |
| if (c->prec < prec1) { |
| if (c->val.len == 0) |
| bf_init(T->ctx, &c->val); |
| func(&c->val, prec1); |
| c->prec = prec1; |
| } else { |
| prec1 = c->prec; |
| } |
| bf_set(T, &c->val); |
| T->sign = sign; |
| if (!bf_can_round(T, prec, flags & BF_RND_MASK, prec1)) { |
| /* and more precision and retry */ |
| ziv_extra_bits = ziv_extra_bits + (ziv_extra_bits / 2); |
| } else { |
| break; |
| } |
| } |
| return bf_round(T, prec, flags); |
| } |
| |
| static void bf_const_free(BFConstCache *c) |
| { |
| bf_delete(&c->val); |
| memset(c, 0, sizeof(*c)); |
| } |
| |
| int bf_const_log2(bf_t *T, limb_t prec, bf_flags_t flags) |
| { |
| bf_context_t *s = T->ctx; |
| return bf_const_get(T, prec, flags, &s->log2_cache, bf_const_log2_internal, 0); |
| } |
| |
| /* return rounded pi * (1 - 2 * sign) */ |
| static int bf_const_pi_signed(bf_t *T, int sign, limb_t prec, bf_flags_t flags) |
| { |
| bf_context_t *s = T->ctx; |
| return bf_const_get(T, prec, flags, &s->pi_cache, bf_const_pi_internal, |
| sign); |
| } |
| |
| int bf_const_pi(bf_t *T, limb_t prec, bf_flags_t flags) |
| { |
| return bf_const_pi_signed(T, 0, prec, flags); |
| } |
| |
| void bf_clear_cache(bf_context_t *s) |
| { |
| #ifdef USE_FFT_MUL |
| fft_clear_cache(s); |
| #endif |
| bf_const_free(&s->log2_cache); |
| bf_const_free(&s->pi_cache); |
| } |
| |
| /* ZivFunc should compute the result 'r' with faithful rounding at |
| precision 'prec'. For efficiency purposes, the final bf_round() |
| does not need to be done in the function. */ |
| typedef int ZivFunc(bf_t *r, const bf_t *a, limb_t prec, void *opaque); |
| |
| static int bf_ziv_rounding(bf_t *r, const bf_t *a, |
| limb_t prec, bf_flags_t flags, |
| ZivFunc *f, void *opaque) |
| { |
| int rnd_mode, ret; |
| slimb_t prec1, ziv_extra_bits; |
| |
| rnd_mode = flags & BF_RND_MASK; |
| if (rnd_mode == BF_RNDF) { |
| /* no need to iterate */ |
| f(r, a, prec, opaque); |
| ret = 0; |
| } else { |
| ziv_extra_bits = 32; |
| for(;;) { |
| prec1 = prec + ziv_extra_bits; |
| ret = f(r, a, prec1, opaque); |
| if (ret & (BF_ST_OVERFLOW | BF_ST_UNDERFLOW | BF_ST_MEM_ERROR)) { |
| /* overflow or underflow should never happen because |
| it indicates the rounding cannot be done correctly, |
| but we do not catch all the cases */ |
| return ret; |
| } |
| /* if the result is exact, we can stop */ |
| if (!(ret & BF_ST_INEXACT)) { |
| ret = 0; |
| break; |
| } |
| if (bf_can_round(r, prec, rnd_mode, prec1)) { |
| ret = BF_ST_INEXACT; |
| break; |
| } |
| ziv_extra_bits = ziv_extra_bits * 2; |
| // printf("ziv_extra_bits=%" PRId64 "\n", (int64_t)ziv_extra_bits); |
| } |
| } |
| if (r->len == 0) |
| return ret; |
| else |
| return __bf_round(r, prec, flags, r->len, ret); |
| } |
| |
| /* add (1 - 2*e_sign) * 2^e */ |
| static int bf_add_epsilon(bf_t *r, const bf_t *a, slimb_t e, int e_sign, |
| limb_t prec, int flags) |
| { |
| bf_t T_s, *T = &T_s; |
| int ret; |
| /* small argument case: result = 1 + epsilon * sign(x) */ |
| bf_init(a->ctx, T); |
| bf_set_ui(T, 1); |
| T->sign = e_sign; |
| T->expn += e; |
| ret = bf_add(r, r, T, prec, flags); |
| bf_delete(T); |
| return ret; |
| } |
| |
| /* Compute the exponential using faithful rounding at precision 'prec'. |
| Note: the algorithm is from MPFR */ |
| static int bf_exp_internal(bf_t *r, const bf_t *a, limb_t prec, void *opaque) |
| { |
| bf_context_t *s = r->ctx; |
| bf_t T_s, *T = &T_s; |
| slimb_t n, K, l, i, prec1; |
| |
| assert(r != a); |
| |
| /* argument reduction: |
| T = a - n*log(2) with 0 <= T < log(2) and n integer. |
| */ |
| bf_init(s, T); |
| if (a->expn <= -1) { |
| /* 0 <= abs(a) <= 0.5 */ |
| if (a->sign) |
| n = -1; |
| else |
| n = 0; |
| } else { |
| bf_const_log2(T, LIMB_BITS, BF_RNDZ); |
| bf_div(T, a, T, LIMB_BITS, BF_RNDD); |
| bf_get_limb(&n, T, 0); |
| } |
| |
| K = bf_isqrt((prec + 1) / 2); |
| l = (prec - 1) / K + 1; |
| /* XXX: precision analysis ? */ |
| prec1 = prec + (K + 2 * l + 18) + K + 8; |
| if (a->expn > 0) |
| prec1 += a->expn; |
| // printf("n=%ld K=%ld prec1=%ld\n", n, K, prec1); |
| |
| bf_const_log2(T, prec1, BF_RNDF); |
| bf_mul_si(T, T, n, prec1, BF_RNDN); |
| bf_sub(T, a, T, prec1, BF_RNDN); |
| |
| /* reduce the range of T */ |
| bf_mul_2exp(T, -K, BF_PREC_INF, BF_RNDZ); |
| |
| /* Taylor expansion around zero : |
| 1 + x + x^2/2 + ... + x^n/n! |
| = (1 + x * (1 + x/2 * (1 + ... (x/n)))) |
| */ |
| { |
| bf_t U_s, *U = &U_s; |
| |
| bf_init(s, U); |
| bf_set_ui(r, 1); |
| for(i = l ; i >= 1; i--) { |
| bf_set_ui(U, i); |
| bf_div(U, T, U, prec1, BF_RNDN); |
| bf_mul(r, r, U, prec1, BF_RNDN); |
| bf_add_si(r, r, 1, prec1, BF_RNDN); |
| } |
| bf_delete(U); |
| } |
| bf_delete(T); |
| |
| /* undo the range reduction */ |
| for(i = 0; i < K; i++) { |
| bf_mul(r, r, r, prec1, BF_RNDN | BF_FLAG_EXT_EXP); |
| } |
| |
| /* undo the argument reduction */ |
| bf_mul_2exp(r, n, BF_PREC_INF, BF_RNDZ | BF_FLAG_EXT_EXP); |
| |
| return BF_ST_INEXACT; |
| } |
| |
| /* crude overflow and underflow tests for exp(a). a_low <= a <= a_high */ |
| static int check_exp_underflow_overflow(bf_context_t *s, bf_t *r, |
| const bf_t *a_low, const bf_t *a_high, |
| limb_t prec, bf_flags_t flags) |
| { |
| bf_t T_s, *T = &T_s; |
| bf_t log2_s, *log2 = &log2_s; |
| slimb_t e_min, e_max; |
| |
| if (a_high->expn <= 0) |
| return 0; |
| |
| e_max = (limb_t)1 << (bf_get_exp_bits(flags) - 1); |
| e_min = -e_max + 3; |
| if (flags & BF_FLAG_SUBNORMAL) |
| e_min -= (prec - 1); |
| |
| bf_init(s, T); |
| bf_init(s, log2); |
| bf_const_log2(log2, LIMB_BITS, BF_RNDU); |
| bf_mul_ui(T, log2, e_max, LIMB_BITS, BF_RNDU); |
| /* a_low > e_max * log(2) implies exp(a) > e_max */ |
| if (bf_cmp_lt(T, a_low) > 0) { |
| /* overflow */ |
| bf_delete(T); |
| bf_delete(log2); |
| return bf_set_overflow(r, 0, prec, flags); |
| } |
| /* a_high < (e_min - 2) * log(2) implies exp(a) < (e_min - 2) */ |
| bf_const_log2(log2, LIMB_BITS, BF_RNDD); |
| bf_mul_si(T, log2, e_min - 2, LIMB_BITS, BF_RNDD); |
| if (bf_cmp_lt(a_high, T)) { |
| int rnd_mode = flags & BF_RND_MASK; |
| |
| /* underflow */ |
| bf_delete(T); |
| bf_delete(log2); |
| if (rnd_mode == BF_RNDU) { |
| /* set the smallest value */ |
| bf_set_ui(r, 1); |
| r->expn = e_min; |
| } else { |
| bf_set_zero(r, 0); |
| } |
| return BF_ST_UNDERFLOW | BF_ST_INEXACT; |
| } |
| bf_delete(log2); |
| bf_delete(T); |
| return 0; |
| } |
| |
| int bf_exp(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags) |
| { |
| bf_context_t *s = r->ctx; |
| int ret; |
| assert(r != a); |
| if (a->len == 0) { |
| if (a->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| } else if (a->expn == BF_EXP_INF) { |
| if (a->sign) |
| bf_set_zero(r, 0); |
| else |
| bf_set_inf(r, 0); |
| } else { |
| bf_set_ui(r, 1); |
| } |
| return 0; |
| } |
| |
| ret = check_exp_underflow_overflow(s, r, a, a, prec, flags); |
| if (ret) |
| return ret; |
| if (a->expn < 0 && (-a->expn) >= (prec + 2)) { |
| /* small argument case: result = 1 + epsilon * sign(x) */ |
| bf_set_ui(r, 1); |
| return bf_add_epsilon(r, r, -(prec + 2), a->sign, prec, flags); |
| } |
| |
| return bf_ziv_rounding(r, a, prec, flags, bf_exp_internal, NULL); |
| } |
| |
| static int bf_log_internal(bf_t *r, const bf_t *a, limb_t prec, void *opaque) |
| { |
| bf_context_t *s = r->ctx; |
| bf_t T_s, *T = &T_s; |
| bf_t U_s, *U = &U_s; |
| bf_t V_s, *V = &V_s; |
| slimb_t n, prec1, l, i, K; |
| |
| assert(r != a); |
| |
| bf_init(s, T); |
| /* argument reduction 1 */ |
| /* T=a*2^n with 2/3 <= T <= 4/3 */ |
| { |
| bf_t U_s, *U = &U_s; |
| bf_set(T, a); |
| n = T->expn; |
| T->expn = 0; |
| /* U= ~ 2/3 */ |
| bf_init(s, U); |
| bf_set_ui(U, 0xaaaaaaaa); |
| U->expn = 0; |
| if (bf_cmp_lt(T, U)) { |
| T->expn++; |
| n--; |
| } |
| bf_delete(U); |
| } |
| // printf("n=%ld\n", n); |
| // bf_print_str("T", T); |
| |
| /* XXX: precision analysis */ |
| /* number of iterations for argument reduction 2 */ |
| K = bf_isqrt((prec + 1) / 2); |
| /* order of Taylor expansion */ |
| l = prec / (2 * K) + 1; |
| /* precision of the intermediate computations */ |
| prec1 = prec + K + 2 * l + 32; |
| |
| bf_init(s, U); |
| bf_init(s, V); |
| |
| /* Note: cancellation occurs here, so we use more precision (XXX: |
| reduce the precision by computing the exact cancellation) */ |
| bf_add_si(T, T, -1, BF_PREC_INF, BF_RNDN); |
| |
| /* argument reduction 2 */ |
| for(i = 0; i < K; i++) { |
| /* T = T / (1 + sqrt(1 + T)) */ |
| bf_add_si(U, T, 1, prec1, BF_RNDN); |
| bf_sqrt(V, U, prec1, BF_RNDF); |
| bf_add_si(U, V, 1, prec1, BF_RNDN); |
| bf_div(T, T, U, prec1, BF_RNDN); |
| } |
| |
| { |
| bf_t Y_s, *Y = &Y_s; |
| bf_t Y2_s, *Y2 = &Y2_s; |
| bf_init(s, Y); |
| bf_init(s, Y2); |
| |
| /* compute ln(1+x) = ln((1+y)/(1-y)) with y=x/(2+x) |
| = y + y^3/3 + ... + y^(2*l + 1) / (2*l+1) |
| with Y=Y^2 |
| = y*(1+Y/3+Y^2/5+...) = y*(1+Y*(1/3+Y*(1/5 + ...))) |
| */ |
| bf_add_si(Y, T, 2, prec1, BF_RNDN); |
| bf_div(Y, T, Y, prec1, BF_RNDN); |
| |
| bf_mul(Y2, Y, Y, prec1, BF_RNDN); |
| bf_set_ui(r, 0); |
| for(i = l; i >= 1; i--) { |
| bf_set_ui(U, 1); |
| bf_set_ui(V, 2 * i + 1); |
| bf_div(U, U, V, prec1, BF_RNDN); |
| bf_add(r, r, U, prec1, BF_RNDN); |
| bf_mul(r, r, Y2, prec1, BF_RNDN); |
| } |
| bf_add_si(r, r, 1, prec1, BF_RNDN); |
| bf_mul(r, r, Y, prec1, BF_RNDN); |
| bf_delete(Y); |
| bf_delete(Y2); |
| } |
| bf_delete(V); |
| bf_delete(U); |
| |
| /* multiplication by 2 for the Taylor expansion and undo the |
| argument reduction 2*/ |
| bf_mul_2exp(r, K + 1, BF_PREC_INF, BF_RNDZ); |
| |
| /* undo the argument reduction 1 */ |
| bf_const_log2(T, prec1, BF_RNDF); |
| bf_mul_si(T, T, n, prec1, BF_RNDN); |
| bf_add(r, r, T, prec1, BF_RNDN); |
| |
| bf_delete(T); |
| return BF_ST_INEXACT; |
| } |
| |
| int bf_log(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags) |
| { |
| bf_context_t *s = r->ctx; |
| bf_t T_s, *T = &T_s; |
| |
| assert(r != a); |
| if (a->len == 0) { |
| if (a->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| return 0; |
| } else if (a->expn == BF_EXP_INF) { |
| if (a->sign) { |
| bf_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else { |
| bf_set_inf(r, 0); |
| return 0; |
| } |
| } else { |
| bf_set_inf(r, 1); |
| return 0; |
| } |
| } |
| if (a->sign) { |
| bf_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } |
| bf_init(s, T); |
| bf_set_ui(T, 1); |
| if (bf_cmp_eq(a, T)) { |
| bf_set_zero(r, 0); |
| bf_delete(T); |
| return 0; |
| } |
| bf_delete(T); |
| |
| return bf_ziv_rounding(r, a, prec, flags, bf_log_internal, NULL); |
| } |
| |
| /* x and y finite and x > 0 */ |
| static int bf_pow_generic(bf_t *r, const bf_t *x, limb_t prec, void *opaque) |
| { |
| bf_context_t *s = r->ctx; |
| const bf_t *y = opaque; |
| bf_t T_s, *T = &T_s; |
| limb_t prec1; |
| |
| bf_init(s, T); |
| /* XXX: proof for the added precision */ |
| prec1 = prec + 32; |
| bf_log(T, x, prec1, BF_RNDF | BF_FLAG_EXT_EXP); |
| bf_mul(T, T, y, prec1, BF_RNDF | BF_FLAG_EXT_EXP); |
| if (bf_is_nan(T)) |
| bf_set_nan(r); |
| else |
| bf_exp_internal(r, T, prec1, NULL); /* no overflow/underlow test needed */ |
| bf_delete(T); |
| return BF_ST_INEXACT; |
| } |
| |
| /* x and y finite, x > 0, y integer and y fits on one limb */ |
| static int bf_pow_int(bf_t *r, const bf_t *x, limb_t prec, void *opaque) |
| { |
| bf_context_t *s = r->ctx; |
| const bf_t *y = opaque; |
| bf_t T_s, *T = &T_s; |
| limb_t prec1; |
| int ret; |
| slimb_t y1; |
| |
| bf_get_limb(&y1, y, 0); |
| if (y1 < 0) |
| y1 = -y1; |
| /* XXX: proof for the added precision */ |
| prec1 = prec + ceil_log2(y1) * 2 + 8; |
| ret = bf_pow_ui(r, x, y1 < 0 ? -y1 : y1, prec1, BF_RNDN | BF_FLAG_EXT_EXP); |
| if (y->sign) { |
| bf_init(s, T); |
| bf_set_ui(T, 1); |
| ret |= bf_div(r, T, r, prec1, BF_RNDN | BF_FLAG_EXT_EXP); |
| bf_delete(T); |
| } |
| return ret; |
| } |
| |
| /* x must be a finite non zero float. Return TRUE if there is a |
| floating point number r such as x=r^(2^n) and return this floating |
| point number 'r'. Otherwise return FALSE and r is undefined. */ |
| static BOOL check_exact_power2n(bf_t *r, const bf_t *x, slimb_t n) |
| { |
| bf_context_t *s = r->ctx; |
| bf_t T_s, *T = &T_s; |
| slimb_t e, i, er; |
| limb_t v; |
| |
| /* x = m*2^e with m odd integer */ |
| e = bf_get_exp_min(x); |
| /* fast check on the exponent */ |
| if (n > (LIMB_BITS - 1)) { |
| if (e != 0) |
| return FALSE; |
| er = 0; |
| } else { |
| if ((e & (((limb_t)1 << n) - 1)) != 0) |
| return FALSE; |
| er = e >> n; |
| } |
| /* every perfect odd square = 1 modulo 8 */ |
| v = get_bits(x->tab, x->len, x->len * LIMB_BITS - x->expn + e); |
| if ((v & 7) != 1) |
| return FALSE; |
| |
| bf_init(s, T); |
| bf_set(T, x); |
| T->expn -= e; |
| for(i = 0; i < n; i++) { |
| if (i != 0) |
| bf_set(T, r); |
| if (bf_sqrtrem(r, NULL, T) != 0) |
| return FALSE; |
| } |
| r->expn += er; |
| return TRUE; |
| } |
| |
| /* prec = BF_PREC_INF is accepted for x and y integers and y >= 0 */ |
| int bf_pow(bf_t *r, const bf_t *x, const bf_t *y, limb_t prec, bf_flags_t flags) |
| { |
| bf_context_t *s = r->ctx; |
| bf_t T_s, *T = &T_s; |
| bf_t ytmp_s; |
| BOOL y_is_int, y_is_odd; |
| int r_sign, ret, rnd_mode; |
| slimb_t y_emin; |
| |
| if (x->len == 0 || y->len == 0) { |
| if (y->expn == BF_EXP_ZERO) { |
| /* pow(x, 0) = 1 */ |
| bf_set_ui(r, 1); |
| } else if (x->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| } else { |
| int cmp_x_abs_1; |
| bf_set_ui(r, 1); |
| cmp_x_abs_1 = bf_cmpu(x, r); |
| if (cmp_x_abs_1 == 0 && (flags & BF_POW_JS_QUIRKS) && |
| (y->expn >= BF_EXP_INF)) { |
| bf_set_nan(r); |
| } else if (cmp_x_abs_1 == 0 && |
| (!x->sign || y->expn != BF_EXP_NAN)) { |
| /* pow(1, y) = 1 even if y = NaN */ |
| /* pow(-1, +/-inf) = 1 */ |
| } else if (y->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| } else if (y->expn == BF_EXP_INF) { |
| if (y->sign == (cmp_x_abs_1 > 0)) { |
| bf_set_zero(r, 0); |
| } else { |
| bf_set_inf(r, 0); |
| } |
| } else { |
| y_emin = bf_get_exp_min(y); |
| y_is_odd = (y_emin == 0); |
| if (y->sign == (x->expn == BF_EXP_ZERO)) { |
| bf_set_inf(r, y_is_odd & x->sign); |
| if (y->sign) { |
| /* pow(0, y) with y < 0 */ |
| return BF_ST_DIVIDE_ZERO; |
| } |
| } else { |
| bf_set_zero(r, y_is_odd & x->sign); |
| } |
| } |
| } |
| return 0; |
| } |
| bf_init(s, T); |
| bf_set(T, x); |
| y_emin = bf_get_exp_min(y); |
| y_is_int = (y_emin >= 0); |
| rnd_mode = flags & BF_RND_MASK; |
| if (x->sign) { |
| if (!y_is_int) { |
| bf_set_nan(r); |
| bf_delete(T); |
| return BF_ST_INVALID_OP; |
| } |
| y_is_odd = (y_emin == 0); |
| r_sign = y_is_odd; |
| /* change the directed rounding mode if the sign of the result |
| is changed */ |
| if (r_sign && (rnd_mode == BF_RNDD || rnd_mode == BF_RNDU)) |
| flags ^= 1; |
| bf_neg(T); |
| } else { |
| r_sign = 0; |
| } |
| |
| bf_set_ui(r, 1); |
| if (bf_cmp_eq(T, r)) { |
| /* abs(x) = 1: nothing more to do */ |
| ret = 0; |
| } else { |
| /* check the overflow/underflow cases */ |
| { |
| bf_t al_s, *al = &al_s; |
| bf_t ah_s, *ah = &ah_s; |
| limb_t precl = LIMB_BITS; |
| |
| bf_init(s, al); |
| bf_init(s, ah); |
| /* compute bounds of log(abs(x)) * y with a low precision */ |
| /* XXX: compute bf_log() once */ |
| /* XXX: add a fast test before this slow test */ |
| bf_log(al, T, precl, BF_RNDD); |
| bf_log(ah, T, precl, BF_RNDU); |
| bf_mul(al, al, y, precl, BF_RNDD ^ y->sign); |
| bf_mul(ah, ah, y, precl, BF_RNDU ^ y->sign); |
| ret = check_exp_underflow_overflow(s, r, al, ah, prec, flags); |
| bf_delete(al); |
| bf_delete(ah); |
| if (ret) |
| goto done; |
| } |
| |
| if (y_is_int) { |
| slimb_t T_bits, e; |
| int_pow: |
| T_bits = T->expn - bf_get_exp_min(T); |
| if (T_bits == 1) { |
| /* pow(2^b, y) = 2^(b*y) */ |
| bf_mul_si(T, y, T->expn - 1, LIMB_BITS, BF_RNDZ); |
| bf_get_limb(&e, T, 0); |
| bf_set_ui(r, 1); |
| ret = bf_mul_2exp(r, e, prec, flags); |
| } else if (prec == BF_PREC_INF) { |
| slimb_t y1; |
| /* specific case for infinite precision (integer case) */ |
| bf_get_limb(&y1, y, 0); |
| assert(!y->sign); |
| /* x must be an integer, so abs(x) >= 2 */ |
| if (y1 >= ((slimb_t)1 << BF_EXP_BITS_MAX)) { |
| bf_delete(T); |
| return bf_set_overflow(r, 0, BF_PREC_INF, flags); |
| } |
| ret = bf_pow_ui(r, T, y1, BF_PREC_INF, BF_RNDZ); |
| } else { |
| if (y->expn <= 31) { |
| /* small enough power: use exponentiation in all cases */ |
| } else if (y->sign) { |
| /* cannot be exact */ |
| goto general_case; |
| } else { |
| if (rnd_mode == BF_RNDF) |
| goto general_case; /* no need to track exact results */ |
| /* see if the result has a chance to be exact: |
| if x=a*2^b (a odd), x^y=a^y*2^(b*y) |
| x^y needs a precision of at least floor_log2(a)*y bits |
| */ |
| bf_mul_si(r, y, T_bits - 1, LIMB_BITS, BF_RNDZ); |
| bf_get_limb(&e, r, 0); |
| if (prec < e) |
| goto general_case; |
| } |
| ret = bf_ziv_rounding(r, T, prec, flags, bf_pow_int, (void *)y); |
| } |
| } else { |
| if (rnd_mode != BF_RNDF) { |
| bf_t *y1; |
| if (y_emin < 0 && check_exact_power2n(r, T, -y_emin)) { |
| /* the problem is reduced to a power to an integer */ |
| #if 0 |
| printf("\nn=%" PRId64 "\n", -(int64_t)y_emin); |
| bf_print_str("T", T); |
| bf_print_str("r", r); |
| #endif |
| bf_set(T, r); |
| y1 = &ytmp_s; |
| y1->tab = y->tab; |
| y1->len = y->len; |
| y1->sign = y->sign; |
| y1->expn = y->expn - y_emin; |
| y = y1; |
| goto int_pow; |
| } |
| } |
| general_case: |
| ret = bf_ziv_rounding(r, T, prec, flags, bf_pow_generic, (void *)y); |
| } |
| } |
| done: |
| bf_delete(T); |
| r->sign = r_sign; |
| return ret; |
| } |
| |
| /* compute sqrt(-2*x-x^2) to get |sin(x)| from cos(x) - 1. */ |
| static void bf_sqrt_sin(bf_t *r, const bf_t *x, limb_t prec1) |
| { |
| bf_context_t *s = r->ctx; |
| bf_t T_s, *T = &T_s; |
| bf_init(s, T); |
| bf_set(T, x); |
| bf_mul(r, T, T, prec1, BF_RNDN); |
| bf_mul_2exp(T, 1, BF_PREC_INF, BF_RNDZ); |
| bf_add(T, T, r, prec1, BF_RNDN); |
| bf_neg(T); |
| bf_sqrt(r, T, prec1, BF_RNDF); |
| bf_delete(T); |
| } |
| |
| static int bf_sincos(bf_t *s, bf_t *c, const bf_t *a, limb_t prec) |
| { |
| bf_context_t *s1 = a->ctx; |
| bf_t T_s, *T = &T_s; |
| bf_t U_s, *U = &U_s; |
| bf_t r_s, *r = &r_s; |
| slimb_t K, prec1, i, l, mod, prec2; |
| int is_neg; |
| |
| assert(c != a && s != a); |
| |
| bf_init(s1, T); |
| bf_init(s1, U); |
| bf_init(s1, r); |
| |
| /* XXX: precision analysis */ |
| K = bf_isqrt(prec / 2); |
| l = prec / (2 * K) + 1; |
| prec1 = prec + 2 * K + l + 8; |
| |
| /* after the modulo reduction, -pi/4 <= T <= pi/4 */ |
| if (a->expn <= -1) { |
| /* abs(a) <= 0.25: no modulo reduction needed */ |
| bf_set(T, a); |
| mod = 0; |
| } else { |
| slimb_t cancel; |
| cancel = 0; |
| for(;;) { |
| prec2 = prec1 + a->expn + cancel; |
| bf_const_pi(U, prec2, BF_RNDF); |
| bf_mul_2exp(U, -1, BF_PREC_INF, BF_RNDZ); |
| bf_remquo(&mod, T, a, U, prec2, BF_RNDN, BF_RNDN); |
| // printf("T.expn=%ld prec2=%ld\n", T->expn, prec2); |
| if (mod == 0 || (T->expn != BF_EXP_ZERO && |
| (T->expn + prec2) >= (prec1 - 1))) |
| break; |
| /* increase the number of bits until the precision is good enough */ |
| cancel = bf_max(-T->expn, (cancel + 1) * 3 / 2); |
| } |
| mod &= 3; |
| } |
| |
| is_neg = T->sign; |
| |
| /* compute cosm1(x) = cos(x) - 1 */ |
| bf_mul(T, T, T, prec1, BF_RNDN); |
| bf_mul_2exp(T, -2 * K, BF_PREC_INF, BF_RNDZ); |
| |
| /* Taylor expansion: |
| -x^2/2 + x^4/4! - x^6/6! + ... |
| */ |
| bf_set_ui(r, 1); |
| for(i = l ; i >= 1; i--) { |
| bf_set_ui(U, 2 * i - 1); |
| bf_mul_ui(U, U, 2 * i, BF_PREC_INF, BF_RNDZ); |
| bf_div(U, T, U, prec1, BF_RNDN); |
| bf_mul(r, r, U, prec1, BF_RNDN); |
| bf_neg(r); |
| if (i != 1) |
| bf_add_si(r, r, 1, prec1, BF_RNDN); |
| } |
| bf_delete(U); |
| |
| /* undo argument reduction: |
| cosm1(2*x)= 2*(2*cosm1(x)+cosm1(x)^2) |
| */ |
| for(i = 0; i < K; i++) { |
| bf_mul(T, r, r, prec1, BF_RNDN); |
| bf_mul_2exp(r, 1, BF_PREC_INF, BF_RNDZ); |
| bf_add(r, r, T, prec1, BF_RNDN); |
| bf_mul_2exp(r, 1, BF_PREC_INF, BF_RNDZ); |
| } |
| bf_delete(T); |
| |
| if (c) { |
| if ((mod & 1) == 0) { |
| bf_add_si(c, r, 1, prec1, BF_RNDN); |
| } else { |
| bf_sqrt_sin(c, r, prec1); |
| c->sign = is_neg ^ 1; |
| } |
| c->sign ^= mod >> 1; |
| } |
| if (s) { |
| if ((mod & 1) == 0) { |
| bf_sqrt_sin(s, r, prec1); |
| s->sign = is_neg; |
| } else { |
| bf_add_si(s, r, 1, prec1, BF_RNDN); |
| } |
| s->sign ^= mod >> 1; |
| } |
| bf_delete(r); |
| return BF_ST_INEXACT; |
| } |
| |
| static int bf_cos_internal(bf_t *r, const bf_t *a, limb_t prec, void *opaque) |
| { |
| return bf_sincos(NULL, r, a, prec); |
| } |
| |
| int bf_cos(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags) |
| { |
| if (a->len == 0) { |
| if (a->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| return 0; |
| } else if (a->expn == BF_EXP_INF) { |
| bf_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else { |
| bf_set_ui(r, 1); |
| return 0; |
| } |
| } |
| |
| /* small argument case: result = 1+r(x) with r(x) = -x^2/2 + |
| O(X^4). We assume r(x) < 2^(2*EXP(x) - 1). */ |
| if (a->expn < 0) { |
| slimb_t e; |
| e = 2 * a->expn - 1; |
| if (e < -(prec + 2)) { |
| bf_set_ui(r, 1); |
| return bf_add_epsilon(r, r, e, 1, prec, flags); |
| } |
| } |
| |
| return bf_ziv_rounding(r, a, prec, flags, bf_cos_internal, NULL); |
| } |
| |
| static int bf_sin_internal(bf_t *r, const bf_t *a, limb_t prec, void *opaque) |
| { |
| return bf_sincos(r, NULL, a, prec); |
| } |
| |
| int bf_sin(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags) |
| { |
| if (a->len == 0) { |
| if (a->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| return 0; |
| } else if (a->expn == BF_EXP_INF) { |
| bf_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else { |
| bf_set_zero(r, a->sign); |
| return 0; |
| } |
| } |
| |
| /* small argument case: result = x+r(x) with r(x) = -x^3/6 + |
| O(X^5). We assume r(x) < 2^(3*EXP(x) - 2). */ |
| if (a->expn < 0) { |
| slimb_t e; |
| e = sat_add(2 * a->expn, a->expn - 2); |
| if (e < a->expn - bf_max(prec + 2, a->len * LIMB_BITS + 2)) { |
| bf_set(r, a); |
| return bf_add_epsilon(r, r, e, 1 - a->sign, prec, flags); |
| } |
| } |
| |
| return bf_ziv_rounding(r, a, prec, flags, bf_sin_internal, NULL); |
| } |
| |
| static int bf_tan_internal(bf_t *r, const bf_t *a, limb_t prec, void *opaque) |
| { |
| bf_context_t *s = r->ctx; |
| bf_t T_s, *T = &T_s; |
| limb_t prec1; |
| |
| /* XXX: precision analysis */ |
| prec1 = prec + 8; |
| bf_init(s, T); |
| bf_sincos(r, T, a, prec1); |
| bf_div(r, r, T, prec1, BF_RNDF); |
| bf_delete(T); |
| return BF_ST_INEXACT; |
| } |
| |
| int bf_tan(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags) |
| { |
| assert(r != a); |
| if (a->len == 0) { |
| if (a->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| return 0; |
| } else if (a->expn == BF_EXP_INF) { |
| bf_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else { |
| bf_set_zero(r, a->sign); |
| return 0; |
| } |
| } |
| |
| /* small argument case: result = x+r(x) with r(x) = x^3/3 + |
| O(X^5). We assume r(x) < 2^(3*EXP(x) - 1). */ |
| if (a->expn < 0) { |
| slimb_t e; |
| e = sat_add(2 * a->expn, a->expn - 1); |
| if (e < a->expn - bf_max(prec + 2, a->len * LIMB_BITS + 2)) { |
| bf_set(r, a); |
| return bf_add_epsilon(r, r, e, a->sign, prec, flags); |
| } |
| } |
| |
| return bf_ziv_rounding(r, a, prec, flags, bf_tan_internal, NULL); |
| } |
| |
| /* if add_pi2 is true, add pi/2 to the result (used for acos(x) to |
| avoid cancellation) */ |
| static int bf_atan_internal(bf_t *r, const bf_t *a, limb_t prec, |
| void *opaque) |
| { |
| bf_context_t *s = r->ctx; |
| BOOL add_pi2 = (BOOL)(intptr_t)opaque; |
| bf_t T_s, *T = &T_s; |
| bf_t U_s, *U = &U_s; |
| bf_t V_s, *V = &V_s; |
| bf_t X2_s, *X2 = &X2_s; |
| int cmp_1; |
| slimb_t prec1, i, K, l; |
| |
| /* XXX: precision analysis */ |
| K = bf_isqrt((prec + 1) / 2); |
| l = prec / (2 * K) + 1; |
| prec1 = prec + K + 2 * l + 32; |
| // printf("prec=%d K=%d l=%d prec1=%d\n", (int)prec, (int)K, (int)l, (int)prec1); |
| |
| bf_init(s, T); |
| cmp_1 = (a->expn >= 1); /* a >= 1 */ |
| if (cmp_1) { |
| bf_set_ui(T, 1); |
| bf_div(T, T, a, prec1, BF_RNDN); |
| } else { |
| bf_set(T, a); |
| } |
| |
| /* abs(T) <= 1 */ |
| |
| /* argument reduction */ |
| |
| bf_init(s, U); |
| bf_init(s, V); |
| bf_init(s, X2); |
| for(i = 0; i < K; i++) { |
| /* T = T / (1 + sqrt(1 + T^2)) */ |
| bf_mul(U, T, T, prec1, BF_RNDN); |
| bf_add_si(U, U, 1, prec1, BF_RNDN); |
| bf_sqrt(V, U, prec1, BF_RNDN); |
| bf_add_si(V, V, 1, prec1, BF_RNDN); |
| bf_div(T, T, V, prec1, BF_RNDN); |
| } |
| |
| /* Taylor series: |
| x - x^3/3 + ... + (-1)^ l * y^(2*l + 1) / (2*l+1) |
| */ |
| bf_mul(X2, T, T, prec1, BF_RNDN); |
| bf_set_ui(r, 0); |
| for(i = l; i >= 1; i--) { |
| bf_set_si(U, 1); |
| bf_set_ui(V, 2 * i + 1); |
| bf_div(U, U, V, prec1, BF_RNDN); |
| bf_neg(r); |
| bf_add(r, r, U, prec1, BF_RNDN); |
| bf_mul(r, r, X2, prec1, BF_RNDN); |
| } |
| bf_neg(r); |
| bf_add_si(r, r, 1, prec1, BF_RNDN); |
| bf_mul(r, r, T, prec1, BF_RNDN); |
| |
| /* undo the argument reduction */ |
| bf_mul_2exp(r, K, BF_PREC_INF, BF_RNDZ); |
| |
| bf_delete(U); |
| bf_delete(V); |
| bf_delete(X2); |
| |
| i = add_pi2; |
| if (cmp_1 > 0) { |
| /* undo the inversion : r = sign(a)*PI/2 - r */ |
| bf_neg(r); |
| i += 1 - 2 * a->sign; |
| } |
| /* add i*(pi/2) with -1 <= i <= 2 */ |
| if (i != 0) { |
| bf_const_pi(T, prec1, BF_RNDF); |
| if (i != 2) |
| bf_mul_2exp(T, -1, BF_PREC_INF, BF_RNDZ); |
| T->sign = (i < 0); |
| bf_add(r, T, r, prec1, BF_RNDN); |
| } |
| |
| bf_delete(T); |
| return BF_ST_INEXACT; |
| } |
| |
| int bf_atan(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags) |
| { |
| bf_context_t *s = r->ctx; |
| bf_t T_s, *T = &T_s; |
| int res; |
| |
| if (a->len == 0) { |
| if (a->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| return 0; |
| } else if (a->expn == BF_EXP_INF) { |
| /* -PI/2 or PI/2 */ |
| bf_const_pi_signed(r, a->sign, prec, flags); |
| bf_mul_2exp(r, -1, BF_PREC_INF, BF_RNDZ); |
| return BF_ST_INEXACT; |
| } else { |
| bf_set_zero(r, a->sign); |
| return 0; |
| } |
| } |
| |
| bf_init(s, T); |
| bf_set_ui(T, 1); |
| res = bf_cmpu(a, T); |
| bf_delete(T); |
| if (res == 0) { |
| /* short cut: abs(a) == 1 -> +/-pi/4 */ |
| bf_const_pi_signed(r, a->sign, prec, flags); |
| bf_mul_2exp(r, -2, BF_PREC_INF, BF_RNDZ); |
| return BF_ST_INEXACT; |
| } |
| |
| /* small argument case: result = x+r(x) with r(x) = -x^3/3 + |
| O(X^5). We assume r(x) < 2^(3*EXP(x) - 1). */ |
| if (a->expn < 0) { |
| slimb_t e; |
| e = sat_add(2 * a->expn, a->expn - 1); |
| if (e < a->expn - bf_max(prec + 2, a->len * LIMB_BITS + 2)) { |
| bf_set(r, a); |
| return bf_add_epsilon(r, r, e, 1 - a->sign, prec, flags); |
| } |
| } |
| |
| return bf_ziv_rounding(r, a, prec, flags, bf_atan_internal, (void *)FALSE); |
| } |
| |
| static int bf_atan2_internal(bf_t *r, const bf_t *y, limb_t prec, void *opaque) |
| { |
| bf_context_t *s = r->ctx; |
| const bf_t *x = opaque; |
| bf_t T_s, *T = &T_s; |
| limb_t prec1; |
| int ret; |
| |
| if (y->expn == BF_EXP_NAN || x->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| return 0; |
| } |
| |
| /* compute atan(y/x) assumming inf/inf = 1 and 0/0 = 0 */ |
| bf_init(s, T); |
| prec1 = prec + 32; |
| if (y->expn == BF_EXP_INF && x->expn == BF_EXP_INF) { |
| bf_set_ui(T, 1); |
| T->sign = y->sign ^ x->sign; |
| } else if (y->expn == BF_EXP_ZERO && x->expn == BF_EXP_ZERO) { |
| bf_set_zero(T, y->sign ^ x->sign); |
| } else { |
| bf_div(T, y, x, prec1, BF_RNDF); |
| } |
| ret = bf_atan(r, T, prec1, BF_RNDF); |
| |
| if (x->sign) { |
| /* if x < 0 (it includes -0), return sign(y)*pi + atan(y/x) */ |
| bf_const_pi(T, prec1, BF_RNDF); |
| T->sign = y->sign; |
| bf_add(r, r, T, prec1, BF_RNDN); |
| ret |= BF_ST_INEXACT; |
| } |
| |
| bf_delete(T); |
| return ret; |
| } |
| |
| int bf_atan2(bf_t *r, const bf_t *y, const bf_t *x, |
| limb_t prec, bf_flags_t flags) |
| { |
| return bf_ziv_rounding(r, y, prec, flags, bf_atan2_internal, (void *)x); |
| } |
| |
| static int bf_asin_internal(bf_t *r, const bf_t *a, limb_t prec, void *opaque) |
| { |
| bf_context_t *s = r->ctx; |
| BOOL is_acos = (BOOL)(intptr_t)opaque; |
| bf_t T_s, *T = &T_s; |
| limb_t prec1, prec2; |
| |
| /* asin(x) = atan(x/sqrt(1-x^2)) |
| acos(x) = pi/2 - asin(x) */ |
| prec1 = prec + 8; |
| /* increase the precision in x^2 to compensate the cancellation in |
| (1-x^2) if x is close to 1 */ |
| /* XXX: use less precision when possible */ |
| if (a->expn >= 0) |
| prec2 = BF_PREC_INF; |
| else |
| prec2 = prec1; |
| bf_init(s, T); |
| bf_mul(T, a, a, prec2, BF_RNDN); |
| bf_neg(T); |
| bf_add_si(T, T, 1, prec2, BF_RNDN); |
| |
| bf_sqrt(r, T, prec1, BF_RNDN); |
| bf_div(T, a, r, prec1, BF_RNDN); |
| if (is_acos) |
| bf_neg(T); |
| bf_atan_internal(r, T, prec1, (void *)(intptr_t)is_acos); |
| bf_delete(T); |
| return BF_ST_INEXACT; |
| } |
| |
| int bf_asin(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags) |
| { |
| bf_context_t *s = r->ctx; |
| bf_t T_s, *T = &T_s; |
| int res; |
| |
| if (a->len == 0) { |
| if (a->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| return 0; |
| } else if (a->expn == BF_EXP_INF) { |
| bf_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else { |
| bf_set_zero(r, a->sign); |
| return 0; |
| } |
| } |
| bf_init(s, T); |
| bf_set_ui(T, 1); |
| res = bf_cmpu(a, T); |
| bf_delete(T); |
| if (res > 0) { |
| bf_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } |
| |
| /* small argument case: result = x+r(x) with r(x) = x^3/6 + |
| O(X^5). We assume r(x) < 2^(3*EXP(x) - 2). */ |
| if (a->expn < 0) { |
| slimb_t e; |
| e = sat_add(2 * a->expn, a->expn - 2); |
| if (e < a->expn - bf_max(prec + 2, a->len * LIMB_BITS + 2)) { |
| bf_set(r, a); |
| return bf_add_epsilon(r, r, e, a->sign, prec, flags); |
| } |
| } |
| |
| return bf_ziv_rounding(r, a, prec, flags, bf_asin_internal, (void *)FALSE); |
| } |
| |
| int bf_acos(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags) |
| { |
| bf_context_t *s = r->ctx; |
| bf_t T_s, *T = &T_s; |
| int res; |
| |
| if (a->len == 0) { |
| if (a->expn == BF_EXP_NAN) { |
| bf_set_nan(r); |
| return 0; |
| } else if (a->expn == BF_EXP_INF) { |
| bf_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else { |
| bf_const_pi(r, prec, flags); |
| bf_mul_2exp(r, -1, BF_PREC_INF, BF_RNDZ); |
| return BF_ST_INEXACT; |
| } |
| } |
| bf_init(s, T); |
| bf_set_ui(T, 1); |
| res = bf_cmpu(a, T); |
| bf_delete(T); |
| if (res > 0) { |
| bf_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else if (res == 0 && a->sign == 0) { |
| bf_set_zero(r, 0); |
| return 0; |
| } |
| |
| return bf_ziv_rounding(r, a, prec, flags, bf_asin_internal, (void *)TRUE); |
| } |
| |
| /***************************************************************/ |
| /* decimal floating point numbers */ |
| |
| #ifdef USE_BF_DEC |
| |
| #define adddq(r1, r0, a1, a0) \ |
| do { \ |
| limb_t __t = r0; \ |
| r0 += (a0); \ |
| r1 += (a1) + (r0 < __t); \ |
| } while (0) |
| |
| #define subdq(r1, r0, a1, a0) \ |
| do { \ |
| limb_t __t = r0; \ |
| r0 -= (a0); \ |
| r1 -= (a1) + (r0 > __t); \ |
| } while (0) |
| |
| #if LIMB_BITS == 64 |
| |
| /* Note: we assume __int128 is available */ |
| #define muldq(r1, r0, a, b) \ |
| do { \ |
| unsigned __int128 __t; \ |
| __t = (unsigned __int128)(a) * (unsigned __int128)(b); \ |
| r0 = __t; \ |
| r1 = __t >> 64; \ |
| } while (0) |
| |
| #define divdq(q, r, a1, a0, b) \ |
| do { \ |
| unsigned __int128 __t; \ |
| limb_t __b = (b); \ |
| __t = ((unsigned __int128)(a1) << 64) | (a0); \ |
| q = __t / __b; \ |
| r = __t % __b; \ |
| } while (0) |
| |
| #else |
| |
| #define muldq(r1, r0, a, b) \ |
| do { \ |
| uint64_t __t; \ |
| __t = (uint64_t)(a) * (uint64_t)(b); \ |
| r0 = __t; \ |
| r1 = __t >> 32; \ |
| } while (0) |
| |
| #define divdq(q, r, a1, a0, b) \ |
| do { \ |
| uint64_t __t; \ |
| limb_t __b = (b); \ |
| __t = ((uint64_t)(a1) << 32) | (a0); \ |
| q = __t / __b; \ |
| r = __t % __b; \ |
| } while (0) |
| |
| #endif /* LIMB_BITS != 64 */ |
| |
| static inline __maybe_unused limb_t shrd(limb_t low, limb_t high, long shift) |
| { |
| if (shift != 0) |
| low = (low >> shift) | (high << (LIMB_BITS - shift)); |
| return low; |
| } |
| |
| static inline __maybe_unused limb_t shld(limb_t a1, limb_t a0, long shift) |
| { |
| if (shift != 0) |
| return (a1 << shift) | (a0 >> (LIMB_BITS - shift)); |
| else |
| return a1; |
| } |
| |
| #if LIMB_DIGITS == 19 |
| |
| /* WARNING: hardcoded for b = 1e19. It is assumed that: |
| 0 <= a1 < 2^63 */ |
| #define divdq_base(q, r, a1, a0)\ |
| do {\ |
| uint64_t __a0, __a1, __t0, __t1, __b = BF_DEC_BASE; \ |
| __a0 = a0;\ |
| __a1 = a1;\ |
| __t0 = __a1;\ |
| __t0 = shld(__t0, __a0, 1);\ |
| muldq(q, __t1, __t0, UINT64_C(17014118346046923173)); \ |
| muldq(__t1, __t0, q, __b);\ |
| subdq(__a1, __a0, __t1, __t0);\ |
| subdq(__a1, __a0, 1, __b * 2); \ |
| __t0 = (slimb_t)__a1 >> 1; \ |
| q += 2 + __t0;\ |
| adddq(__a1, __a0, 0, __b & __t0);\ |
| q += __a1; \ |
| __a0 += __b & __a1; \ |
| r = __a0;\ |
| } while(0) |
| |
| #elif LIMB_DIGITS == 9 |
| |
| /* WARNING: hardcoded for b = 1e9. It is assumed that: |
| 0 <= a1 < 2^29 */ |
| #define divdq_base(q, r, a1, a0)\ |
| do {\ |
| uint32_t __t0, __t1, __b = BF_DEC_BASE; \ |
| __t0 = a1;\ |
| __t1 = a0;\ |
| __t0 = (__t0 << 3) | (__t1 >> (32 - 3)); \ |
| muldq(q, __t1, __t0, 2305843009U);\ |
| r = a0 - q * __b;\ |
| __t1 = (r >= __b);\ |
| q += __t1;\ |
| if (__t1)\ |
| r -= __b;\ |
| } while(0) |
| |
| #endif |
| |
| /* fast integer division by a fixed constant */ |
| |
| typedef struct FastDivData { |
| limb_t m1; /* multiplier */ |
| int8_t shift1; |
| int8_t shift2; |
| } FastDivData; |
| |
| /* From "Division by Invariant Integers using Multiplication" by |
| Torborn Granlund and Peter L. Montgomery */ |
| /* d must be != 0 */ |
| static inline __maybe_unused void fast_udiv_init(FastDivData *s, limb_t d) |
| { |
| int l; |
| limb_t q, r, m1; |
| if (d == 1) |
| l = 0; |
| else |
| l = 64 - clz64(d - 1); |
| divdq(q, r, ((limb_t)1 << l) - d, 0, d); |
| (void)r; |
| m1 = q + 1; |
| // printf("d=%lu l=%d m1=0x%016lx\n", d, l, m1); |
| s->m1 = m1; |
| s->shift1 = l; |
| if (s->shift1 > 1) |
| s->shift1 = 1; |
| s->shift2 = l - 1; |
| if (s->shift2 < 0) |
| s->shift2 = 0; |
| } |
| |
| static inline limb_t fast_udiv(limb_t a, const FastDivData *s) |
| { |
| limb_t t0, t1; |
| muldq(t1, t0, s->m1, a); |
| t0 = (a - t1) >> s->shift1; |
| return (t1 + t0) >> s->shift2; |
| } |
| |
| /* contains 10^i */ |
| const limb_t mp_pow_dec[LIMB_DIGITS + 1] = { |
| 1U, |
| 10U, |
| 100U, |
| 1000U, |
| 10000U, |
| 100000U, |
| 1000000U, |
| 10000000U, |
| 100000000U, |
| 1000000000U, |
| #if LIMB_BITS == 64 |
| 10000000000U, |
| 100000000000U, |
| 1000000000000U, |
| 10000000000000U, |
| 100000000000000U, |
| 1000000000000000U, |
| 10000000000000000U, |
| 100000000000000000U, |
| 1000000000000000000U, |
| 10000000000000000000U, |
| #endif |
| }; |
| |
| /* precomputed from fast_udiv_init(10^i) */ |
| static const FastDivData mp_pow_div[LIMB_DIGITS + 1] = { |
| #if LIMB_BITS == 32 |
| { 0x00000001, 0, 0 }, |
| { 0x9999999a, 1, 3 }, |
| { 0x47ae147b, 1, 6 }, |
| { 0x0624dd30, 1, 9 }, |
| { 0xa36e2eb2, 1, 13 }, |
| { 0x4f8b588f, 1, 16 }, |
| { 0x0c6f7a0c, 1, 19 }, |
| { 0xad7f29ac, 1, 23 }, |
| { 0x5798ee24, 1, 26 }, |
| { 0x12e0be83, 1, 29 }, |
| #else |
| { 0x0000000000000001, 0, 0 }, |
| { 0x999999999999999a, 1, 3 }, |
| { 0x47ae147ae147ae15, 1, 6 }, |
| { 0x0624dd2f1a9fbe77, 1, 9 }, |
| { 0xa36e2eb1c432ca58, 1, 13 }, |
| { 0x4f8b588e368f0847, 1, 16 }, |
| { 0x0c6f7a0b5ed8d36c, 1, 19 }, |
| { 0xad7f29abcaf48579, 1, 23 }, |
| { 0x5798ee2308c39dfa, 1, 26 }, |
| { 0x12e0be826d694b2f, 1, 29 }, |
| { 0xb7cdfd9d7bdbab7e, 1, 33 }, |
| { 0x5fd7fe17964955fe, 1, 36 }, |
| { 0x19799812dea11198, 1, 39 }, |
| { 0xc25c268497681c27, 1, 43 }, |
| { 0x6849b86a12b9b01f, 1, 46 }, |
| { 0x203af9ee756159b3, 1, 49 }, |
| { 0xcd2b297d889bc2b7, 1, 53 }, |
| { 0x70ef54646d496893, 1, 56 }, |
| { 0x2725dd1d243aba0f, 1, 59 }, |
| { 0xd83c94fb6d2ac34d, 1, 63 }, |
| #endif |
| }; |
| |
| /* divide by 10^shift with 0 <= shift <= LIMB_DIGITS */ |
| static inline limb_t fast_shr_dec(limb_t a, int shift) |
| { |
| return fast_udiv(a, &mp_pow_div[shift]); |
| } |
| |
| /* division and remainder by 10^shift */ |
| #define fast_shr_rem_dec(q, r, a, shift) q = fast_shr_dec(a, shift), r = a - q * mp_pow_dec[shift] |
| |
| limb_t mp_add_dec(limb_t *res, const limb_t *op1, const limb_t *op2, |
| mp_size_t n, limb_t carry) |
| { |
| limb_t base = BF_DEC_BASE; |
| mp_size_t i; |
| limb_t k, a, v; |
| |
| k=carry; |
| for(i=0;i<n;i++) { |
| /* XXX: reuse the trick in add_mod */ |
| v = op1[i]; |
| a = v + op2[i] + k - base; |
| k = a <= v; |
| if (!k) |
| a += base; |
| res[i]=a; |
| } |
| return k; |
| } |
| |
| limb_t mp_add_ui_dec(limb_t *tab, limb_t b, mp_size_t n) |
| { |
| limb_t base = BF_DEC_BASE; |
| mp_size_t i; |
| limb_t k, a, v; |
| |
| k=b; |
| for(i=0;i<n;i++) { |
| v = tab[i]; |
| a = v + k - base; |
| k = a <= v; |
| if (!k) |
| a += base; |
| tab[i] = a; |
| if (k == 0) |
| break; |
| } |
| return k; |
| } |
| |
| limb_t mp_sub_dec(limb_t *res, const limb_t *op1, const limb_t *op2, |
| mp_size_t n, limb_t carry) |
| { |
| limb_t base = BF_DEC_BASE; |
| mp_size_t i; |
| limb_t k, v, a; |
| |
| k=carry; |
| for(i=0;i<n;i++) { |
| v = op1[i]; |
| a = v - op2[i] - k; |
| k = a > v; |
| if (k) |
| a += base; |
| res[i] = a; |
| } |
| return k; |
| } |
| |
| limb_t mp_sub_ui_dec(limb_t *tab, limb_t b, mp_size_t n) |
| { |
| limb_t base = BF_DEC_BASE; |
| mp_size_t i; |
| limb_t k, v, a; |
| |
| k=b; |
| for(i=0;i<n;i++) { |
| v = tab[i]; |
| a = v - k; |
| k = a > v; |
| if (k) |
| a += base; |
| tab[i]=a; |
| if (k == 0) |
| break; |
| } |
| return k; |
| } |
| |
| /* taba[] = taba[] * b + l. 0 <= b, l <= base - 1. Return the high carry */ |
| limb_t mp_mul1_dec(limb_t *tabr, const limb_t *taba, mp_size_t n, |
| limb_t b, limb_t l) |
| { |
| mp_size_t i; |
| limb_t t0, t1, r; |
| |
| for(i = 0; i < n; i++) { |
| muldq(t1, t0, taba[i], b); |
| adddq(t1, t0, 0, l); |
| divdq_base(l, r, t1, t0); |
| tabr[i] = r; |
| } |
| return l; |
| } |
| |
| /* tabr[] += taba[] * b. 0 <= b <= base - 1. Return the value to add |
| to the high word */ |
| limb_t mp_add_mul1_dec(limb_t *tabr, const limb_t *taba, mp_size_t n, |
| limb_t b) |
| { |
| mp_size_t i; |
| limb_t l, t0, t1, r; |
| |
| l = 0; |
| for(i = 0; i < n; i++) { |
| muldq(t1, t0, taba[i], b); |
| adddq(t1, t0, 0, l); |
| adddq(t1, t0, 0, tabr[i]); |
| divdq_base(l, r, t1, t0); |
| tabr[i] = r; |
| } |
| return l; |
| } |
| |
| /* tabr[] -= taba[] * b. 0 <= b <= base - 1. Return the value to |
| substract to the high word. */ |
| limb_t mp_sub_mul1_dec(limb_t *tabr, const limb_t *taba, mp_size_t n, |
| limb_t b) |
| { |
| limb_t base = BF_DEC_BASE; |
| mp_size_t i; |
| limb_t l, t0, t1, r, a, v, c; |
| |
| /* XXX: optimize */ |
| l = 0; |
| for(i = 0; i < n; i++) { |
| muldq(t1, t0, taba[i], b); |
| adddq(t1, t0, 0, l); |
| divdq_base(l, r, t1, t0); |
| v = tabr[i]; |
| a = v - r; |
| c = a > v; |
| if (c) |
| a += base; |
| /* never bigger than base because r = 0 when l = base - 1 */ |
| l += c; |
| tabr[i] = a; |
| } |
| return l; |
| } |
| |
| /* size of the result : op1_size + op2_size. */ |
| void mp_mul_basecase_dec(limb_t *result, |
| const limb_t *op1, mp_size_t op1_size, |
| const limb_t *op2, mp_size_t op2_size) |
| { |
| mp_size_t i; |
| limb_t r; |
| |
| result[op1_size] = mp_mul1_dec(result, op1, op1_size, op2[0], 0); |
| |
| for(i=1;i<op2_size;i++) { |
| r = mp_add_mul1_dec(result + i, op1, op1_size, op2[i]); |
| result[i + op1_size] = r; |
| } |
| } |
| |
| /* taba[] = (taba[] + r*base^na) / b. 0 <= b < base. 0 <= r < |
| b. Return the remainder. */ |
| limb_t mp_div1_dec(limb_t *tabr, const limb_t *taba, mp_size_t na, |
| limb_t b, limb_t r) |
| { |
| limb_t base = BF_DEC_BASE; |
| mp_size_t i; |
| limb_t t0, t1, q; |
| int shift; |
| |
| #if (BF_DEC_BASE % 2) == 0 |
| if (b == 2) { |
| limb_t base_div2; |
| /* Note: only works if base is even */ |
| base_div2 = base >> 1; |
| if (r) |
| r = base_div2; |
| for(i = na - 1; i >= 0; i--) { |
| t0 = taba[i]; |
| tabr[i] = (t0 >> 1) + r; |
| r = 0; |
| if (t0 & 1) |
| r = base_div2; |
| } |
| if (r) |
| r = 1; |
| } else |
| #endif |
| if (na >= UDIV1NORM_THRESHOLD) { |
| shift = clz(b); |
| if (shift == 0) { |
| /* normalized case: b >= 2^(LIMB_BITS-1) */ |
| limb_t b_inv; |
| b_inv = udiv1norm_init(b); |
| for(i = na - 1; i >= 0; i--) { |
| muldq(t1, t0, r, base); |
| adddq(t1, t0, 0, taba[i]); |
| q = udiv1norm(&r, t1, t0, b, b_inv); |
| tabr[i] = q; |
| } |
| } else { |
| limb_t b_inv; |
| b <<= shift; |
| b_inv = udiv1norm_init(b); |
| for(i = na - 1; i >= 0; i--) { |
| muldq(t1, t0, r, base); |
| adddq(t1, t0, 0, taba[i]); |
| t1 = (t1 << shift) | (t0 >> (LIMB_BITS - shift)); |
| t0 <<= shift; |
| q = udiv1norm(&r, t1, t0, b, b_inv); |
| r >>= shift; |
| tabr[i] = q; |
| } |
| } |
| } else { |
| for(i = na - 1; i >= 0; i--) { |
| muldq(t1, t0, r, base); |
| adddq(t1, t0, 0, taba[i]); |
| divdq(q, r, t1, t0, b); |
| tabr[i] = q; |
| } |
| } |
| return r; |
| } |
| |
| static __maybe_unused void mp_print_str_dec(const char *str, |
| const limb_t *tab, slimb_t n) |
| { |
| slimb_t i; |
| printf("%s=", str); |
| for(i = n - 1; i >= 0; i--) { |
| if (i != n - 1) |
| printf("_"); |
| printf("%0*" PRIu_LIMB, LIMB_DIGITS, tab[i]); |
| } |
| printf("\n"); |
| } |
| |
| static __maybe_unused void mp_print_str_h_dec(const char *str, |
| const limb_t *tab, slimb_t n, |
| limb_t high) |
| { |
| slimb_t i; |
| printf("%s=", str); |
| printf("%0*" PRIu_LIMB, LIMB_DIGITS, high); |
| for(i = n - 1; i >= 0; i--) { |
| printf("_"); |
| printf("%0*" PRIu_LIMB, LIMB_DIGITS, tab[i]); |
| } |
| printf("\n"); |
| } |
| |
| //#define DEBUG_DIV_SLOW |
| |
| #define DIV_STATIC_ALLOC_LEN 16 |
| |
| /* return q = a / b and r = a % b. |
| |
| taba[na] must be allocated if tabb1[nb - 1] < B / 2. tabb1[nb - 1] |
| must be != zero. na must be >= nb. 's' can be NULL if tabb1[nb - 1] |
| >= B / 2. |
| |
| The remainder is is returned in taba and contains nb libms. tabq |
| contains na - nb + 1 limbs. No overlap is permitted. |
| |
| Running time of the standard method: (na - nb + 1) * nb |
| Return 0 if OK, -1 if memory alloc error |
| */ |
| /* XXX: optimize */ |
| static int mp_div_dec(bf_context_t *s, limb_t *tabq, |
| limb_t *taba, mp_size_t na, |
| const limb_t *tabb1, mp_size_t nb) |
| { |
| limb_t base = BF_DEC_BASE; |
| limb_t r, mult, t0, t1, a, c, q, v, *tabb; |
| mp_size_t i, j; |
| limb_t static_tabb[DIV_STATIC_ALLOC_LEN]; |
| |
| #ifdef DEBUG_DIV_SLOW |
| mp_print_str_dec("a", taba, na); |
| mp_print_str_dec("b", tabb1, nb); |
| #endif |
| |
| /* normalize tabb */ |
| r = tabb1[nb - 1]; |
| assert(r != 0); |
| i = na - nb; |
| if (r >= BF_DEC_BASE / 2) { |
| mult = 1; |
| tabb = (limb_t *)tabb1; |
| q = 1; |
| for(j = nb - 1; j >= 0; j--) { |
| if (taba[i + j] != tabb[j]) { |
| if (taba[i + j] < tabb[j]) |
| q = 0; |
| break; |
| } |
| } |
| tabq[i] = q; |
| if (q) { |
| mp_sub_dec(taba + i, taba + i, tabb, nb, 0); |
| } |
| i--; |
| } else { |
| mult = base / (r + 1); |
| if (likely(nb <= DIV_STATIC_ALLOC_LEN)) { |
| tabb = static_tabb; |
| } else { |
| tabb = bf_malloc(s, sizeof(limb_t) * nb); |
| if (!tabb) |
| return -1; |
| } |
| mp_mul1_dec(tabb, tabb1, nb, mult, 0); |
| taba[na] = mp_mul1_dec(taba, taba, na, mult, 0); |
| } |
| |
| #ifdef DEBUG_DIV_SLOW |
| printf("mult=" FMT_LIMB "\n", mult); |
| mp_print_str_dec("a_norm", taba, na + 1); |
| mp_print_str_dec("b_norm", tabb, nb); |
| #endif |
| |
| for(; i >= 0; i--) { |
| if (unlikely(taba[i + nb] >= tabb[nb - 1])) { |
| /* XXX: check if it is really possible */ |
| q = base - 1; |
| } else { |
| muldq(t1, t0, taba[i + nb], base); |
| adddq(t1, t0, 0, taba[i + nb - 1]); |
| divdq(q, r, t1, t0, tabb[nb - 1]); |
| } |
| // printf("i=%d q1=%ld\n", i, q); |
| |
| r = mp_sub_mul1_dec(taba + i, tabb, nb, q); |
| // mp_dump("r1", taba + i, nb, bd); |
| // printf("r2=%ld\n", r); |
| |
| v = taba[i + nb]; |
| a = v - r; |
| c = a > v; |
| if (c) |
| a += base; |
| taba[i + nb] = a; |
| |
| if (c != 0) { |
| /* negative result */ |
| for(;;) { |
| q--; |
| c = mp_add_dec(taba + i, taba + i, tabb, nb, 0); |
| /* propagate carry and test if positive result */ |
| if (c != 0) { |
| if (++taba[i + nb] == base) { |
| break; |
| } |
| } |
| } |
| } |
| tabq[i] = q; |
| } |
| |
| #ifdef DEBUG_DIV_SLOW |
| mp_print_str_dec("q", tabq, na - nb + 1); |
| mp_print_str_dec("r", taba, nb); |
| #endif |
| |
| /* remove the normalization */ |
| if (mult != 1) { |
| mp_div1_dec(taba, taba, nb, mult, 0); |
| if (unlikely(tabb != static_tabb)) |
| bf_free(s, tabb); |
| } |
| return 0; |
| } |
| |
| /* divide by 10^shift */ |
| static limb_t mp_shr_dec(limb_t *tab_r, const limb_t *tab, mp_size_t n, |
| limb_t shift, limb_t high) |
| { |
| mp_size_t i; |
| limb_t l, a, q, r; |
| |
| assert(shift >= 1 && shift < LIMB_DIGITS); |
| l = high; |
| for(i = n - 1; i >= 0; i--) { |
| a = tab[i]; |
| fast_shr_rem_dec(q, r, a, shift); |
| tab_r[i] = q + l * mp_pow_dec[LIMB_DIGITS - shift]; |
| l = r; |
| } |
| return l; |
| } |
| |
| /* multiply by 10^shift */ |
| static limb_t mp_shl_dec(limb_t *tab_r, const limb_t *tab, mp_size_t n, |
| limb_t shift, limb_t low) |
| { |
| mp_size_t i; |
| limb_t l, a, q, r; |
| |
| assert(shift >= 1 && shift < LIMB_DIGITS); |
| l = low; |
| for(i = 0; i < n; i++) { |
| a = tab[i]; |
| fast_shr_rem_dec(q, r, a, LIMB_DIGITS - shift); |
| tab_r[i] = r * mp_pow_dec[shift] + l; |
| l = q; |
| } |
| return l; |
| } |
| |
| static limb_t mp_sqrtrem2_dec(limb_t *tabs, limb_t *taba) |
| { |
| int k; |
| dlimb_t a, b, r; |
| limb_t taba1[2], s, r0, r1; |
| |
| /* convert to binary and normalize */ |
| a = (dlimb_t)taba[1] * BF_DEC_BASE + taba[0]; |
| k = clz(a >> LIMB_BITS) & ~1; |
| b = a << k; |
| taba1[0] = b; |
| taba1[1] = b >> LIMB_BITS; |
| mp_sqrtrem2(&s, taba1); |
| s >>= (k >> 1); |
| /* convert the remainder back to decimal */ |
| r = a - (dlimb_t)s * (dlimb_t)s; |
| divdq_base(r1, r0, r >> LIMB_BITS, r); |
| taba[0] = r0; |
| tabs[0] = s; |
| return r1; |
| } |
| |
| //#define DEBUG_SQRTREM_DEC |
| |
| /* tmp_buf must contain (n / 2 + 1 limbs) */ |
| static limb_t mp_sqrtrem_rec_dec(limb_t *tabs, limb_t *taba, limb_t n, |
| limb_t *tmp_buf) |
| { |
| limb_t l, h, rh, ql, qh, c, i; |
| |
| if (n == 1) |
| return mp_sqrtrem2_dec(tabs, taba); |
| #ifdef DEBUG_SQRTREM_DEC |
| mp_print_str_dec("a", taba, 2 * n); |
| #endif |
| l = n / 2; |
| h = n - l; |
| qh = mp_sqrtrem_rec_dec(tabs + l, taba + 2 * l, h, tmp_buf); |
| #ifdef DEBUG_SQRTREM_DEC |
| mp_print_str_dec("s1", tabs + l, h); |
| mp_print_str_h_dec("r1", taba + 2 * l, h, qh); |
| mp_print_str_h_dec("r2", taba + l, n, qh); |
| #endif |
| |
| /* the remainder is in taba + 2 * l. Its high bit is in qh */ |
| if (qh) { |
| mp_sub_dec(taba + 2 * l, taba + 2 * l, tabs + l, h, 0); |
| } |
| /* instead of dividing by 2*s, divide by s (which is normalized) |
| and update q and r */ |
| mp_div_dec(NULL, tmp_buf, taba + l, n, tabs + l, h); |
| qh += tmp_buf[l]; |
| for(i = 0; i < l; i++) |
| tabs[i] = tmp_buf[i]; |
| ql = mp_div1_dec(tabs, tabs, l, 2, qh & 1); |
| qh = qh >> 1; /* 0 or 1 */ |
| if (ql) |
| rh = mp_add_dec(taba + l, taba + l, tabs + l, h, 0); |
| else |
| rh = 0; |
| #ifdef DEBUG_SQRTREM_DEC |
| mp_print_str_h_dec("q", tabs, l, qh); |
| mp_print_str_h_dec("u", taba + l, h, rh); |
| #endif |
| |
| mp_add_ui_dec(tabs + l, qh, h); |
| #ifdef DEBUG_SQRTREM_DEC |
| mp_print_str_dec("s2", tabs, n); |
| #endif |
| |
| /* q = qh, tabs[l - 1 ... 0], r = taba[n - 1 ... l] */ |
| /* subtract q^2. if qh = 1 then q = B^l, so we can take shortcuts */ |
| if (qh) { |
| c = qh; |
| } else { |
| mp_mul_basecase_dec(taba + n, tabs, l, tabs, l); |
| c = mp_sub_dec(taba, taba, taba + n, 2 * l, 0); |
| } |
| rh -= mp_sub_ui_dec(taba + 2 * l, c, n - 2 * l); |
| if ((slimb_t)rh < 0) { |
| mp_sub_ui_dec(tabs, 1, n); |
| rh += mp_add_mul1_dec(taba, tabs, n, 2); |
| rh += mp_add_ui_dec(taba, 1, n); |
| } |
| return rh; |
| } |
| |
| /* 'taba' has 2*n limbs with n >= 1 and taba[2*n-1] >= B/4. Return (s, |
| r) with s=floor(sqrt(a)) and r=a-s^2. 0 <= r <= 2 * s. tabs has n |
| limbs. r is returned in the lower n limbs of taba. Its r[n] is the |
| returned value of the function. */ |
| int mp_sqrtrem_dec(bf_context_t *s, limb_t *tabs, limb_t *taba, limb_t n) |
| { |
| limb_t tmp_buf1[8]; |
| limb_t *tmp_buf; |
| mp_size_t n2; |
| n2 = n / 2 + 1; |
| if (n2 <= countof(tmp_buf1)) { |
| tmp_buf = tmp_buf1; |
| } else { |
| tmp_buf = bf_malloc(s, sizeof(limb_t) * n2); |
| if (!tmp_buf) |
| return -1; |
| } |
| taba[n] = mp_sqrtrem_rec_dec(tabs, taba, n, tmp_buf); |
| if (tmp_buf != tmp_buf1) |
| bf_free(s, tmp_buf); |
| return 0; |
| } |
| |
| /* return the number of leading zero digits, from 0 to LIMB_DIGITS */ |
| static int clz_dec(limb_t a) |
| { |
| if (a == 0) |
| return LIMB_DIGITS; |
| switch(LIMB_BITS - 1 - clz(a)) { |
| case 0: /* 1-1 */ |
| return LIMB_DIGITS - 1; |
| case 1: /* 2-3 */ |
| return LIMB_DIGITS - 1; |
| case 2: /* 4-7 */ |
| return LIMB_DIGITS - 1; |
| case 3: /* 8-15 */ |
| if (a < 10) |
| return LIMB_DIGITS - 1; |
| else |
| return LIMB_DIGITS - 2; |
| case 4: /* 16-31 */ |
| return LIMB_DIGITS - 2; |
| case 5: /* 32-63 */ |
| return LIMB_DIGITS - 2; |
| case 6: /* 64-127 */ |
| if (a < 100) |
| return LIMB_DIGITS - 2; |
| else |
| return LIMB_DIGITS - 3; |
| case 7: /* 128-255 */ |
| return LIMB_DIGITS - 3; |
| case 8: /* 256-511 */ |
| return LIMB_DIGITS - 3; |
| case 9: /* 512-1023 */ |
| if (a < 1000) |
| return LIMB_DIGITS - 3; |
| else |
| return LIMB_DIGITS - 4; |
| case 10: /* 1024-2047 */ |
| return LIMB_DIGITS - 4; |
| case 11: /* 2048-4095 */ |
| return LIMB_DIGITS - 4; |
| case 12: /* 4096-8191 */ |
| return LIMB_DIGITS - 4; |
| case 13: /* 8192-16383 */ |
| if (a < 10000) |
| return LIMB_DIGITS - 4; |
| else |
| return LIMB_DIGITS - 5; |
| case 14: /* 16384-32767 */ |
| return LIMB_DIGITS - 5; |
| case 15: /* 32768-65535 */ |
| return LIMB_DIGITS - 5; |
| case 16: /* 65536-131071 */ |
| if (a < 100000) |
| return LIMB_DIGITS - 5; |
| else |
| return LIMB_DIGITS - 6; |
| case 17: /* 131072-262143 */ |
| return LIMB_DIGITS - 6; |
| case 18: /* 262144-524287 */ |
| return LIMB_DIGITS - 6; |
| case 19: /* 524288-1048575 */ |
| if (a < 1000000) |
| return LIMB_DIGITS - 6; |
| else |
| return LIMB_DIGITS - 7; |
| case 20: /* 1048576-2097151 */ |
| return LIMB_DIGITS - 7; |
| case 21: /* 2097152-4194303 */ |
| return LIMB_DIGITS - 7; |
| case 22: /* 4194304-8388607 */ |
| return LIMB_DIGITS - 7; |
| case 23: /* 8388608-16777215 */ |
| if (a < 10000000) |
| return LIMB_DIGITS - 7; |
| else |
| return LIMB_DIGITS - 8; |
| case 24: /* 16777216-33554431 */ |
| return LIMB_DIGITS - 8; |
| case 25: /* 33554432-67108863 */ |
| return LIMB_DIGITS - 8; |
| case 26: /* 67108864-134217727 */ |
| if (a < 100000000) |
| return LIMB_DIGITS - 8; |
| else |
| return LIMB_DIGITS - 9; |
| #if LIMB_BITS == 64 |
| case 27: /* 134217728-268435455 */ |
| return LIMB_DIGITS - 9; |
| case 28: /* 268435456-536870911 */ |
| return LIMB_DIGITS - 9; |
| case 29: /* 536870912-1073741823 */ |
| if (a < 1000000000) |
| return LIMB_DIGITS - 9; |
| else |
| return LIMB_DIGITS - 10; |
| case 30: /* 1073741824-2147483647 */ |
| return LIMB_DIGITS - 10; |
| case 31: /* 2147483648-4294967295 */ |
| return LIMB_DIGITS - 10; |
| case 32: /* 4294967296-8589934591 */ |
| return LIMB_DIGITS - 10; |
| case 33: /* 8589934592-17179869183 */ |
| if (a < 10000000000) |
| return LIMB_DIGITS - 10; |
| else |
| return LIMB_DIGITS - 11; |
| case 34: /* 17179869184-34359738367 */ |
| return LIMB_DIGITS - 11; |
| case 35: /* 34359738368-68719476735 */ |
| return LIMB_DIGITS - 11; |
| case 36: /* 68719476736-137438953471 */ |
| if (a < 100000000000) |
| return LIMB_DIGITS - 11; |
| else |
| return LIMB_DIGITS - 12; |
| case 37: /* 137438953472-274877906943 */ |
| return LIMB_DIGITS - 12; |
| case 38: /* 274877906944-549755813887 */ |
| return LIMB_DIGITS - 12; |
| case 39: /* 549755813888-1099511627775 */ |
| if (a < 1000000000000) |
| return LIMB_DIGITS - 12; |
| else |
| return LIMB_DIGITS - 13; |
| case 40: /* 1099511627776-2199023255551 */ |
| return LIMB_DIGITS - 13; |
| case 41: /* 2199023255552-4398046511103 */ |
| return LIMB_DIGITS - 13; |
| case 42: /* 4398046511104-8796093022207 */ |
| return LIMB_DIGITS - 13; |
| case 43: /* 8796093022208-17592186044415 */ |
| if (a < 10000000000000) |
| return LIMB_DIGITS - 13; |
| else |
| return LIMB_DIGITS - 14; |
| case 44: /* 17592186044416-35184372088831 */ |
| return LIMB_DIGITS - 14; |
| case 45: /* 35184372088832-70368744177663 */ |
| return LIMB_DIGITS - 14; |
| case 46: /* 70368744177664-140737488355327 */ |
| if (a < 100000000000000) |
| return LIMB_DIGITS - 14; |
| else |
| return LIMB_DIGITS - 15; |
| case 47: /* 140737488355328-281474976710655 */ |
| return LIMB_DIGITS - 15; |
| case 48: /* 281474976710656-562949953421311 */ |
| return LIMB_DIGITS - 15; |
| case 49: /* 562949953421312-1125899906842623 */ |
| if (a < 1000000000000000) |
| return LIMB_DIGITS - 15; |
| else |
| return LIMB_DIGITS - 16; |
| case 50: /* 1125899906842624-2251799813685247 */ |
| return LIMB_DIGITS - 16; |
| case 51: /* 2251799813685248-4503599627370495 */ |
| return LIMB_DIGITS - 16; |
| case 52: /* 4503599627370496-9007199254740991 */ |
| return LIMB_DIGITS - 16; |
| case 53: /* 9007199254740992-18014398509481983 */ |
| if (a < 10000000000000000) |
| return LIMB_DIGITS - 16; |
| else |
| return LIMB_DIGITS - 17; |
| case 54: /* 18014398509481984-36028797018963967 */ |
| return LIMB_DIGITS - 17; |
| case 55: /* 36028797018963968-72057594037927935 */ |
| return LIMB_DIGITS - 17; |
| case 56: /* 72057594037927936-144115188075855871 */ |
| if (a < 100000000000000000) |
| return LIMB_DIGITS - 17; |
| else |
| return LIMB_DIGITS - 18; |
| case 57: /* 144115188075855872-288230376151711743 */ |
| return LIMB_DIGITS - 18; |
| case 58: /* 288230376151711744-576460752303423487 */ |
| return LIMB_DIGITS - 18; |
| case 59: /* 576460752303423488-1152921504606846975 */ |
| if (a < 1000000000000000000) |
| return LIMB_DIGITS - 18; |
| else |
| return LIMB_DIGITS - 19; |
| #endif |
| default: |
| return 0; |
| } |
| } |
| |
| /* for debugging */ |
| void bfdec_print_str(const char *str, const bfdec_t *a) |
| { |
| slimb_t i; |
| printf("%s=", str); |
| |
| if (a->expn == BF_EXP_NAN) { |
| printf("NaN"); |
| } else { |
| if (a->sign) |
| putchar('-'); |
| if (a->expn == BF_EXP_ZERO) { |
| putchar('0'); |
| } else if (a->expn == BF_EXP_INF) { |
| printf("Inf"); |
| } else { |
| printf("0."); |
| for(i = a->len - 1; i >= 0; i--) |
| printf("%0*" PRIu_LIMB, LIMB_DIGITS, a->tab[i]); |
| printf("e%" PRId_LIMB, a->expn); |
| } |
| } |
| printf("\n"); |
| } |
| |
| /* return != 0 if one digit between 0 and bit_pos inclusive is not zero. */ |
| static inline limb_t scan_digit_nz(const bfdec_t *r, slimb_t bit_pos) |
| { |
| slimb_t pos; |
| limb_t v, q; |
| int shift; |
| |
| if (bit_pos < 0) |
| return 0; |
| pos = (limb_t)bit_pos / LIMB_DIGITS; |
| shift = (limb_t)bit_pos % LIMB_DIGITS; |
| fast_shr_rem_dec(q, v, r->tab[pos], shift + 1); |
| (void)q; |
| if (v != 0) |
| return 1; |
| pos--; |
| while (pos >= 0) { |
| if (r->tab[pos] != 0) |
| return 1; |
| pos--; |
| } |
| return 0; |
| } |
| |
| static limb_t get_digit(const limb_t *tab, limb_t len, slimb_t pos) |
| { |
| slimb_t i; |
| int shift; |
| i = floor_div(pos, LIMB_DIGITS); |
| if (i < 0 || i >= len) |
| return 0; |
| shift = pos - i * LIMB_DIGITS; |
| return fast_shr_dec(tab[i], shift) % 10; |
| } |
| |
| #if 0 |
| static limb_t get_digits(const limb_t *tab, limb_t len, slimb_t pos) |
| { |
| limb_t a0, a1; |
| int shift; |
| slimb_t i; |
| |
| i = floor_div(pos, LIMB_DIGITS); |
| shift = pos - i * LIMB_DIGITS; |
| if (i >= 0 && i < len) |
| a0 = tab[i]; |
| else |
| a0 = 0; |
| if (shift == 0) { |
| return a0; |
| } else { |
| i++; |
| if (i >= 0 && i < len) |
| a1 = tab[i]; |
| else |
| a1 = 0; |
| return fast_shr_dec(a0, shift) + |
| fast_urem(a1, &mp_pow_div[LIMB_DIGITS - shift]) * |
| mp_pow_dec[shift]; |
| } |
| } |
| #endif |
| |
| /* return the addend for rounding. Note that prec can be <= 0 for bf_rint() */ |
| static int bfdec_get_rnd_add(int *pret, const bfdec_t *r, limb_t l, |
| slimb_t prec, int rnd_mode) |
| { |
| int add_one, inexact; |
| limb_t digit1, digit0; |
| |
| // bfdec_print_str("get_rnd_add", r); |
| if (rnd_mode == BF_RNDF) { |
| digit0 = 1; /* faithful rounding does not honor the INEXACT flag */ |
| } else { |
| /* starting limb for bit 'prec + 1' */ |
| digit0 = scan_digit_nz(r, l * LIMB_DIGITS - 1 - bf_max(0, prec + 1)); |
| } |
| |
| /* get the digit at 'prec' */ |
| digit1 = get_digit(r->tab, l, l * LIMB_DIGITS - 1 - prec); |
| inexact = (digit1 | digit0) != 0; |
| |
| add_one = 0; |
| switch(rnd_mode) { |
| case BF_RNDZ: |
| break; |
| case BF_RNDN: |
| if (digit1 == 5) { |
| if (digit0) { |
| add_one = 1; |
| } else { |
| /* round to even */ |
| add_one = |
| get_digit(r->tab, l, l * LIMB_DIGITS - 1 - (prec - 1)) & 1; |
| } |
| } else if (digit1 > 5) { |
| add_one = 1; |
| } |
| break; |
| case BF_RNDD: |
| case BF_RNDU: |
| if (r->sign == (rnd_mode == BF_RNDD)) |
| add_one = inexact; |
| break; |
| case BF_RNDNA: |
| case BF_RNDF: |
| add_one = (digit1 >= 5); |
| break; |
| case BF_RNDA: |
| add_one = inexact; |
| break; |
| default: |
| abort(); |
| } |
| |
| if (inexact) |
| *pret |= BF_ST_INEXACT; |
| return add_one; |
| } |
| |
| /* round to prec1 bits assuming 'r' is non zero and finite. 'r' is |
| assumed to have length 'l' (1 <= l <= r->len). prec1 can be |
| BF_PREC_INF. BF_FLAG_SUBNORMAL is not supported. Cannot fail with |
| BF_ST_MEM_ERROR. |
| */ |
| static int __bfdec_round(bfdec_t *r, limb_t prec1, bf_flags_t flags, limb_t l) |
| { |
| int shift, add_one, rnd_mode, ret; |
| slimb_t i, bit_pos, pos, e_min, e_max, e_range, prec; |
| |
| /* XXX: align to IEEE 754 2008 for decimal numbers ? */ |
| e_range = (limb_t)1 << (bf_get_exp_bits(flags) - 1); |
| e_min = -e_range + 3; |
| e_max = e_range; |
| |
| if (flags & BF_FLAG_RADPNT_PREC) { |
| /* 'prec' is the precision after the decimal point */ |
| if (prec1 != BF_PREC_INF) |
| prec = r->expn + prec1; |
| else |
| prec = prec1; |
| } else if (unlikely(r->expn < e_min) && (flags & BF_FLAG_SUBNORMAL)) { |
| /* restrict the precision in case of potentially subnormal |
| result */ |
| assert(prec1 != BF_PREC_INF); |
| prec = prec1 - (e_min - r->expn); |
| } else { |
| prec = prec1; |
| } |
| |
| /* round to prec bits */ |
| rnd_mode = flags & BF_RND_MASK; |
| ret = 0; |
| add_one = bfdec_get_rnd_add(&ret, r, l, prec, rnd_mode); |
| |
| if (prec <= 0) { |
| if (add_one) { |
| bfdec_resize(r, 1); /* cannot fail because r is non zero */ |
| r->tab[0] = BF_DEC_BASE / 10; |
| r->expn += 1 - prec; |
| ret |= BF_ST_UNDERFLOW | BF_ST_INEXACT; |
| return ret; |
| } else { |
| goto underflow; |
| } |
| } else if (add_one) { |
| limb_t carry; |
| |
| /* add one starting at digit 'prec - 1' */ |
| bit_pos = l * LIMB_DIGITS - 1 - (prec - 1); |
| pos = bit_pos / LIMB_DIGITS; |
| carry = mp_pow_dec[bit_pos % LIMB_DIGITS]; |
| carry = mp_add_ui_dec(r->tab + pos, carry, l - pos); |
| if (carry) { |
| /* shift right by one digit */ |
| mp_shr_dec(r->tab + pos, r->tab + pos, l - pos, 1, 1); |
| r->expn++; |
| } |
| } |
| |
| /* check underflow */ |
| if (unlikely(r->expn < e_min)) { |
| if (flags & BF_FLAG_SUBNORMAL) { |
| /* if inexact, also set the underflow flag */ |
| if (ret & BF_ST_INEXACT) |
| ret |= BF_ST_UNDERFLOW; |
| } else { |
| underflow: |
| bfdec_set_zero(r, r->sign); |
| ret |= BF_ST_UNDERFLOW | BF_ST_INEXACT; |
| return ret; |
| } |
| } |
| |
| /* check overflow */ |
| if (unlikely(r->expn > e_max)) { |
| bfdec_set_inf(r, r->sign); |
| ret |= BF_ST_OVERFLOW | BF_ST_INEXACT; |
| return ret; |
| } |
| |
| /* keep the bits starting at 'prec - 1' */ |
| bit_pos = l * LIMB_DIGITS - 1 - (prec - 1); |
| i = floor_div(bit_pos, LIMB_DIGITS); |
| if (i >= 0) { |
| shift = smod(bit_pos, LIMB_DIGITS); |
| if (shift != 0) { |
| r->tab[i] = fast_shr_dec(r->tab[i], shift) * |
| mp_pow_dec[shift]; |
| } |
| } else { |
| i = 0; |
| } |
| /* remove trailing zeros */ |
| while (r->tab[i] == 0) |
| i++; |
| if (i > 0) { |
| l -= i; |
| memmove(r->tab, r->tab + i, l * sizeof(limb_t)); |
| } |
| bfdec_resize(r, l); /* cannot fail */ |
| return ret; |
| } |
| |
| /* Cannot fail with BF_ST_MEM_ERROR. */ |
| int bfdec_round(bfdec_t *r, limb_t prec, bf_flags_t flags) |
| { |
| if (r->len == 0) |
| return 0; |
| return __bfdec_round(r, prec, flags, r->len); |
| } |
| |
| /* 'r' must be a finite number. Cannot fail with BF_ST_MEM_ERROR. */ |
| int bfdec_normalize_and_round(bfdec_t *r, limb_t prec1, bf_flags_t flags) |
| { |
| limb_t l, v; |
| int shift, ret; |
| |
| // bfdec_print_str("bf_renorm", r); |
| l = r->len; |
| while (l > 0 && r->tab[l - 1] == 0) |
| l--; |
| if (l == 0) { |
| /* zero */ |
| r->expn = BF_EXP_ZERO; |
| bfdec_resize(r, 0); /* cannot fail */ |
| ret = 0; |
| } else { |
| r->expn -= (r->len - l) * LIMB_DIGITS; |
| /* shift to have the MSB set to '1' */ |
| v = r->tab[l - 1]; |
| shift = clz_dec(v); |
| if (shift != 0) { |
| mp_shl_dec(r->tab, r->tab, l, shift, 0); |
| r->expn -= shift; |
| } |
| ret = __bfdec_round(r, prec1, flags, l); |
| } |
| // bf_print_str("r_final", r); |
| return ret; |
| } |
| |
| int bfdec_set_ui(bfdec_t *r, uint64_t v) |
| { |
| #if LIMB_BITS == 32 |
| if (v >= BF_DEC_BASE * BF_DEC_BASE) { |
| if (bfdec_resize(r, 3)) |
| goto fail; |
| r->tab[0] = v % BF_DEC_BASE; |
| v /= BF_DEC_BASE; |
| r->tab[1] = v % BF_DEC_BASE; |
| r->tab[2] = v / BF_DEC_BASE; |
| r->expn = 3 * LIMB_DIGITS; |
| } else |
| #endif |
| if (v >= BF_DEC_BASE) { |
| if (bfdec_resize(r, 2)) |
| goto fail; |
| r->tab[0] = v % BF_DEC_BASE; |
| r->tab[1] = v / BF_DEC_BASE; |
| r->expn = 2 * LIMB_DIGITS; |
| } else { |
| if (bfdec_resize(r, 1)) |
| goto fail; |
| r->tab[0] = v; |
| r->expn = LIMB_DIGITS; |
| } |
| r->sign = 0; |
| return bfdec_normalize_and_round(r, BF_PREC_INF, 0); |
| fail: |
| bfdec_set_nan(r); |
| return BF_ST_MEM_ERROR; |
| } |
| |
| int bfdec_set_si(bfdec_t *r, int64_t v) |
| { |
| int ret; |
| if (v < 0) { |
| ret = bfdec_set_ui(r, -v); |
| r->sign = 1; |
| } else { |
| ret = bfdec_set_ui(r, v); |
| } |
| return ret; |
| } |
| |
| static int bfdec_add_internal(bfdec_t *r, const bfdec_t *a, const bfdec_t *b, limb_t prec, bf_flags_t flags, int b_neg) |
| { |
| bf_context_t *s = r->ctx; |
| int is_sub, cmp_res, a_sign, b_sign, ret; |
| |
| a_sign = a->sign; |
| b_sign = b->sign ^ b_neg; |
| is_sub = a_sign ^ b_sign; |
| cmp_res = bfdec_cmpu(a, b); |
| if (cmp_res < 0) { |
| const bfdec_t *tmp; |
| tmp = a; |
| a = b; |
| b = tmp; |
| a_sign = b_sign; /* b_sign is never used later */ |
| } |
| /* abs(a) >= abs(b) */ |
| if (cmp_res == 0 && is_sub && a->expn < BF_EXP_INF) { |
| /* zero result */ |
| bfdec_set_zero(r, (flags & BF_RND_MASK) == BF_RNDD); |
| ret = 0; |
| } else if (a->len == 0 || b->len == 0) { |
| ret = 0; |
| if (a->expn >= BF_EXP_INF) { |
| if (a->expn == BF_EXP_NAN) { |
| /* at least one operand is NaN */ |
| bfdec_set_nan(r); |
| ret = 0; |
| } else if (b->expn == BF_EXP_INF && is_sub) { |
| /* infinities with different signs */ |
| bfdec_set_nan(r); |
| ret = BF_ST_INVALID_OP; |
| } else { |
| bfdec_set_inf(r, a_sign); |
| } |
| } else { |
| /* at least one zero and not subtract */ |
| if (bfdec_set(r, a)) |
| return BF_ST_MEM_ERROR; |
| r->sign = a_sign; |
| goto renorm; |
| } |
| } else { |
| slimb_t d, a_offset, b_offset, i, r_len; |
| limb_t carry; |
| limb_t *b1_tab; |
| int b_shift; |
| mp_size_t b1_len; |
| |
| d = a->expn - b->expn; |
| |
| /* XXX: not efficient in time and memory if the precision is |
| not infinite */ |
| r_len = bf_max(a->len, b->len + (d + LIMB_DIGITS - 1) / LIMB_DIGITS); |
| if (bfdec_resize(r, r_len)) |
| goto fail; |
| r->sign = a_sign; |
| r->expn = a->expn; |
| |
| a_offset = r_len - a->len; |
| for(i = 0; i < a_offset; i++) |
| r->tab[i] = 0; |
| for(i = 0; i < a->len; i++) |
| r->tab[a_offset + i] = a->tab[i]; |
| |
| b_shift = d % LIMB_DIGITS; |
| if (b_shift == 0) { |
| b1_len = b->len; |
| b1_tab = (limb_t *)b->tab; |
| } else { |
| b1_len = b->len + 1; |
| b1_tab = bf_malloc(s, sizeof(limb_t) * b1_len); |
| if (!b1_tab) |
| goto fail; |
| b1_tab[0] = mp_shr_dec(b1_tab + 1, b->tab, b->len, b_shift, 0) * |
| mp_pow_dec[LIMB_DIGITS - b_shift]; |
| } |
| b_offset = r_len - (b->len + (d + LIMB_DIGITS - 1) / LIMB_DIGITS); |
| |
| if (is_sub) { |
| carry = mp_sub_dec(r->tab + b_offset, r->tab + b_offset, |
| b1_tab, b1_len, 0); |
| if (carry != 0) { |
| carry = mp_sub_ui_dec(r->tab + b_offset + b1_len, carry, |
| r_len - (b_offset + b1_len)); |
| assert(carry == 0); |
| } |
| } else { |
| carry = mp_add_dec(r->tab + b_offset, r->tab + b_offset, |
| b1_tab, b1_len, 0); |
| if (carry != 0) { |
| carry = mp_add_ui_dec(r->tab + b_offset + b1_len, carry, |
| r_len - (b_offset + b1_len)); |
| } |
| if (carry != 0) { |
| if (bfdec_resize(r, r_len + 1)) { |
| if (b_shift != 0) |
| bf_free(s, b1_tab); |
| goto fail; |
| } |
| r->tab[r_len] = 1; |
| r->expn += LIMB_DIGITS; |
| } |
| } |
| if (b_shift != 0) |
| bf_free(s, b1_tab); |
| renorm: |
| ret = bfdec_normalize_and_round(r, prec, flags); |
| } |
| return ret; |
| fail: |
| bfdec_set_nan(r); |
| return BF_ST_MEM_ERROR; |
| } |
| |
| static int __bfdec_add(bfdec_t *r, const bfdec_t *a, const bfdec_t *b, limb_t prec, |
| bf_flags_t flags) |
| { |
| return bfdec_add_internal(r, a, b, prec, flags, 0); |
| } |
| |
| static int __bfdec_sub(bfdec_t *r, const bfdec_t *a, const bfdec_t *b, limb_t prec, |
| bf_flags_t flags) |
| { |
| return bfdec_add_internal(r, a, b, prec, flags, 1); |
| } |
| |
| int bfdec_add(bfdec_t *r, const bfdec_t *a, const bfdec_t *b, limb_t prec, |
| bf_flags_t flags) |
| { |
| return bf_op2((bf_t *)r, (bf_t *)a, (bf_t *)b, prec, flags, |
| (bf_op2_func_t *)__bfdec_add); |
| } |
| |
| int bfdec_sub(bfdec_t *r, const bfdec_t *a, const bfdec_t *b, limb_t prec, |
| bf_flags_t flags) |
| { |
| return bf_op2((bf_t *)r, (bf_t *)a, (bf_t *)b, prec, flags, |
| (bf_op2_func_t *)__bfdec_sub); |
| } |
| |
| int bfdec_mul(bfdec_t *r, const bfdec_t *a, const bfdec_t *b, limb_t prec, |
| bf_flags_t flags) |
| { |
| int ret, r_sign; |
| |
| if (a->len < b->len) { |
| const bfdec_t *tmp = a; |
| a = b; |
| b = tmp; |
| } |
| r_sign = a->sign ^ b->sign; |
| /* here b->len <= a->len */ |
| if (b->len == 0) { |
| if (a->expn == BF_EXP_NAN || b->expn == BF_EXP_NAN) { |
| bfdec_set_nan(r); |
| ret = 0; |
| } else if (a->expn == BF_EXP_INF || b->expn == BF_EXP_INF) { |
| if ((a->expn == BF_EXP_INF && b->expn == BF_EXP_ZERO) || |
| (a->expn == BF_EXP_ZERO && b->expn == BF_EXP_INF)) { |
| bfdec_set_nan(r); |
| ret = BF_ST_INVALID_OP; |
| } else { |
| bfdec_set_inf(r, r_sign); |
| ret = 0; |
| } |
| } else { |
| bfdec_set_zero(r, r_sign); |
| ret = 0; |
| } |
| } else { |
| bfdec_t tmp, *r1 = NULL; |
| limb_t a_len, b_len; |
| limb_t *a_tab, *b_tab; |
| |
| a_len = a->len; |
| b_len = b->len; |
| a_tab = a->tab; |
| b_tab = b->tab; |
| |
| if (r == a || r == b) { |
| bfdec_init(r->ctx, &tmp); |
| r1 = r; |
| r = &tmp; |
| } |
| if (bfdec_resize(r, a_len + b_len)) { |
| bfdec_set_nan(r); |
| ret = BF_ST_MEM_ERROR; |
| goto done; |
| } |
| mp_mul_basecase_dec(r->tab, a_tab, a_len, b_tab, b_len); |
| r->sign = r_sign; |
| r->expn = a->expn + b->expn; |
| ret = bfdec_normalize_and_round(r, prec, flags); |
| done: |
| if (r == &tmp) |
| bfdec_move(r1, &tmp); |
| } |
| return ret; |
| } |
| |
| int bfdec_mul_si(bfdec_t *r, const bfdec_t *a, int64_t b1, limb_t prec, |
| bf_flags_t flags) |
| { |
| bfdec_t b; |
| int ret; |
| bfdec_init(r->ctx, &b); |
| ret = bfdec_set_si(&b, b1); |
| ret |= bfdec_mul(r, a, &b, prec, flags); |
| bfdec_delete(&b); |
| return ret; |
| } |
| |
| int bfdec_add_si(bfdec_t *r, const bfdec_t *a, int64_t b1, limb_t prec, |
| bf_flags_t flags) |
| { |
| bfdec_t b; |
| int ret; |
| |
| bfdec_init(r->ctx, &b); |
| ret = bfdec_set_si(&b, b1); |
| ret |= bfdec_add(r, a, &b, prec, flags); |
| bfdec_delete(&b); |
| return ret; |
| } |
| |
| static int __bfdec_div(bfdec_t *r, const bfdec_t *a, const bfdec_t *b, |
| limb_t prec, bf_flags_t flags) |
| { |
| int ret, r_sign; |
| limb_t n, nb, precl; |
| |
| r_sign = a->sign ^ b->sign; |
| if (a->expn >= BF_EXP_INF || b->expn >= BF_EXP_INF) { |
| if (a->expn == BF_EXP_NAN || b->expn == BF_EXP_NAN) { |
| bfdec_set_nan(r); |
| return 0; |
| } else if (a->expn == BF_EXP_INF && b->expn == BF_EXP_INF) { |
| bfdec_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else if (a->expn == BF_EXP_INF) { |
| bfdec_set_inf(r, r_sign); |
| return 0; |
| } else { |
| bfdec_set_zero(r, r_sign); |
| return 0; |
| } |
| } else if (a->expn == BF_EXP_ZERO) { |
| if (b->expn == BF_EXP_ZERO) { |
| bfdec_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else { |
| bfdec_set_zero(r, r_sign); |
| return 0; |
| } |
| } else if (b->expn == BF_EXP_ZERO) { |
| bfdec_set_inf(r, r_sign); |
| return BF_ST_DIVIDE_ZERO; |
| } |
| |
| nb = b->len; |
| if (prec == BF_PREC_INF) { |
| /* infinite precision: return BF_ST_INVALID_OP if not an exact |
| result */ |
| /* XXX: check */ |
| precl = nb + 1; |
| } else if (flags & BF_FLAG_RADPNT_PREC) { |
| /* number of digits after the decimal point */ |
| /* XXX: check (2 extra digits for rounding + 2 digits) */ |
| precl = (bf_max(a->expn - b->expn, 0) + 2 + |
| prec + 2 + LIMB_DIGITS - 1) / LIMB_DIGITS; |
| } else { |
| /* number of limbs of the quotient (2 extra digits for rounding) */ |
| precl = (prec + 2 + LIMB_DIGITS - 1) / LIMB_DIGITS; |
| } |
| n = bf_max(a->len, precl); |
| |
| { |
| limb_t *taba, na, i; |
| slimb_t d; |
| |
| na = n + nb; |
| taba = bf_malloc(r->ctx, (na + 1) * sizeof(limb_t)); |
| if (!taba) |
| goto fail; |
| d = na - a->len; |
| memset(taba, 0, d * sizeof(limb_t)); |
| memcpy(taba + d, a->tab, a->len * sizeof(limb_t)); |
| if (bfdec_resize(r, n + 1)) |
| goto fail1; |
| if (mp_div_dec(r->ctx, r->tab, taba, na, b->tab, nb)) { |
| fail1: |
| bf_free(r->ctx, taba); |
| goto fail; |
| } |
| /* see if non zero remainder */ |
| for(i = 0; i < nb; i++) { |
| if (taba[i] != 0) |
| break; |
| } |
| bf_free(r->ctx, taba); |
| if (i != nb) { |
| if (prec == BF_PREC_INF) { |
| bfdec_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else { |
| r->tab[0] |= 1; |
| } |
| } |
| r->expn = a->expn - b->expn + LIMB_DIGITS; |
| r->sign = r_sign; |
| ret = bfdec_normalize_and_round(r, prec, flags); |
| } |
| return ret; |
| fail: |
| bfdec_set_nan(r); |
| return BF_ST_MEM_ERROR; |
| } |
| |
| int bfdec_div(bfdec_t *r, const bfdec_t *a, const bfdec_t *b, limb_t prec, |
| bf_flags_t flags) |
| { |
| return bf_op2((bf_t *)r, (bf_t *)a, (bf_t *)b, prec, flags, |
| (bf_op2_func_t *)__bfdec_div); |
| } |
| |
| /* a and b must be finite numbers with a >= 0 and b > 0. 'q' is the |
| integer defined as floor(a/b) and r = a - q * b. */ |
| static void bfdec_tdivremu(bf_context_t *s, bfdec_t *q, bfdec_t *r, |
| const bfdec_t *a, const bfdec_t *b) |
| { |
| if (bfdec_cmpu(a, b) < 0) { |
| bfdec_set_ui(q, 0); |
| bfdec_set(r, a); |
| } else { |
| bfdec_div(q, a, b, 0, BF_RNDZ | BF_FLAG_RADPNT_PREC); |
| bfdec_mul(r, q, b, BF_PREC_INF, BF_RNDZ); |
| bfdec_sub(r, a, r, BF_PREC_INF, BF_RNDZ); |
| } |
| } |
| |
| /* division and remainder. |
| |
| rnd_mode is the rounding mode for the quotient. The additional |
| rounding mode BF_RND_EUCLIDIAN is supported. |
| |
| 'q' is an integer. 'r' is rounded with prec and flags (prec can be |
| BF_PREC_INF). |
| */ |
| int bfdec_divrem(bfdec_t *q, bfdec_t *r, const bfdec_t *a, const bfdec_t *b, |
| limb_t prec, bf_flags_t flags, int rnd_mode) |
| { |
| bf_context_t *s = q->ctx; |
| bfdec_t a1_s, *a1 = &a1_s; |
| bfdec_t b1_s, *b1 = &b1_s; |
| bfdec_t r1_s, *r1 = &r1_s; |
| int q_sign, res; |
| BOOL is_ceil, is_rndn; |
| |
| assert(q != a && q != b); |
| assert(r != a && r != b); |
| assert(q != r); |
| |
| if (a->len == 0 || b->len == 0) { |
| bfdec_set_zero(q, 0); |
| if (a->expn == BF_EXP_NAN || b->expn == BF_EXP_NAN) { |
| bfdec_set_nan(r); |
| return 0; |
| } else if (a->expn == BF_EXP_INF || b->expn == BF_EXP_ZERO) { |
| bfdec_set_nan(r); |
| return BF_ST_INVALID_OP; |
| } else { |
| bfdec_set(r, a); |
| return bfdec_round(r, prec, flags); |
| } |
| } |
| |
| q_sign = a->sign ^ b->sign; |
| is_rndn = (rnd_mode == BF_RNDN || rnd_mode == BF_RNDNA); |
| switch(rnd_mode) { |
| default: |
| case BF_RNDZ: |
| case BF_RNDN: |
| case BF_RNDNA: |
| is_ceil = FALSE; |
| break; |
| case BF_RNDD: |
| is_ceil = q_sign; |
| break; |
| case BF_RNDU: |
| is_ceil = q_sign ^ 1; |
| break; |
| case BF_RNDA: |
| is_ceil = TRUE; |
| break; |
| case BF_DIVREM_EUCLIDIAN: |
| is_ceil = a->sign; |
| break; |
| } |
| |
| a1->expn = a->expn; |
| a1->tab = a->tab; |
| a1->len = a->len; |
| a1->sign = 0; |
| |
| b1->expn = b->expn; |
| b1->tab = b->tab; |
| b1->len = b->len; |
| b1->sign = 0; |
| |
| // bfdec_print_str("a1", a1); |
| // bfdec_print_str("b1", b1); |
| /* XXX: could improve to avoid having a large 'q' */ |
| bfdec_tdivremu(s, q, r, a1, b1); |
| if (bfdec_is_nan(q) || bfdec_is_nan(r)) |
| goto fail; |
| // bfdec_print_str("q", q); |
| // bfdec_print_str("r", r); |
| |
| if (r->len != 0) { |
| if (is_rndn) { |
| bfdec_init(s, r1); |
| if (bfdec_set(r1, r)) |
| goto fail; |
| if (bfdec_mul_si(r1, r1, 2, BF_PREC_INF, BF_RNDZ)) { |
| bfdec_delete(r1); |
| goto fail; |
| } |
| res = bfdec_cmpu(r1, b); |
| bfdec_delete(r1); |
| if (res > 0 || |
| (res == 0 && |
| (rnd_mode == BF_RNDNA || |
| (get_digit(q->tab, q->len, q->len * LIMB_DIGITS - q->expn) & 1) != 0))) { |
| goto do_sub_r; |
| } |
| } else if (is_ceil) { |
| do_sub_r: |
| res = bfdec_add_si(q, q, 1, BF_PREC_INF, BF_RNDZ); |
| res |= bfdec_sub(r, r, b1, BF_PREC_INF, BF_RNDZ); |
| if (res & BF_ST_MEM_ERROR) |
| goto fail; |
| } |
| } |
| |
| r->sign ^= a->sign; |
| q->sign = q_sign; |
| return bfdec_round(r, prec, flags); |
| fail: |
| bfdec_set_nan(q); |
| bfdec_set_nan(r); |
| return BF_ST_MEM_ERROR; |
| } |
| |
| int bfdec_rem(bfdec_t *r, const bfdec_t *a, const bfdec_t *b, limb_t prec, |
| bf_flags_t flags, int rnd_mode) |
| { |
| bfdec_t q_s, *q = &q_s; |
| int ret; |
| |
| bfdec_init(r->ctx, q); |
| ret = bfdec_divrem(q, r, a, b, prec, flags, rnd_mode); |
| bfdec_delete(q); |
| return ret; |
| } |
| |
| /* convert to integer (infinite precision) */ |
| int bfdec_rint(bfdec_t *r, int rnd_mode) |
| { |
| return bfdec_round(r, 0, rnd_mode | BF_FLAG_RADPNT_PREC); |
| } |
| |
| int bfdec_sqrt(bfdec_t *r, const bfdec_t *a, limb_t prec, bf_flags_t flags) |
| { |
| bf_context_t *s = a->ctx; |
| int ret, k; |
| limb_t *a1, v; |
| slimb_t n, n1, prec1; |
| limb_t res; |
| |
| assert(r != a); |
| |
| if (a->len == 0) { |
| if (a->expn == BF_EXP_NAN) { |
| bfdec_set_nan(r); |
| } else if (a->expn == BF_EXP_INF && a->sign) { |
| goto invalid_op; |
| } else { |
| bfdec_set(r, a); |
| } |
| ret = 0; |
| } else if (a->sign || prec == BF_PREC_INF) { |
| invalid_op: |
| bfdec_set_nan(r); |
| ret = BF_ST_INVALID_OP; |
| } else { |
| if (flags & BF_FLAG_RADPNT_PREC) { |
| prec1 = bf_max(floor_div(a->expn + 1, 2) + prec, 1); |
| } else { |
| prec1 = prec; |
| } |
| /* convert the mantissa to an integer with at least 2 * |
| prec + 4 digits */ |
| n = (2 * (prec1 + 2) + 2 * LIMB_DIGITS - 1) / (2 * LIMB_DIGITS); |
| if (bfdec_resize(r, n)) |
| goto fail; |
| a1 = bf_malloc(s, sizeof(limb_t) * 2 * n); |
| if (!a1) |
| goto fail; |
| n1 = bf_min(2 * n, a->len); |
| memset(a1, 0, (2 * n - n1) * sizeof(limb_t)); |
| memcpy(a1 + 2 * n - n1, a->tab + a->len - n1, n1 * sizeof(limb_t)); |
| if (a->expn & 1) { |
| res = mp_shr_dec(a1, a1, 2 * n, 1, 0); |
| } else { |
| res = 0; |
| } |
| /* normalize so that a1 >= B^(2*n)/4. Not need for n = 1 |
| because mp_sqrtrem2_dec already does it */ |
| k = 0; |
| if (n > 1) { |
| v = a1[2 * n - 1]; |
| while (v < BF_DEC_BASE / 4) { |
| k++; |
| v *= 4; |
| } |
| if (k != 0) |
| mp_mul1_dec(a1, a1, 2 * n, 1 << (2 * k), 0); |
| } |
| if (mp_sqrtrem_dec(s, r->tab, a1, n)) { |
| bf_free(s, a1); |
| goto fail; |
| } |
| if (k != 0) |
| mp_div1_dec(r->tab, r->tab, n, 1 << k, 0); |
| if (!res) { |
| res = mp_scan_nz(a1, n + 1); |
| } |
| bf_free(s, a1); |
| if (!res) { |
| res = mp_scan_nz(a->tab, a->len - n1); |
| } |
| if (res != 0) |
| r->tab[0] |= 1; |
| r->sign = 0; |
| r->expn = (a->expn + 1) >> 1; |
| ret = bfdec_round(r, prec, flags); |
| } |
| return ret; |
| fail: |
| bfdec_set_nan(r); |
| return BF_ST_MEM_ERROR; |
| } |
| |
| /* The rounding mode is always BF_RNDZ. Return BF_ST_OVERFLOW if there |
| is an overflow and 0 otherwise. No memory error is possible. */ |
| int bfdec_get_int32(int *pres, const bfdec_t *a) |
| { |
| uint32_t v; |
| int ret; |
| if (a->expn >= BF_EXP_INF) { |
| ret = 0; |
| if (a->expn == BF_EXP_INF) { |
| v = (uint32_t)INT32_MAX + a->sign; |
| /* XXX: return overflow ? */ |
| } else { |
| v = INT32_MAX; |
| } |
| } else if (a->expn <= 0) { |
| v = 0; |
| ret = 0; |
| } else if (a->expn <= 9) { |
| v = fast_shr_dec(a->tab[a->len - 1], LIMB_DIGITS - a->expn); |
| if (a->sign) |
| v = -v; |
| ret = 0; |
| } else if (a->expn == 10) { |
| uint64_t v1; |
| uint32_t v_max; |
| #if LIMB_BITS == 64 |
| v1 = fast_shr_dec(a->tab[a->len - 1], LIMB_DIGITS - a->expn); |
| #else |
| v1 = (uint64_t)a->tab[a->len - 1] * 10 + |
| get_digit(a->tab, a->len, (a->len - 1) * LIMB_DIGITS - 1); |
| #endif |
| v_max = (uint32_t)INT32_MAX + a->sign; |
| if (v1 > v_max) { |
| v = v_max; |
| ret = BF_ST_OVERFLOW; |
| } else { |
| v = v1; |
| if (a->sign) |
| v = -v; |
| ret = 0; |
| } |
| } else { |
| v = (uint32_t)INT32_MAX + a->sign; |
| ret = BF_ST_OVERFLOW; |
| } |
| *pres = v; |
| return ret; |
| } |
| |
| /* power to an integer with infinite precision */ |
| int bfdec_pow_ui(bfdec_t *r, const bfdec_t *a, limb_t b) |
| { |
| int ret, n_bits, i; |
| |
| assert(r != a); |
| if (b == 0) |
| return bfdec_set_ui(r, 1); |
| ret = bfdec_set(r, a); |
| n_bits = LIMB_BITS - clz(b); |
| for(i = n_bits - 2; i >= 0; i--) { |
| ret |= bfdec_mul(r, r, r, BF_PREC_INF, BF_RNDZ); |
| if ((b >> i) & 1) |
| ret |= bfdec_mul(r, r, a, BF_PREC_INF, BF_RNDZ); |
| } |
| return ret; |
| } |
| |
| char *bfdec_ftoa(size_t *plen, const bfdec_t *a, limb_t prec, bf_flags_t flags) |
| { |
| return bf_ftoa_internal(plen, (const bf_t *)a, 10, prec, flags, TRUE); |
| } |
| |
| int bfdec_atof(bfdec_t *r, const char *str, const char **pnext, |
| limb_t prec, bf_flags_t flags) |
| { |
| slimb_t dummy_exp; |
| return bf_atof_internal((bf_t *)r, &dummy_exp, str, pnext, 10, prec, |
| flags, TRUE); |
| } |
| |
| #endif /* USE_BF_DEC */ |
| |
| #ifdef USE_FFT_MUL |
| /***************************************************************/ |
| /* Integer multiplication with FFT */ |
| |
| /* or LIMB_BITS at bit position 'pos' in tab */ |
| static inline void put_bits(limb_t *tab, limb_t len, slimb_t pos, limb_t val) |
| { |
| limb_t i; |
| int p; |
| |
| i = pos >> LIMB_LOG2_BITS; |
| p = pos & (LIMB_BITS - 1); |
| if (i < len) |
| tab[i] |= val << p; |
| if (p != 0) { |
| i++; |
| if (i < len) { |
| tab[i] |= val >> (LIMB_BITS - p); |
| } |
| } |
| } |
| |
| #if defined(__AVX2__) |
| |
| typedef double NTTLimb; |
| |
| /* we must have: modulo >= 1 << NTT_MOD_LOG2_MIN */ |
| #define NTT_MOD_LOG2_MIN 50 |
| #define NTT_MOD_LOG2_MAX 51 |
| #define NB_MODS 5 |
| #define NTT_PROOT_2EXP 39 |
| static const int ntt_int_bits[NB_MODS] = { 254, 203, 152, 101, 50, }; |
| |
| static const limb_t ntt_mods[NB_MODS] = { 0x00073a8000000001, 0x0007858000000001, 0x0007a38000000001, 0x0007a68000000001, 0x0007fd8000000001, |
| }; |
| |
| static const limb_t ntt_proot[2][NB_MODS] = { |
| { 0x00056198d44332c8, 0x0002eb5d640aad39, 0x00047e31eaa35fd0, 0x0005271ac118a150, 0x00075e0ce8442bd5, }, |
| { 0x000461169761bcc5, 0x0002dac3cb2da688, 0x0004abc97751e3bf, 0x000656778fc8c485, 0x0000dc6469c269fa, }, |
| }; |
| |
| static const limb_t ntt_mods_cr[NB_MODS * (NB_MODS - 1) / 2] = { |
| 0x00020e4da740da8e, 0x0004c3dc09c09c1d, 0x000063bd097b4271, 0x000799d8f18f18fd, |
| 0x0005384222222264, 0x000572b07c1f07fe, 0x00035cd08888889a, |
| 0x00066015555557e3, 0x000725960b60b623, |
| 0x0002fc1fa1d6ce12, |
| }; |
| |
| #else |
| |
| typedef limb_t NTTLimb; |
| |
| #if LIMB_BITS == 64 |
| |
| #define NTT_MOD_LOG2_MIN 61 |
| #define NTT_MOD_LOG2_MAX 62 |
| #define NB_MODS 5 |
| #define NTT_PROOT_2EXP 51 |
| static const int ntt_int_bits[NB_MODS] = { 307, 246, 185, 123, 61, }; |
| |
| static const limb_t ntt_mods[NB_MODS] = { 0x28d8000000000001, 0x2a88000000000001, 0x2ed8000000000001, 0x3508000000000001, 0x3aa8000000000001, |
| }; |
| |
| static const limb_t ntt_proot[2][NB_MODS] = { |
| { 0x1b8ea61034a2bea7, 0x21a9762de58206fb, 0x02ca782f0756a8ea, 0x278384537a3e50a1, 0x106e13fee74ce0ab, }, |
| { 0x233513af133e13b8, 0x1d13140d1c6f75f1, 0x12cde57f97e3eeda, 0x0d6149e23cbe654f, 0x36cd204f522a1379, }, |
| }; |
| |
| static const limb_t ntt_mods_cr[NB_MODS * (NB_MODS - 1) / 2] = { |
| 0x08a9ed097b425eea, 0x18a44aaaaaaaaab3, 0x2493f57f57f57f5d, 0x126b8d0649a7f8d4, |
| 0x09d80ed7303b5ccc, 0x25b8bcf3cf3cf3d5, 0x2ce6ce63398ce638, |
| 0x0e31fad40a57eb59, 0x02a3529fd4a7f52f, |
| 0x3a5493e93e93e94a, |
| }; |
| |
| #elif LIMB_BITS == 32 |
| |
| /* we must have: modulo >= 1 << NTT_MOD_LOG2_MIN */ |
| #define NTT_MOD_LOG2_MIN 29 |
| #define NTT_MOD_LOG2_MAX 30 |
| #define NB_MODS 5 |
| #define NTT_PROOT_2EXP 20 |
| static const int ntt_int_bits[NB_MODS] = { 148, 119, 89, 59, 29, }; |
| |
| static const limb_t ntt_mods[NB_MODS] = { 0x0000000032b00001, 0x0000000033700001, 0x0000000036d00001, 0x0000000037300001, 0x000000003e500001, |
| }; |
| |
| static const limb_t ntt_proot[2][NB_MODS] = { |
| { 0x0000000032525f31, 0x0000000005eb3b37, 0x00000000246eda9f, 0x0000000035f25901, 0x00000000022f5768, }, |
| { 0x00000000051eba1a, 0x00000000107be10e, 0x000000001cd574e0, 0x00000000053806e6, 0x000000002cd6bf98, }, |
| }; |
| |
| static const limb_t ntt_mods_cr[NB_MODS * (NB_MODS - 1) / 2] = { |
| 0x000000000449559a, 0x000000001eba6ca9, 0x000000002ec18e46, 0x000000000860160b, |
| 0x000000000d321307, 0x000000000bf51120, 0x000000000f662938, |
| 0x000000000932ab3e, 0x000000002f40eef8, |
| 0x000000002e760905, |
| }; |
| |
| #endif /* LIMB_BITS */ |
| |
| #endif /* !AVX2 */ |
| |
| #if defined(__AVX2__) |
| #define NTT_TRIG_K_MAX 18 |
| #else |
| #define NTT_TRIG_K_MAX 19 |
| #endif |
| |
| typedef struct BFNTTState { |
| bf_context_t *ctx; |
| |
| /* used for mul_mod_fast() */ |
| limb_t ntt_mods_div[NB_MODS]; |
| |
| limb_t ntt_proot_pow[NB_MODS][2][NTT_PROOT_2EXP + 1]; |
| limb_t ntt_proot_pow_inv[NB_MODS][2][NTT_PROOT_2EXP + 1]; |
| NTTLimb *ntt_trig[NB_MODS][2][NTT_TRIG_K_MAX + 1]; |
| /* 1/2^n mod m */ |
| limb_t ntt_len_inv[NB_MODS][NTT_PROOT_2EXP + 1][2]; |
| #if defined(__AVX2__) |
| __m256d ntt_mods_cr_vec[NB_MODS * (NB_MODS - 1) / 2]; |
| __m256d ntt_mods_vec[NB_MODS]; |
| __m256d ntt_mods_inv_vec[NB_MODS]; |
| #else |
| limb_t ntt_mods_cr_inv[NB_MODS * (NB_MODS - 1) / 2]; |
| #endif |
| } BFNTTState; |
| |
| static NTTLimb *get_trig(BFNTTState *s, int k, int inverse, int m_idx); |
| |
| /* add modulo with up to (LIMB_BITS-1) bit modulo */ |
| static inline limb_t add_mod(limb_t a, limb_t b, limb_t m) |
| { |
| limb_t r; |
| r = a + b; |
| if (r >= m) |
| r -= m; |
| return r; |
| } |
| |
| /* sub modulo with up to LIMB_BITS bit modulo */ |
| static inline limb_t sub_mod(limb_t a, limb_t b, limb_t m) |
| { |
| limb_t r; |
| r = a - b; |
| if (r > a) |
| r += m; |
| return r; |
| } |
| |
| /* return (r0+r1*B) mod m |
| precondition: 0 <= r0+r1*B < 2^(64+NTT_MOD_LOG2_MIN) |
| */ |
| static inline limb_t mod_fast(dlimb_t r, |
| limb_t m, limb_t m_inv) |
| { |
| limb_t a1, q, t0, r1, r0; |
| |
| a1 = r >> NTT_MOD_LOG2_MIN; |
| |
| q = ((dlimb_t)a1 * m_inv) >> LIMB_BITS; |
| r = r - (dlimb_t)q * m - m * 2; |
| r1 = r >> LIMB_BITS; |
| t0 = (slimb_t)r1 >> 1; |
| r += m & t0; |
| r0 = r; |
| r1 = r >> LIMB_BITS; |
| r0 += m & r1; |
| return r0; |
| } |
| |
| /* faster version using precomputed modulo inverse. |
| precondition: 0 <= a * b < 2^(64+NTT_MOD_LOG2_MIN) */ |
| static inline limb_t mul_mod_fast(limb_t a, limb_t b, |
| limb_t m, limb_t m_inv) |
| { |
| dlimb_t r; |
| r = (dlimb_t)a * (dlimb_t)b; |
| return mod_fast(r, m, m_inv); |
| } |
| |
| static inline limb_t init_mul_mod_fast(limb_t m) |
| { |
| dlimb_t t; |
| assert(m < (limb_t)1 << NTT_MOD_LOG2_MAX); |
| assert(m >= (limb_t)1 << NTT_MOD_LOG2_MIN); |
| t = (dlimb_t)1 << (LIMB_BITS + NTT_MOD_LOG2_MIN); |
| return t / m; |
| } |
| |
| /* Faster version used when the multiplier is constant. 0 <= a < 2^64, |
| 0 <= b < m. */ |
| static inline limb_t mul_mod_fast2(limb_t a, limb_t b, |
| limb_t m, limb_t b_inv) |
| { |
| limb_t r, q; |
| |
| q = ((dlimb_t)a * (dlimb_t)b_inv) >> LIMB_BITS; |
| r = a * b - q * m; |
| if (r >= m) |
| r -= m; |
| return r; |
| } |
| |
| /* Faster version used when the multiplier is constant. 0 <= a < 2^64, |
| 0 <= b < m. Let r = a * b mod m. The return value is 'r' or 'r + |
| m'. */ |
| static inline limb_t mul_mod_fast3(limb_t a, limb_t b, |
| limb_t m, limb_t b_inv) |
| { |
| limb_t r, q; |
| |
| q = ((dlimb_t)a * (dlimb_t)b_inv) >> LIMB_BITS; |
| r = a * b - q * m; |
| return r; |
| } |
| |
| static inline limb_t init_mul_mod_fast2(limb_t b, limb_t m) |
| { |
| return ((dlimb_t)b << LIMB_BITS) / m; |
| } |
| |
| #ifdef __AVX2__ |
| |
| static inline limb_t ntt_limb_to_int(NTTLimb a, limb_t m) |
| { |
| slimb_t v; |
| v = a; |
| if (v < 0) |
| v += m; |
| if (v >= m) |
| v -= m; |
| return v; |
| } |
| |
| static inline NTTLimb int_to_ntt_limb(limb_t a, limb_t m) |
| { |
| return (slimb_t)a; |
| } |
| |
| static inline NTTLimb int_to_ntt_limb2(limb_t a, limb_t m) |
| { |
| if (a >= (m / 2)) |
| a -= m; |
| return (slimb_t)a; |
| } |
| |
| /* return r + m if r < 0 otherwise r. */ |
| static inline __m256d ntt_mod1(__m256d r, __m256d m) |
| { |
| return _mm256_blendv_pd(r, r + m, r); |
| } |
| |
| /* input: abs(r) < 2 * m. Output: abs(r) < m */ |
| static inline __m256d ntt_mod(__m256d r, __m256d mf, __m256d m2f) |
| { |
| return _mm256_blendv_pd(r, r + m2f, r) - mf; |
| } |
| |
| /* input: abs(a*b) < 2 * m^2, output: abs(r) < m */ |
| static inline __m256d ntt_mul_mod(__m256d a, __m256d b, __m256d mf, |
| __m256d m_inv) |
| { |
| __m256d r, q, ab1, ab0, qm0, qm1; |
| ab1 = a * b; |
| q = _mm256_round_pd(ab1 * m_inv, 0); /* round to nearest */ |
| qm1 = q * mf; |
| qm0 = _mm256_fmsub_pd(q, mf, qm1); /* low part */ |
| ab0 = _mm256_fmsub_pd(a, b, ab1); /* low part */ |
| r = (ab1 - qm1) + (ab0 - qm0); |
| return r; |
| } |
| |
| static void *bf_aligned_malloc(bf_context_t *s, size_t size, size_t align) |
| { |
| void *ptr; |
| void **ptr1; |
| ptr = bf_malloc(s, size + sizeof(void *) + align - 1); |
| if (!ptr) |
| return NULL; |
| ptr1 = (void **)(((uintptr_t)ptr + sizeof(void *) + align - 1) & |
| ~(align - 1)); |
| ptr1[-1] = ptr; |
| return ptr1; |
| } |
| |
| static void bf_aligned_free(bf_context_t *s, void *ptr) |
| { |
| if (!ptr) |
| return; |
| bf_free(s, ((void **)ptr)[-1]); |
| } |
| |
| static void *ntt_malloc(BFNTTState *s, size_t size) |
| { |
| return bf_aligned_malloc(s->ctx, size, 64); |
| } |
| |
| static void ntt_free(BFNTTState *s, void *ptr) |
| { |
| bf_aligned_free(s->ctx, ptr); |
| } |
| |
| static no_inline int ntt_fft(BFNTTState *s, |
| NTTLimb *out_buf, NTTLimb *in_buf, |
| NTTLimb *tmp_buf, int fft_len_log2, |
| int inverse, int m_idx) |
| { |
| limb_t nb_blocks, fft_per_block, p, k, n, stride_in, i, j; |
| NTTLimb *tab_in, *tab_out, *tmp, *trig; |
| __m256d m_inv, mf, m2f, c, a0, a1, b0, b1; |
| limb_t m; |
| int l; |
| |
| m = ntt_mods[m_idx]; |
| |
| m_inv = _mm256_set1_pd(1.0 / (double)m); |
| mf = _mm256_set1_pd(m); |
| m2f = _mm256_set1_pd(m * 2); |
| |
| n = (limb_t)1 << fft_len_log2; |
| assert(n >= 8); |
| stride_in = n / 2; |
| |
| tab_in = in_buf; |
| tab_out = tmp_buf; |
| trig = get_trig(s, fft_len_log2, inverse, m_idx); |
| if (!trig) |
| return -1; |
| p = 0; |
| for(k = 0; k < stride_in; k += 4) { |
| a0 = _mm256_load_pd(&tab_in[k]); |
| a1 = _mm256_load_pd(&tab_in[k + stride_in]); |
| c = _mm256_load_pd(trig); |
| trig += 4; |
| b0 = ntt_mod(a0 + a1, mf, m2f); |
| b1 = ntt_mul_mod(a0 - a1, c, mf, m_inv); |
| a0 = _mm256_permute2f128_pd(b0, b1, 0x20); |
| a1 = _mm256_permute2f128_pd(b0, b1, 0x31); |
| a0 = _mm256_permute4x64_pd(a0, 0xd8); |
| a1 = _mm256_permute4x64_pd(a1, 0xd8); |
| _mm256_store_pd(&tab_out[p], a0); |
| _mm256_store_pd(&tab_out[p + 4], a1); |
| p += 2 * 4; |
| } |
| tmp = tab_in; |
| tab_in = tab_out; |
| tab_out = tmp; |
| |
| trig = get_trig(s, fft_len_log2 - 1, inverse, m_idx); |
| if (!trig) |
| return -1; |
| p = 0; |
| for(k = 0; k < stride_in; k += 4) { |
| a0 = _mm256_load_pd(&tab_in[k]); |
| a1 = _mm256_load_pd(&tab_in[k + stride_in]); |
| c = _mm256_setr_pd(trig[0], trig[0], trig[1], trig[1]); |
| trig += 2; |
| b0 = ntt_mod(a0 + a1, mf, m2f); |
| b1 = ntt_mul_mod(a0 - a1, c, mf, m_inv); |
| a0 = _mm256_permute2f128_pd(b0, b1, 0x20); |
| a1 = _mm256_permute2f128_pd(b0, b1, 0x31); |
| _mm256_store_pd(&tab_out[p], a0); |
| _mm256_store_pd(&tab_out[p + 4], a1); |
| p += 2 * 4; |
| } |
| tmp = tab_in; |
| tab_in = tab_out; |
| tab_out = tmp; |
| |
| nb_blocks = n / 4; |
| fft_per_block = 4; |
| |
| l = fft_len_log2 - 2; |
| while (nb_blocks != 2) { |
| nb_blocks >>= 1; |
| p = 0; |
| k = 0; |
| trig = get_trig(s, l, inverse, m_idx); |
| if (!trig) |
| return -1; |
| for(i = 0; i < nb_blocks; i++) { |
| c = _mm256_set1_pd(trig[0]); |
| trig++; |
| for(j = 0; j < fft_per_block; j += 4) { |
| a0 = _mm256_load_pd(&tab_in[k + j]); |
| a1 = _mm256_load_pd(&tab_in[k + j + stride_in]); |
| b0 = ntt_mod(a0 + a1, mf, m2f); |
| b1 = ntt_mul_mod(a0 - a1, c, mf, m_inv); |
| _mm256_store_pd(&tab_out[p + j], b0); |
| _mm256_store_pd(&tab_out[p + j + fft_per_block], b1); |
| } |
| k += fft_per_block; |
| p += 2 * fft_per_block; |
| } |
| fft_per_block <<= 1; |
| l--; |
| tmp = tab_in; |
| tab_in = tab_out; |
| tab_out = tmp; |
| } |
| |
| tab_out = out_buf; |
| for(k = 0; k < stride_in; k += 4) { |
| a0 = _mm256_load_pd(&tab_in[k]); |
| a1 = _mm256_load_pd(&tab_in[k + stride_in]); |
| b0 = ntt_mod(a0 + a1, mf, m2f); |
| b1 = ntt_mod(a0 - a1, mf, m2f); |
| _mm256_store_pd(&tab_out[k], b0); |
| _mm256_store_pd(&tab_out[k + stride_in], b1); |
| } |
| return 0; |
| } |
| |
| static void ntt_vec_mul(BFNTTState *s, |
| NTTLimb *tab1, NTTLimb *tab2, limb_t fft_len_log2, |
| int k_tot, int m_idx) |
| { |
| limb_t i, c_inv, n, m; |
| __m256d m_inv, mf, a, b, c; |
| |
| m = ntt_mods[m_idx]; |
| c_inv = s->ntt_len_inv[m_idx][k_tot][0]; |
| m_inv = _mm256_set1_pd(1.0 / (double)m); |
| mf = _mm256_set1_pd(m); |
| c = _mm256_set1_pd(int_to_ntt_limb(c_inv, m)); |
| n = (limb_t)1 << fft_len_log2; |
| for(i = 0; i < n; i += 4) { |
| a = _mm256_load_pd(&tab1[i]); |
| b = _mm256_load_pd(&tab2[i]); |
| a = ntt_mul_mod(a, b, mf, m_inv); |
| a = ntt_mul_mod(a, c, mf, m_inv); |
| _mm256_store_pd(&tab1[i], a); |
| } |
| } |
| |
| static no_inline void mul_trig(NTTLimb *buf, |
| limb_t n, limb_t c1, limb_t m, limb_t m_inv1) |
| { |
| limb_t i, c2, c3, c4; |
| __m256d c, c_mul, a0, mf, m_inv; |
| assert(n >= 2); |
| |
| mf = _mm256_set1_pd(m); |
| m_inv = _mm256_set1_pd(1.0 / (double)m); |
| |
| c2 = mul_mod_fast(c1, c1, m, m_inv1); |
| c3 = mul_mod_fast(c2, c1, m, m_inv1); |
| c4 = mul_mod_fast(c2, c2, m, m_inv1); |
| c = _mm256_setr_pd(1, int_to_ntt_limb(c1, m), |
| int_to_ntt_limb(c2, m), int_to_ntt_limb(c3, m)); |
| c_mul = _mm256_set1_pd(int_to_ntt_limb(c4, m)); |
| for(i = 0; i < n; i += 4) { |
| a0 = _mm256_load_pd(&buf[i]); |
| a0 = ntt_mul_mod(a0, c, mf, m_inv); |
| _mm256_store_pd(&buf[i], a0); |
| c = ntt_mul_mod(c, c_mul, mf, m_inv); |
| } |
| } |
| |
| #else |
| |
| static void *ntt_malloc(BFNTTState *s, size_t size) |
| { |
| return bf_malloc(s->ctx, size); |
| } |
| |
| static void ntt_free(BFNTTState *s, void *ptr) |
| { |
| bf_free(s->ctx, ptr); |
| } |
| |
| static inline limb_t ntt_limb_to_int(NTTLimb a, limb_t m) |
| { |
| if (a >= m) |
| a -= m; |
| return a; |
| } |
| |
| static inline NTTLimb int_to_ntt_limb(slimb_t a, limb_t m) |
| { |
| return a; |
| } |
| |
| static no_inline int ntt_fft(BFNTTState *s, NTTLimb *out_buf, NTTLimb *in_buf, |
| NTTLimb *tmp_buf, int fft_len_log2, |
| int inverse, int m_idx) |
| { |
| limb_t nb_blocks, fft_per_block, p, k, n, stride_in, i, j, m, m2; |
| NTTLimb *tab_in, *tab_out, *tmp, a0, a1, b0, b1, c, *trig, c_inv; |
| int l; |
| |
| m = ntt_mods[m_idx]; |
| m2 = 2 * m; |
| n = (limb_t)1 << fft_len_log2; |
| nb_blocks = n; |
| fft_per_block = 1; |
| stride_in = n / 2; |
| tab_in = in_buf; |
| tab_out = tmp_buf; |
| l = fft_len_log2; |
| while (nb_blocks != 2) { |
| nb_blocks >>= 1; |
| p = 0; |
| k = 0; |
| trig = get_trig(s, l, inverse, m_idx); |
| if (!trig) |
| return -1; |
| for(i = 0; i < nb_blocks; i++) { |
| c = trig[0]; |
| c_inv = trig[1]; |
| trig += 2; |
| for(j = 0; j < fft_per_block; j++) { |
| a0 = tab_in[k + j]; |
| a1 = tab_in[k + j + stride_in]; |
| b0 = add_mod(a0, a1, m2); |
| b1 = a0 - a1 + m2; |
| b1 = mul_mod_fast3(b1, c, m, c_inv); |
| tab_out[p + j] = b0; |
| tab_out[p + j + fft_per_block] = b1; |
| } |
| k += fft_per_block; |
| p += 2 * fft_per_block; |
| } |
| fft_per_block <<= 1; |
| l--; |
| tmp = tab_in; |
| tab_in = tab_out; |
| tab_out = tmp; |
| } |
| /* no twiddle in last step */ |
| tab_out = out_buf; |
| for(k = 0; k < stride_in; k++) { |
| a0 = tab_in[k]; |
| a1 = tab_in[k + stride_in]; |
| b0 = add_mod(a0, a1, m2); |
| b1 = sub_mod(a0, a1, m2); |
| tab_out[k] = b0; |
| tab_out[k + stride_in] = b1; |
| } |
| return 0; |
| } |
| |
| static void ntt_vec_mul(BFNTTState *s, |
| NTTLimb *tab1, NTTLimb *tab2, int fft_len_log2, |
| int k_tot, int m_idx) |
| { |
| limb_t i, norm, norm_inv, a, n, m, m_inv; |
| |
| m = ntt_mods[m_idx]; |
| m_inv = s->ntt_mods_div[m_idx]; |
| norm = s->ntt_len_inv[m_idx][k_tot][0]; |
| norm_inv = s->ntt_len_inv[m_idx][k_tot][1]; |
| n = (limb_t)1 << fft_len_log2; |
| for(i = 0; i < n; i++) { |
| a = tab1[i]; |
| /* need to reduce the range so that the product is < |
| 2^(LIMB_BITS+NTT_MOD_LOG2_MIN) */ |
| if (a >= m) |
| a -= m; |
| a = mul_mod_fast(a, tab2[i], m, m_inv); |
| a = mul_mod_fast3(a, norm, m, norm_inv); |
| tab1[i] = a; |
| } |
| } |
| |
| static no_inline void mul_trig(NTTLimb *buf, |
| limb_t n, limb_t c_mul, limb_t m, limb_t m_inv) |
| { |
| limb_t i, c0, c_mul_inv; |
| |
| c0 = 1; |
| c_mul_inv = init_mul_mod_fast2(c_mul, m); |
| for(i = 0; i < n; i++) { |
| buf[i] = mul_mod_fast(buf[i], c0, m, m_inv); |
| c0 = mul_mod_fast2(c0, c_mul, m, c_mul_inv); |
| } |
| } |
| |
| #endif /* !AVX2 */ |
| |
| static no_inline NTTLimb *get_trig(BFNTTState *s, |
| int k, int inverse, int m_idx) |
| { |
| NTTLimb *tab; |
| limb_t i, n2, c, c_mul, m, c_mul_inv; |
| |
| if (k > NTT_TRIG_K_MAX) |
| return NULL; |
| |
| tab = s->ntt_trig[m_idx][inverse][k]; |
| if (tab) |
| return tab; |
| n2 = (limb_t)1 << (k - 1); |
| m = ntt_mods[m_idx]; |
| #ifdef __AVX2__ |
| tab = ntt_malloc(s, sizeof(NTTLimb) * n2); |
| #else |
| tab = ntt_malloc(s, sizeof(NTTLimb) * n2 * 2); |
| #endif |
| if (!tab) |
| return NULL; |
| c = 1; |
| c_mul = s->ntt_proot_pow[m_idx][inverse][k]; |
| c_mul_inv = s->ntt_proot_pow_inv[m_idx][inverse][k]; |
| for(i = 0; i < n2; i++) { |
| #ifdef __AVX2__ |
| tab[i] = int_to_ntt_limb2(c, m); |
| #else |
| tab[2 * i] = int_to_ntt_limb(c, m); |
| tab[2 * i + 1] = init_mul_mod_fast2(c, m); |
| #endif |
| c = mul_mod_fast2(c, c_mul, m, c_mul_inv); |
| } |
| s->ntt_trig[m_idx][inverse][k] = tab; |
| return tab; |
| } |
| |
| void fft_clear_cache(bf_context_t *s1) |
| { |
| int m_idx, inverse, k; |
| BFNTTState *s = s1->ntt_state; |
| if (s) { |
| for(m_idx = 0; m_idx < NB_MODS; m_idx++) { |
| for(inverse = 0; inverse < 2; inverse++) { |
| for(k = 0; k < NTT_TRIG_K_MAX + 1; k++) { |
| if (s->ntt_trig[m_idx][inverse][k]) { |
| ntt_free(s, s->ntt_trig[m_idx][inverse][k]); |
| s->ntt_trig[m_idx][inverse][k] = NULL; |
| } |
| } |
| } |
| } |
| #if defined(__AVX2__) |
| bf_aligned_free(s1, s); |
| #else |
| bf_free(s1, s); |
| #endif |
| s1->ntt_state = NULL; |
| } |
| } |
| |
| #define STRIP_LEN 16 |
| |
| /* dst = buf1, src = buf2 */ |
| static int ntt_fft_partial(BFNTTState *s, NTTLimb *buf1, |
| int k1, int k2, limb_t n1, limb_t n2, int inverse, |
| limb_t m_idx) |
| { |
| limb_t i, j, c_mul, c0, m, m_inv, strip_len, l; |
| NTTLimb *buf2, *buf3; |
| |
| buf2 = NULL; |
| buf3 = ntt_malloc(s, sizeof(NTTLimb) * n1); |
| if (!buf3) |
| goto fail; |
| if (k2 == 0) { |
| if (ntt_fft(s, buf1, buf1, buf3, k1, inverse, m_idx)) |
| goto fail; |
| } else { |
| strip_len = STRIP_LEN; |
| buf2 = ntt_malloc(s, sizeof(NTTLimb) * n1 * strip_len); |
| if (!buf2) |
| goto fail; |
| m = ntt_mods[m_idx]; |
| m_inv = s->ntt_mods_div[m_idx]; |
| c0 = s->ntt_proot_pow[m_idx][inverse][k1 + k2]; |
| c_mul = 1; |
| assert((n2 % strip_len) == 0); |
| for(j = 0; j < n2; j += strip_len) { |
| for(i = 0; i < n1; i++) { |
| for(l = 0; l < strip_len; l++) { |
| buf2[i + l * n1] = buf1[i * n2 + (j + l)]; |
| } |
| } |
| for(l = 0; l < strip_len; l++) { |
| if (inverse) |
| mul_trig(buf2 + l * n1, n1, c_mul, m, m_inv); |
| if (ntt_fft(s, buf2 + l * n1, buf2 + l * n1, buf3, k1, inverse, m_idx)) |
| goto fail; |
| if (!inverse) |
| mul_trig(buf2 + l * n1, n1, c_mul, m, m_inv); |
| c_mul = mul_mod_fast(c_mul, c0, m, m_inv); |
| } |
| |
| for(i = 0; i < n1; i++) { |
| for(l = 0; l < strip_len; l++) { |
| buf1[i * n2 + (j + l)] = buf2[i + l *n1]; |
| } |
| } |
| } |
| ntt_free(s, buf2); |
| } |
| ntt_free(s, buf3); |
| return 0; |
| fail: |
| ntt_free(s, buf2); |
| ntt_free(s, buf3); |
| return -1; |
| } |
| |
| |
| /* dst = buf1, src = buf2, tmp = buf3 */ |
| static int ntt_conv(BFNTTState *s, NTTLimb *buf1, NTTLimb *buf2, |
| int k, int k_tot, limb_t m_idx) |
| { |
| limb_t n1, n2, i; |
| int k1, k2; |
| |
| if (k <= NTT_TRIG_K_MAX) { |
| k1 = k; |
| } else { |
| /* recursive split of the FFT */ |
| k1 = bf_min(k / 2, NTT_TRIG_K_MAX); |
| } |
| k2 = k - k1; |
| n1 = (limb_t)1 << k1; |
| n2 = (limb_t)1 << k2; |
| |
| if (ntt_fft_partial(s, buf1, k1, k2, n1, n2, 0, m_idx)) |
| return -1; |
| if (ntt_fft_partial(s, buf2, k1, k2, n1, n2, 0, m_idx)) |
| return -1; |
| if (k2 == 0) { |
| ntt_vec_mul(s, buf1, buf2, k, k_tot, m_idx); |
| } else { |
| for(i = 0; i < n1; i++) { |
| ntt_conv(s, buf1 + i * n2, buf2 + i * n2, k2, k_tot, m_idx); |
| } |
| } |
| if (ntt_fft_partial(s, buf1, k1, k2, n1, n2, 1, m_idx)) |
| return -1; |
| return 0; |
| } |
| |
| |
| static no_inline void limb_to_ntt(BFNTTState *s, |
| NTTLimb *tabr, limb_t fft_len, |
| const limb_t *taba, limb_t a_len, int dpl, |
| int first_m_idx, int nb_mods) |
| { |
| slimb_t i, n; |
| dlimb_t a, b; |
| int j, shift; |
| limb_t base_mask1, a0, a1, a2, r, m, m_inv; |
| |
| #if 0 |
| for(i = 0; i < a_len; i++) { |
| printf("%" PRId64 ": " FMT_LIMB "\n", |
| (int64_t)i, taba[i]); |
| } |
| #endif |
| memset(tabr, 0, sizeof(NTTLimb) * fft_len * nb_mods); |
| shift = dpl & (LIMB_BITS - 1); |
| if (shift == 0) |
| base_mask1 = -1; |
| else |
| base_mask1 = ((limb_t)1 << shift) - 1; |
| n = bf_min(fft_len, (a_len * LIMB_BITS + dpl - 1) / dpl); |
| for(i = 0; i < n; i++) { |
| a0 = get_bits(taba, a_len, i * dpl); |
| if (dpl <= LIMB_BITS) { |
| a0 &= base_mask1; |
| a = a0; |
| } else { |
| a1 = get_bits(taba, a_len, i * dpl + LIMB_BITS); |
| if (dpl <= (LIMB_BITS + NTT_MOD_LOG2_MIN)) { |
| a = a0 | ((dlimb_t)(a1 & base_mask1) << LIMB_BITS); |
| } else { |
| if (dpl > 2 * LIMB_BITS) { |
| a2 = get_bits(taba, a_len, i * dpl + LIMB_BITS * 2) & |
| base_mask1; |
| } else { |
| a1 &= base_mask1; |
| a2 = 0; |
| } |
| // printf("a=0x%016lx%016lx%016lx\n", a2, a1, a0); |
| a = (a0 >> (LIMB_BITS - NTT_MOD_LOG2_MAX + NTT_MOD_LOG2_MIN)) | |
| ((dlimb_t)a1 << (NTT_MOD_LOG2_MAX - NTT_MOD_LOG2_MIN)) | |
| ((dlimb_t)a2 << (LIMB_BITS + NTT_MOD_LOG2_MAX - NTT_MOD_LOG2_MIN)); |
| a0 &= ((limb_t)1 << (LIMB_BITS - NTT_MOD_LOG2_MAX + NTT_MOD_LOG2_MIN)) - 1; |
| } |
| } |
| for(j = 0; j < nb_mods; j++) { |
| m = ntt_mods[first_m_idx + j]; |
| m_inv = s->ntt_mods_div[first_m_idx + j]; |
| r = mod_fast(a, m, m_inv); |
| if (dpl > (LIMB_BITS + NTT_MOD_LOG2_MIN)) { |
| b = ((dlimb_t)r << (LIMB_BITS - NTT_MOD_LOG2_MAX + NTT_MOD_LOG2_MIN)) | a0; |
| r = mod_fast(b, m, m_inv); |
| } |
| tabr[i + j * fft_len] = int_to_ntt_limb(r, m); |
| } |
| } |
| } |
| |
| #if defined(__AVX2__) |
| |
| #define VEC_LEN 4 |
| |
| typedef union { |
| __m256d v; |
| double d[4]; |
| } VecUnion; |
| |
| static no_inline void ntt_to_limb(BFNTTState *s, limb_t *tabr, limb_t r_len, |
| const NTTLimb *buf, int fft_len_log2, int dpl, |
| int nb_mods) |
| { |
| const limb_t *mods = ntt_mods + NB_MODS - nb_mods; |
| const __m256d *mods_cr_vec, *mf, *m_inv; |
| VecUnion y[NB_MODS]; |
| limb_t u[NB_MODS], carry[NB_MODS], fft_len, base_mask1, r; |
| slimb_t i, len, pos; |
| int j, k, l, shift, n_limb1, p; |
| dlimb_t t; |
| |
| j = NB_MODS * (NB_MODS - 1) / 2 - nb_mods * (nb_mods - 1) / 2; |
| mods_cr_vec = s->ntt_mods_cr_vec + j; |
| mf = s->ntt_mods_vec + NB_MODS - nb_mods; |
| m_inv = s->ntt_mods_inv_vec + NB_MODS - nb_mods; |
| |
| shift = dpl & (LIMB_BITS - 1); |
| if (shift == 0) |
| base_mask1 = -1; |
| else |
| base_mask1 = ((limb_t)1 << shift) - 1; |
| n_limb1 = ((unsigned)dpl - 1) / LIMB_BITS; |
| for(j = 0; j < NB_MODS; j++) |
| carry[j] = 0; |
| for(j = 0; j < NB_MODS; j++) |
| u[j] = 0; /* avoid warnings */ |
| memset(tabr, 0, sizeof(limb_t) * r_len); |
| fft_len = (limb_t)1 << fft_len_log2; |
| len = bf_min(fft_len, (r_len * LIMB_BITS + dpl - 1) / dpl); |
| len = (len + VEC_LEN - 1) & ~(VEC_LEN - 1); |
| i = 0; |
| while (i < len) { |
| for(j = 0; j < nb_mods; j++) |
| y[j].v = *(__m256d *)&buf[i + fft_len * j]; |
| |
| /* Chinese remainder to get mixed radix representation */ |
| l = 0; |
| for(j = 0; j < nb_mods - 1; j++) { |
| y[j].v = ntt_mod1(y[j].v, mf[j]); |
| for(k = j + 1; k < nb_mods; k++) { |
| y[k].v = ntt_mul_mod(y[k].v - y[j].v, |
| mods_cr_vec[l], mf[k], m_inv[k]); |
| l++; |
| } |
| } |
| y[j].v = ntt_mod1(y[j].v, mf[j]); |
| |
| for(p = 0; p < VEC_LEN; p++) { |
| /* back to normal representation */ |
| u[0] = (int64_t)y[nb_mods - 1].d[p]; |
| l = 1; |
| for(j = nb_mods - 2; j >= 1; j--) { |
| r = (int64_t)y[j].d[p]; |
| for(k = 0; k < l; k++) { |
| t = (dlimb_t)u[k] * mods[j] + r; |
| r = t >> LIMB_BITS; |
| u[k] = t; |
| } |
| u[l] = r; |
| l++; |
| } |
| /* XXX: for nb_mods = 5, l should be 4 */ |
| |
| /* last step adds the carry */ |
| r = (int64_t)y[0].d[p]; |
| for(k = 0; k < l; k++) { |
| t = (dlimb_t)u[k] * mods[j] + r + carry[k]; |
| r = t >> LIMB_BITS; |
| u[k] = t; |
| } |
| u[l] = r + carry[l]; |
| |
| #if 0 |
| printf("%" PRId64 ": ", i); |
| for(j = nb_mods - 1; j >= 0; j--) { |
| printf(" %019" PRIu64, u[j]); |
| } |
| printf("\n"); |
| #endif |
| |
| /* write the digits */ |
| pos = i * dpl; |
| for(j = 0; j < n_limb1; j++) { |
| put_bits(tabr, r_len, pos, u[j]); |
| pos += LIMB_BITS; |
| } |
| put_bits(tabr, r_len, pos, u[n_limb1] & base_mask1); |
| /* shift by dpl digits and set the carry */ |
| if (shift == 0) { |
| for(j = n_limb1 + 1; j < nb_mods; j++) |
| carry[j - (n_limb1 + 1)] = u[j]; |
| } else { |
| for(j = n_limb1; j < nb_mods - 1; j++) { |
| carry[j - n_limb1] = (u[j] >> shift) | |
| (u[j + 1] << (LIMB_BITS - shift)); |
| } |
| carry[nb_mods - 1 - n_limb1] = u[nb_mods - 1] >> shift; |
| } |
| i++; |
| } |
| } |
| } |
| #else |
| static no_inline void ntt_to_limb(BFNTTState *s, limb_t *tabr, limb_t r_len, |
| const NTTLimb *buf, int fft_len_log2, int dpl, |
| int nb_mods) |
| { |
| const limb_t *mods = ntt_mods + NB_MODS - nb_mods; |
| const limb_t *mods_cr, *mods_cr_inv; |
| limb_t y[NB_MODS], u[NB_MODS], carry[NB_MODS], fft_len, base_mask1, r; |
| slimb_t i, len, pos; |
| int j, k, l, shift, n_limb1; |
| dlimb_t t; |
| |
| j = NB_MODS * (NB_MODS - 1) / 2 - nb_mods * (nb_mods - 1) / 2; |
| mods_cr = ntt_mods_cr + j; |
| mods_cr_inv = s->ntt_mods_cr_inv + j; |
| |
| shift = dpl & (LIMB_BITS - 1); |
| if (shift == 0) |
| base_mask1 = -1; |
| else |
| base_mask1 = ((limb_t)1 << shift) - 1; |
| n_limb1 = ((unsigned)dpl - 1) / LIMB_BITS; |
| for(j = 0; j < NB_MODS; j++) |
| carry[j] = 0; |
| for(j = 0; j < NB_MODS; j++) |
| u[j] = 0; /* avoid warnings */ |
| memset(tabr, 0, sizeof(limb_t) * r_len); |
| fft_len = (limb_t)1 << fft_len_log2; |
| len = bf_min(fft_len, (r_len * LIMB_BITS + dpl - 1) / dpl); |
| for(i = 0; i < len; i++) { |
| for(j = 0; j < nb_mods; j++) { |
| y[j] = ntt_limb_to_int(buf[i + fft_len * j], mods[j]); |
| } |
| |
| /* Chinese remainder to get mixed radix representation */ |
| l = 0; |
| for(j = 0; j < nb_mods - 1; j++) { |
| for(k = j + 1; k < nb_mods; k++) { |
| limb_t m; |
| m = mods[k]; |
| /* Note: there is no overflow in the sub_mod() because |
| the modulos are sorted by increasing order */ |
| y[k] = mul_mod_fast2(y[k] - y[j] + m, |
| mods_cr[l], m, mods_cr_inv[l]); |
| l++; |
| } |
| } |
| |
| /* back to normal representation */ |
| u[0] = y[nb_mods - 1]; |
| l = 1; |
| for(j = nb_mods - 2; j >= 1; j--) { |
| r = y[j]; |
| for(k = 0; k < l; k++) { |
| t = (dlimb_t)u[k] * mods[j] + r; |
| r = t >> LIMB_BITS; |
| u[k] = t; |
| } |
| u[l] = r; |
| l++; |
| } |
| |
| /* last step adds the carry */ |
| r = y[0]; |
| for(k = 0; k < l; k++) { |
| t = (dlimb_t)u[k] * mods[j] + r + carry[k]; |
| r = t >> LIMB_BITS; |
| u[k] = t; |
| } |
| u[l] = r + carry[l]; |
| |
| #if 0 |
| printf("%" PRId64 ": ", (int64_t)i); |
| for(j = nb_mods - 1; j >= 0; j--) { |
| printf(" " FMT_LIMB, u[j]); |
| } |
| printf("\n"); |
| #endif |
| |
| /* write the digits */ |
| pos = i * dpl; |
| for(j = 0; j < n_limb1; j++) { |
| put_bits(tabr, r_len, pos, u[j]); |
| pos += LIMB_BITS; |
| } |
| put_bits(tabr, r_len, pos, u[n_limb1] & base_mask1); |
| /* shift by dpl digits and set the carry */ |
| if (shift == 0) { |
| for(j = n_limb1 + 1; j < nb_mods; j++) |
| carry[j - (n_limb1 + 1)] = u[j]; |
| } else { |
| for(j = n_limb1; j < nb_mods - 1; j++) { |
| carry[j - n_limb1] = (u[j] >> shift) | |
| (u[j + 1] << (LIMB_BITS - shift)); |
| } |
| carry[nb_mods - 1 - n_limb1] = u[nb_mods - 1] >> shift; |
| } |
| } |
| } |
| #endif |
| |
| static int ntt_static_init(bf_context_t *s1) |
| { |
| BFNTTState *s; |
| int inverse, i, j, k, l; |
| limb_t c, c_inv, c_inv2, m, m_inv; |
| |
| if (s1->ntt_state) |
| return 0; |
| #if defined(__AVX2__) |
| s = bf_aligned_malloc(s1, sizeof(*s), 64); |
| #else |
| s = bf_malloc(s1, sizeof(*s)); |
| #endif |
| if (!s) |
| return -1; |
| memset(s, 0, sizeof(*s)); |
| s1->ntt_state = s; |
| s->ctx = s1; |
| |
| for(j = 0; j < NB_MODS; j++) { |
| m = ntt_mods[j]; |
| m_inv = init_mul_mod_fast(m); |
| s->ntt_mods_div[j] = m_inv; |
| #if defined(__AVX2__) |
| s->ntt_mods_vec[j] = _mm256_set1_pd(m); |
| s->ntt_mods_inv_vec[j] = _mm256_set1_pd(1.0 / (double)m); |
| #endif |
| c_inv2 = (m + 1) / 2; /* 1/2 */ |
| c_inv = 1; |
| for(i = 0; i <= NTT_PROOT_2EXP; i++) { |
| s->ntt_len_inv[j][i][0] = c_inv; |
| s->ntt_len_inv[j][i][1] = init_mul_mod_fast2(c_inv, m); |
| c_inv = mul_mod_fast(c_inv, c_inv2, m, m_inv); |
| } |
| |
| for(inverse = 0; inverse < 2; inverse++) { |
| c = ntt_proot[inverse][j]; |
| for(i = 0; i < NTT_PROOT_2EXP; i++) { |
| s->ntt_proot_pow[j][inverse][NTT_PROOT_2EXP - i] = c; |
| s->ntt_proot_pow_inv[j][inverse][NTT_PROOT_2EXP - i] = |
| init_mul_mod_fast2(c, m); |
| c = mul_mod_fast(c, c, m, m_inv); |
| } |
| } |
| } |
| |
| l = 0; |
| for(j = 0; j < NB_MODS - 1; j++) { |
| for(k = j + 1; k < NB_MODS; k++) { |
| #if defined(__AVX2__) |
| s->ntt_mods_cr_vec[l] = _mm256_set1_pd(int_to_ntt_limb2(ntt_mods_cr[l], |
| ntt_mods[k])); |
| #else |
| s->ntt_mods_cr_inv[l] = init_mul_mod_fast2(ntt_mods_cr[l], |
| ntt_mods[k]); |
| #endif |
| l++; |
| } |
| } |
| return 0; |
| } |
| |
| int bf_get_fft_size(int *pdpl, int *pnb_mods, limb_t len) |
| { |
| int dpl, fft_len_log2, n_bits, nb_mods, dpl_found, fft_len_log2_found; |
| int int_bits, nb_mods_found; |
| limb_t cost, min_cost; |
| |
| min_cost = -1; |
| dpl_found = 0; |
| nb_mods_found = 4; |
| fft_len_log2_found = 0; |
| for(nb_mods = 3; nb_mods <= NB_MODS; nb_mods++) { |
| int_bits = ntt_int_bits[NB_MODS - nb_mods]; |
| dpl = bf_min((int_bits - 4) / 2, |
| 2 * LIMB_BITS + 2 * NTT_MOD_LOG2_MIN - NTT_MOD_LOG2_MAX); |
| for(;;) { |
| fft_len_log2 = ceil_log2((len * LIMB_BITS + dpl - 1) / dpl); |
| if (fft_len_log2 > NTT_PROOT_2EXP) |
| goto next; |
| n_bits = fft_len_log2 + 2 * dpl; |
| if (n_bits <= int_bits) { |
| cost = ((limb_t)(fft_len_log2 + 1) << fft_len_log2) * nb_mods; |
| // printf("n=%d dpl=%d: cost=%" PRId64 "\n", nb_mods, dpl, (int64_t)cost); |
| if (cost < min_cost) { |
| min_cost = cost; |
| dpl_found = dpl; |
| nb_mods_found = nb_mods; |
| fft_len_log2_found = fft_len_log2; |
| } |
| break; |
| } |
| dpl--; |
| if (dpl == 0) |
| break; |
| } |
| next: ; |
| } |
| if (!dpl_found) |
| abort(); |
| /* limit dpl if possible to reduce fixed cost of limb/NTT conversion */ |
| if (dpl_found > (LIMB_BITS + NTT_MOD_LOG2_MIN) && |
| ((limb_t)(LIMB_BITS + NTT_MOD_LOG2_MIN) << fft_len_log2_found) >= |
| len * LIMB_BITS) { |
| dpl_found = LIMB_BITS + NTT_MOD_LOG2_MIN; |
| } |
| *pnb_mods = nb_mods_found; |
| *pdpl = dpl_found; |
| return fft_len_log2_found; |
| } |
| |
| /* return 0 if OK, -1 if memory error */ |
| static no_inline int fft_mul(bf_context_t *s1, |
| bf_t *res, limb_t *a_tab, limb_t a_len, |
| limb_t *b_tab, limb_t b_len, int mul_flags) |
| { |
| BFNTTState *s; |
| int dpl, fft_len_log2, j, nb_mods, reduced_mem; |
| slimb_t len, fft_len; |
| NTTLimb *buf1, *buf2, *ptr; |
| #if defined(USE_MUL_CHECK) |
| limb_t ha, hb, hr, h_ref; |
| #endif |
| |
| if (ntt_static_init(s1)) |
| return -1; |
| s = s1->ntt_state; |
| |
| /* find the optimal number of digits per limb (dpl) */ |
| len = a_len + b_len; |
| fft_len_log2 = bf_get_fft_size(&dpl, &nb_mods, len); |
| fft_len = (uint64_t)1 << fft_len_log2; |
| // printf("len=%" PRId64 " fft_len_log2=%d dpl=%d\n", len, fft_len_log2, dpl); |
| #if defined(USE_MUL_CHECK) |
| ha = mp_mod1(a_tab, a_len, BF_CHKSUM_MOD, 0); |
| hb = mp_mod1(b_tab, b_len, BF_CHKSUM_MOD, 0); |
| #endif |
| if ((mul_flags & (FFT_MUL_R_OVERLAP_A | FFT_MUL_R_OVERLAP_B)) == 0) { |
| if (!(mul_flags & FFT_MUL_R_NORESIZE)) |
| bf_resize(res, 0); |
| } else if (mul_flags & FFT_MUL_R_OVERLAP_B) { |
| limb_t *tmp_tab, tmp_len; |
| /* it is better to free 'b' first */ |
| tmp_tab = a_tab; |
| a_tab = b_tab; |
| b_tab = tmp_tab; |
| tmp_len = a_len; |
| a_len = b_len; |
| b_len = tmp_len; |
| } |
| buf1 = ntt_malloc(s, sizeof(NTTLimb) * fft_len * nb_mods); |
| if (!buf1) |
| return -1; |
| limb_to_ntt(s, buf1, fft_len, a_tab, a_len, dpl, |
| NB_MODS - nb_mods, nb_mods); |
| if ((mul_flags & (FFT_MUL_R_OVERLAP_A | FFT_MUL_R_OVERLAP_B)) == |
| FFT_MUL_R_OVERLAP_A) { |
| if (!(mul_flags & FFT_MUL_R_NORESIZE)) |
| bf_resize(res, 0); |
| } |
| reduced_mem = (fft_len_log2 >= 14); |
| if (!reduced_mem) { |
| buf2 = ntt_malloc(s, sizeof(NTTLimb) * fft_len * nb_mods); |
| if (!buf2) |
| goto fail; |
| limb_to_ntt(s, buf2, fft_len, b_tab, b_len, dpl, |
| NB_MODS - nb_mods, nb_mods); |
| if (!(mul_flags & FFT_MUL_R_NORESIZE)) |
| bf_resize(res, 0); /* in case res == b */ |
| } else { |
| buf2 = ntt_malloc(s, sizeof(NTTLimb) * fft_len); |
| if (!buf2) |
| goto fail; |
| } |
| for(j = 0; j < nb_mods; j++) { |
| if (reduced_mem) { |
| limb_to_ntt(s, buf2, fft_len, b_tab, b_len, dpl, |
| NB_MODS - nb_mods + j, 1); |
| ptr = buf2; |
| } else { |
| ptr = buf2 + fft_len * j; |
| } |
| if (ntt_conv(s, buf1 + fft_len * j, ptr, |
| fft_len_log2, fft_len_log2, j + NB_MODS - nb_mods)) |
| goto fail; |
| } |
| if (!(mul_flags & FFT_MUL_R_NORESIZE)) |
| bf_resize(res, 0); /* in case res == b and reduced mem */ |
| ntt_free(s, buf2); |
| buf2 = NULL; |
| if (!(mul_flags & FFT_MUL_R_NORESIZE)) { |
| if (bf_resize(res, len)) |
| goto fail; |
| } |
| ntt_to_limb(s, res->tab, len, buf1, fft_len_log2, dpl, nb_mods); |
| ntt_free(s, buf1); |
| #if defined(USE_MUL_CHECK) |
| hr = mp_mod1(res->tab, len, BF_CHKSUM_MOD, 0); |
| h_ref = mul_mod(ha, hb, BF_CHKSUM_MOD); |
| if (hr != h_ref) { |
| printf("ntt_mul_error: len=%" PRId_LIMB " fft_len_log2=%d dpl=%d nb_mods=%d\n", |
| len, fft_len_log2, dpl, nb_mods); |
| // printf("ha=0x" FMT_LIMB" hb=0x" FMT_LIMB " hr=0x" FMT_LIMB " expected=0x" FMT_LIMB "\n", ha, hb, hr, h_ref); |
| exit(1); |
| } |
| #endif |
| return 0; |
| fail: |
| ntt_free(s, buf1); |
| ntt_free(s, buf2); |
| return -1; |
| } |
| |
| #else /* USE_FFT_MUL */ |
| |
| int bf_get_fft_size(int *pdpl, int *pnb_mods, limb_t len) |
| { |
| return 0; |
| } |
| |
| #endif /* !USE_FFT_MUL */ |