src/runtime/mpn.cpp

/*++
Copyright (c) 2011 Microsoft Corporation

Module Name:

    mpn.cpp

Abstract:

    Multi Precision Natural Numbers

Author:

    Christoph Wintersteiger (cwinter) 2011-11-16.

Revision History:

--*/
#include <stdint.h>
#include "runtime/mpn.h"
#include "runtime/debug.h"
#include "runtime/buffer.h"

#define max(a,b)    (((a) > (b)) ? (a) : (b))

namespace lean {

typedef uint64_t mpn_double_digit;
static_assert(sizeof(mpn_double_digit) == 2 * sizeof(mpn_digit), "size alignment");

static const mpn_digit zero = 0;

int mpn_compare(mpn_digit const * a, size_t const lnga,
                mpn_digit const * b, size_t const lngb) {
    int res = 0;

    size_t j = max(lnga, lngb) - 1;
    for (; j != (size_t)-1 && res == 0; j--) {
        mpn_digit const & u_j = (j < lnga) ? a[j] : zero;
        mpn_digit const & v_j = (j < lngb) ? b[j] : zero;
        if (u_j > v_j)
            res = 1;
        else if (u_j < v_j)
            res = -1;
    }
    return res;
}

void  mpn_add(mpn_digit const * a, size_t const lnga,
              mpn_digit const * b, size_t const lngb,
              mpn_digit * c, size_t const lngc_alloc,
              size_t * plngc) {
    // Essentially Knuth's Algorithm A
    size_t len = max(lnga, lngb);
    lean_assert(lngc_alloc == len+1 && len > 0);
    mpn_digit k = 0;
    mpn_digit r;
    bool c1, c2;
    for (size_t j = 0; j < len; j++) {
        mpn_digit const & u_j = (j < lnga) ? a[j] : zero;
        mpn_digit const & v_j = (j < lngb) ? b[j] : zero;
        r = u_j + v_j; c1 = r < u_j;
        c[j] = r + k;  c2 = c[j] < r;
        k = c1 | c2;
    }
    c[len] = k;
    size_t &os = *plngc;
    for (os = len+1; os > 1 && c[os-1] == 0; ) os--;
    lean_assert(os > 0 && os <= len+1);
}

void mpn_sub(mpn_digit const * a, size_t const lnga,
             mpn_digit const * b, size_t const lngb,
             mpn_digit * c, mpn_digit * pborrow) {
    // Essentially Knuth's Algorithm S
    size_t len = max(lnga, lngb);
    mpn_digit & k = *pborrow; k = 0;
    mpn_digit r;
    bool c1, c2;
    for (size_t j = 0; j < len; j++) {
        mpn_digit const & u_j = (j < lnga) ? a[j] : zero;
        mpn_digit const & v_j = (j < lngb) ? b[j] : zero;
        r = u_j - v_j; c1 = r > u_j;
        c[j] = r - k;  c2 = c[j] > r;
        k = c1 | c2;
    }
}

void mpn_mul(mpn_digit const * a, size_t const lnga,
             mpn_digit const * b, size_t const lngb,
             mpn_digit * c) {
    // Essentially Knuth's Algorithm M.
    // Perhaps implement a more efficient version, see e.g., Knuth, Section 4.3.3.
    size_t i;
    mpn_digit k;

#define DIGIT_BITS (sizeof(mpn_digit)*8)
#define HALF_BITS (sizeof(mpn_digit)*4)

    for (unsigned i = 0; i < lnga; i++)
        c[i] = 0;

    for (size_t j = 0; j < lngb; j++) {
        mpn_digit const & v_j = b[j];
        if (v_j == 0) { // This branch may be omitted according to Knuth.
            c[j+lnga] = 0;
        }
        else {
            k = 0;
            for (i = 0; i < lnga; i++) {
                mpn_digit const & u_i = a[i];
                mpn_double_digit t;
                t = ((mpn_double_digit)u_i * (mpn_double_digit)v_j) +
                    (mpn_double_digit) c[i+j] +
                    (mpn_double_digit) k;

                c[i+j] = (t << DIGIT_BITS) >> DIGIT_BITS;
                k = t >> DIGIT_BITS;
            }
            c[j+lnga] = k;
        }
    }
}

#define MASK_FIRST (~((mpn_digit)(-1) >> 1))
#define FIRST_BITS(N, X) ((X) >> (DIGIT_BITS-(N)))
#define LAST_BITS(N, X) (((X) << (DIGIT_BITS-(N))) >> (DIGIT_BITS-(N)))
#define BASE ((mpn_double_digit)0x01 << DIGIT_BITS)

class  mpn_buffer : public buffer<mpn_digit> {
public:
    mpn_buffer() : buffer<mpn_digit>() {}

    mpn_buffer(size_t nsz, const mpn_digit & elem = 0):buffer<mpn_digit>() {
        for (size_t i = 0; i < nsz; i++) push_back(elem);
    }

    void resize(size_t nsz, const mpn_digit & elem = 0) {
        buffer<mpn_digit>::resize(static_cast<unsigned>(nsz), elem);
    }

    mpn_digit & operator[](size_t idx) {
        return buffer<mpn_digit>::operator[](static_cast<unsigned>(idx));
    }

    const mpn_digit & operator[](size_t idx) const {
        return buffer<mpn_digit>::operator[](static_cast<unsigned>(idx));
    }
};

static size_t div_normalize(mpn_digit const * numer, size_t const lnum,
                            mpn_digit const * denom, size_t const lden,
                            mpn_buffer & n_numer,
                            mpn_buffer & n_denom) {
    size_t d = 0;
    while (lden > 0 && ((denom[lden-1] << d) & MASK_FIRST) == 0) d++;
    lean_assert(d < DIGIT_BITS);

    n_numer.resize(lnum+1);
    n_denom.resize(lden);

    if (d == 0) {
        n_numer[lnum] = 0;
        for (size_t i = 0; i < lnum; i++)
            n_numer[i] = numer[i];
        for (size_t i = 0; i < lden; i++)
            n_denom[i] = denom[i];
    }
    else if (lnum != 0) {
        lean_assert(lden > 0);
        mpn_digit q = FIRST_BITS(d, numer[lnum-1]);
        n_numer[lnum] = q;
        for (size_t i = lnum-1; i > 0; i--)
            n_numer[i] = (numer[i] << d) | FIRST_BITS(d, numer[i-1]);
        n_numer[0] = numer[0] << d;
        for (size_t i = lden-1; i > 0; i--)
            n_denom[i] = denom[i] << d | FIRST_BITS(d, denom[i-1]);
        n_denom[0] = denom[0] << d;
    }
    else {
        d = 0;
    }
    return d;
}

static void div_unnormalize(mpn_buffer & numer, mpn_buffer & denom,
                            size_t const d, mpn_digit * rem) {
    if (d == 0) {
        for (size_t i = 0; i < denom.size(); i++)
            rem[i] = numer[i];
    }
    else {
        for (size_t i = 0; i < denom.size()-1; i++)
            rem[i] = numer[i] >> d | (LAST_BITS(d, numer[i+1]) << (DIGIT_BITS-d));
        rem[denom.size()-1] = numer[denom.size()-1] >> d;
    }
}

static void div_1(mpn_buffer & numer, mpn_digit const denom,
                  mpn_digit * quot) {
    mpn_double_digit q_hat, temp, ms;
    mpn_digit borrow;

    for (size_t j = numer.size()-1; j > 0; j--) {
        temp = (((mpn_double_digit)numer[j]) << DIGIT_BITS) | ((mpn_double_digit)numer[j-1]);
        q_hat = temp / (mpn_double_digit) denom;
        if (q_hat >= BASE) {
            lean_unreachable(); // is this reachable with normalized v?
        }
        lean_assert(q_hat < BASE);
        ms = temp - (q_hat * (mpn_double_digit) denom);
        borrow = ms > temp;
        numer[j-1] = (mpn_digit) ms;
        numer[j] = ms >> DIGIT_BITS;
        quot[j-1] = (mpn_digit) q_hat;
        if (borrow) {
            quot[j-1]--;
            numer[j] = numer[j-1] + denom;
        }
    }
}

static void div_n(mpn_buffer & numer, mpn_buffer const & denom,
                  mpn_digit * quot, mpn_digit * rem,
                  mpn_buffer & ms, mpn_buffer & ab) {
    lean_assert(denom.size() > 1);

    // This is essentially Knuth's Algorithm D.
    size_t m = numer.size() - denom.size();
    size_t n = denom.size();

    lean_assert(numer.size() == m+n);

    ms.resize(n+1);

    mpn_double_digit q_hat, temp, r_hat;
    mpn_digit borrow;

    for (size_t j = m-1; j != (size_t)-1; j--) {
        temp = (((mpn_double_digit)numer[j+n]) << DIGIT_BITS) | ((mpn_double_digit)numer[j+n-1]);
        q_hat = temp / (mpn_double_digit) denom[n-1];
        r_hat = temp % (mpn_double_digit) denom[n-1];
        recheck:
        if (q_hat >= BASE ||
            ((q_hat * denom[n-2]) > ((r_hat << DIGIT_BITS) + numer[j+n-2]))) {
                q_hat--;
                r_hat += denom[n-1];
                if (r_hat < BASE) goto recheck;
        }
        lean_assert(q_hat < BASE);
        // Replace numer[j+n]...numer[j] with
        // numer[j+n]...numer[j] - q * (denom[n-1]...denom[0])
        mpn_digit q_hat_small = (mpn_digit)q_hat;
        mpn_mul(&q_hat_small, 1, denom.data(), n, ms.data());
        mpn_sub(&numer[j], n+1, ms.data(), n+1, &numer[j], &borrow);
        quot[j] = q_hat_small;
        if (borrow) {
            quot[j]--;
            ab.resize(n+2);
            size_t real_size;
            mpn_add(denom.data(), n, &numer[j], n+1, ab.data(), n+2, &real_size);
            for (size_t i = 0; i < n+1; i++)
                numer[j+i] = ab[i];
        }
    }
}

void mpn_div(mpn_digit const * numer, size_t const lnum,
             mpn_digit const * denom, size_t const lden,
             mpn_digit * quot,
             mpn_digit * rem) {

    if (lnum < lden) {
        for (size_t i = 0; i < (lnum-lden+1); i++)
            quot[i] = 0;
        for (size_t i = 0; i < lden; i++)
            rem[i] = (i < lnum) ? numer[i] : 0;
        return;
    }

    bool all_zero = true;
    for (size_t i = 0; i < lden && all_zero; i++)
        if (denom[i] != zero) all_zero = false;

    lean_assert(!all_zero);

    lean_assert(denom[lden-1] != 0);

    if (lnum == 1 && lden == 1) {
        *quot = numer[0] / denom[0];
        *rem  = numer[0] % denom[0];
    }
    else if (lnum < lden || (lnum == lden && numer[lnum-1] < denom[lden-1])) {
        *quot = 0;
        for (size_t i = 0; i < lden; i++)
            rem[i] = (i < lnum) ? numer[i] : 0;
    }
    else  {
        mpn_buffer u, v, t_ms, t_ab;
        size_t d = div_normalize(numer, lnum, denom, lden, u, v);
        if (lden == 1)
            div_1(u, v[0], quot);
        else
            div_n(u, v, quot, rem, t_ms, t_ab);
        div_unnormalize(u, v, d, rem);
    }

#ifdef LEAN_DEBUG
    mpn_buffer temp(lnum+1, 0);
    mpn_mul(quot, lnum-lden+1, denom, lden, temp.data());
    size_t real_size;
    mpn_add(temp.data(), lnum, rem, lden, temp.data(), lnum+1, &real_size);
    bool ok = true;
    for (size_t i = 0; i < lnum && ok; i++)
        if (temp[i] != numer[i]) ok = false;
    if (temp[lnum] != 0) ok = false;
    lean_assert(ok);
#endif
}

char * mpn_to_string(mpn_digit const * a, size_t const lng, char * buf, size_t const lbuf) {
    lean_assert(buf && lbuf > 0);

    if (lng == 1) {
#ifdef _WINDOWS
        sprintf_s(buf, lbuf, "%u", *a);
#else
        snprintf(buf, lbuf, "%u", *a);
#endif
    }
    else {
        mpn_buffer temp(lng, 0), t_numer(lng+1, 0), t_denom(1, 0);
        for (unsigned i = 0; i < lng; i++)
            temp[i] = a[i];

        size_t j = 0;
        mpn_digit rem;
        mpn_digit ten = 10;
        while (!temp.empty() && (temp.size() > 1 || temp[0] != 0)) {
            size_t d = div_normalize(&temp[0], temp.size(), &ten, 1, t_numer, t_denom);
            div_1(t_numer, t_denom[0], &temp[0]);
            div_unnormalize(t_numer, t_denom, d, &rem);
            buf[j++] = '0' + rem;
            while (!temp.empty() && temp.back() == 0)
                temp.pop_back();
        }
        buf[j] = 0;

        j--;
        size_t mid = (j/2) + ((j % 2) ? 1 : 0);
        for (size_t i = 0; i < mid; i++)
            std::swap(buf[i], buf[j-i]);
    }
    return buf;
}
}