| /* @(#)k_rem_pio2.c 5.1 93/09/24 */ |
| /* |
| * ==================================================== |
| * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| * |
| * Developed at SunPro, a Sun Microsystems, Inc. business. |
| * Permission to use, copy, modify, and distribute this |
| * software is freely granted, provided that this notice |
| * is preserved. |
| * ==================================================== |
| */ |
| |
| #if defined(LIBM_SCCS) && !defined(lint) |
| static const char rcsid[] = |
| "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $"; |
| #endif |
| |
| /* |
| * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) |
| * double x[],y[]; int e0,nx,prec; int ipio2[]; |
| * |
| * __kernel_rem_pio2 return the last three digits of N with |
| * y = x - N*pi/2 |
| * so that |y| < pi/2. |
| * |
| * The method is to compute the integer (mod 8) and fraction parts of |
| * (2/pi)*x without doing the full multiplication. In general we |
| * skip the part of the product that are known to be a huge integer ( |
| * more accurately, = 0 mod 8 ). Thus the number of operations are |
| * independent of the exponent of the input. |
| * |
| * (2/pi) is represented by an array of 24-bit integers in ipio2[]. |
| * |
| * Input parameters: |
| * x[] The input value (must be positive) is broken into nx |
| * pieces of 24-bit integers in double precision format. |
| * x[i] will be the i-th 24 bit of x. The scaled exponent |
| * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 |
| * match x's up to 24 bits. |
| * |
| * Example of breaking a double positive z into x[0]+x[1]+x[2]: |
| * e0 = ilogb(z)-23 |
| * z = scalbn(z,-e0) |
| * for i = 0,1,2 |
| * x[i] = floor(z) |
| * z = (z-x[i])*2**24 |
| * |
| * |
| * y[] ouput result in an array of double precision numbers. |
| * The dimension of y[] is: |
| * 24-bit precision 1 |
| * 53-bit precision 2 |
| * 64-bit precision 2 |
| * 113-bit precision 3 |
| * The actual value is the sum of them. Thus for 113-bit |
| * precison, one may have to do something like: |
| * |
| * long double t,w,r_head, r_tail; |
| * t = (long double)y[2] + (long double)y[1]; |
| * w = (long double)y[0]; |
| * r_head = t+w; |
| * r_tail = w - (r_head - t); |
| * |
| * e0 The exponent of x[0] |
| * |
| * nx dimension of x[] |
| * |
| * prec an integer indicating the precision: |
| * 0 24 bits (single) |
| * 1 53 bits (double) |
| * 2 64 bits (extended) |
| * 3 113 bits (quad) |
| * |
| * ipio2[] |
| * integer array, contains the (24*i)-th to (24*i+23)-th |
| * bit of 2/pi after binary point. The corresponding |
| * floating value is |
| * |
| * ipio2[i] * 2^(-24(i+1)). |
| * |
| * External function: |
| * double scalbn(), floor(); |
| * |
| * |
| * Here is the description of some local variables: |
| * |
| * jk jk+1 is the initial number of terms of ipio2[] needed |
| * in the computation. The recommended value is 2,3,4, |
| * 6 for single, double, extended,and quad. |
| * |
| * jz local integer variable indicating the number of |
| * terms of ipio2[] used. |
| * |
| * jx nx - 1 |
| * |
| * jv index for pointing to the suitable ipio2[] for the |
| * computation. In general, we want |
| * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 |
| * is an integer. Thus |
| * e0-3-24*jv >= 0 or (e0-3)/24 >= jv |
| * Hence jv = max(0,(e0-3)/24). |
| * |
| * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. |
| * |
| * q[] double array with integral value, representing the |
| * 24-bits chunk of the product of x and 2/pi. |
| * |
| * q0 the corresponding exponent of q[0]. Note that the |
| * exponent for q[i] would be q0-24*i. |
| * |
| * PIo2[] double precision array, obtained by cutting pi/2 |
| * into 24 bits chunks. |
| * |
| * f[] ipio2[] in floating point |
| * |
| * iq[] integer array by breaking up q[] in 24-bits chunk. |
| * |
| * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] |
| * |
| * ih integer. If >0 it indicates q[] is >= 0.5, hence |
| * it also indicates the *sign* of the result. |
| * |
| */ |
| |
| |
| /* |
| * Constants: |
| * The hexadecimal values are the intended ones for the following |
| * constants. The decimal values may be used, provided that the |
| * compiler will convert from decimal to binary accurately enough |
| * to produce the hexadecimal values shown. |
| */ |
| |
| #include "math_libm.h" |
| #include "math_private.h" |
| |
| #include "SDL_assert.h" |
| |
| libm_hidden_proto(scalbn) |
| libm_hidden_proto(floor) |
| #ifdef __STDC__ |
| static const int init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */ |
| #else |
| static int init_jk[] = { 2, 3, 4, 6 }; |
| #endif |
| |
| #ifdef __STDC__ |
| static const double PIo2[] = { |
| #else |
| static double PIo2[] = { |
| #endif |
| 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ |
| 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ |
| 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ |
| 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ |
| 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ |
| 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ |
| 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ |
| 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ |
| }; |
| |
| #ifdef __STDC__ |
| static const double |
| #else |
| static double |
| #endif |
| zero = 0.0, one = 1.0, two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ |
| twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ |
| |
| #ifdef __STDC__ |
| int attribute_hidden |
| __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, |
| const int32_t * ipio2) |
| #else |
| int attribute_hidden |
| __kernel_rem_pio2(x, y, e0, nx, prec, ipio2) |
| double x[], y[]; |
| int e0, nx, prec; |
| int32_t ipio2[]; |
| #endif |
| { |
| int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih; |
| double z, fw, f[20], fq[20], q[20]; |
| |
| /* initialize jk */ |
| SDL_assert((prec >= 0) && (prec < SDL_arraysize(init_jk))); |
| jk = init_jk[prec]; |
| SDL_assert((jk >= 2) && (jk <= 6)); |
| jp = jk; |
| |
| /* determine jx,jv,q0, note that 3>q0 */ |
| SDL_assert(nx > 0); |
| jx = nx - 1; |
| jv = (e0 - 3) / 24; |
| if (jv < 0) |
| jv = 0; |
| q0 = e0 - 24 * (jv + 1); |
| |
| /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ |
| j = jv - jx; |
| m = jx + jk; |
| for (i = 0; i <= m; i++, j++) |
| f[i] = (j < 0) ? zero : (double) ipio2[j]; |
| |
| /* compute q[0],q[1],...q[jk] */ |
| for (i = 0; i <= jk; i++) { |
| for (j = 0, fw = 0.0; j <= jx; j++) |
| fw += x[j] * f[jx + i - j]; |
| q[i] = fw; |
| } |
| |
| jz = jk; |
| recompute: |
| /* distill q[] into iq[] reversingly */ |
| for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) { |
| fw = (double) ((int32_t) (twon24 * z)); |
| iq[i] = (int32_t) (z - two24 * fw); |
| z = q[j - 1] + fw; |
| } |
| |
| /* compute n */ |
| z = scalbn(z, q0); /* actual value of z */ |
| z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */ |
| n = (int32_t) z; |
| z -= (double) n; |
| ih = 0; |
| if (q0 > 0) { /* need iq[jz-1] to determine n */ |
| i = (iq[jz - 1] >> (24 - q0)); |
| n += i; |
| iq[jz - 1] -= i << (24 - q0); |
| ih = iq[jz - 1] >> (23 - q0); |
| } else if (q0 == 0) |
| ih = iq[jz - 1] >> 23; |
| else if (z >= 0.5) |
| ih = 2; |
| |
| if (ih > 0) { /* q > 0.5 */ |
| n += 1; |
| carry = 0; |
| for (i = 0; i < jz; i++) { /* compute 1-q */ |
| j = iq[i]; |
| if (carry == 0) { |
| if (j != 0) { |
| carry = 1; |
| iq[i] = 0x1000000 - j; |
| } |
| } else |
| iq[i] = 0xffffff - j; |
| } |
| if (q0 > 0) { /* rare case: chance is 1 in 12 */ |
| switch (q0) { |
| case 1: |
| iq[jz - 1] &= 0x7fffff; |
| break; |
| case 2: |
| iq[jz - 1] &= 0x3fffff; |
| break; |
| } |
| } |
| if (ih == 2) { |
| z = one - z; |
| if (carry != 0) |
| z -= scalbn(one, q0); |
| } |
| } |
| |
| /* check if recomputation is needed */ |
| if (z == zero) { |
| j = 0; |
| for (i = jz - 1; i >= jk; i--) |
| j |= iq[i]; |
| if (j == 0) { /* need recomputation */ |
| for (k = 1; iq[jk - k] == 0; k++); /* k = no. of terms needed */ |
| |
| for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */ |
| f[jx + i] = (double) ipio2[jv + i]; |
| for (j = 0, fw = 0.0; j <= jx; j++) |
| fw += x[j] * f[jx + i - j]; |
| q[i] = fw; |
| } |
| jz += k; |
| goto recompute; |
| } |
| } |
| |
| /* chop off zero terms */ |
| if (z == 0.0) { |
| jz -= 1; |
| q0 -= 24; |
| while (iq[jz] == 0) { |
| jz--; |
| q0 -= 24; |
| } |
| } else { /* break z into 24-bit if necessary */ |
| z = scalbn(z, -q0); |
| if (z >= two24) { |
| fw = (double) ((int32_t) (twon24 * z)); |
| iq[jz] = (int32_t) (z - two24 * fw); |
| jz += 1; |
| q0 += 24; |
| iq[jz] = (int32_t) fw; |
| } else |
| iq[jz] = (int32_t) z; |
| } |
| |
| /* convert integer "bit" chunk to floating-point value */ |
| fw = scalbn(one, q0); |
| for (i = jz; i >= 0; i--) { |
| q[i] = fw * (double) iq[i]; |
| fw *= twon24; |
| } |
| |
| /* compute PIo2[0,...,jp]*q[jz,...,0] */ |
| for (i = jz; i >= 0; i--) { |
| for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++) |
| fw += PIo2[k] * q[i + k]; |
| fq[jz - i] = fw; |
| } |
| |
| /* compress fq[] into y[] */ |
| switch (prec) { |
| case 0: |
| fw = 0.0; |
| for (i = jz; i >= 0; i--) |
| fw += fq[i]; |
| y[0] = (ih == 0) ? fw : -fw; |
| break; |
| case 1: |
| case 2: |
| fw = 0.0; |
| for (i = jz; i >= 0; i--) |
| fw += fq[i]; |
| y[0] = (ih == 0) ? fw : -fw; |
| fw = fq[0] - fw; |
| for (i = 1; i <= jz; i++) |
| fw += fq[i]; |
| y[1] = (ih == 0) ? fw : -fw; |
| break; |
| case 3: /* painful */ |
| for (i = jz; i > 0; i--) { |
| fw = fq[i - 1] + fq[i]; |
| fq[i] += fq[i - 1] - fw; |
| fq[i - 1] = fw; |
| } |
| for (i = jz; i > 1; i--) { |
| fw = fq[i - 1] + fq[i]; |
| fq[i] += fq[i - 1] - fw; |
| fq[i - 1] = fw; |
| } |
| for (fw = 0.0, i = jz; i >= 2; i--) |
| fw += fq[i]; |
| if (ih == 0) { |
| y[0] = fq[0]; |
| y[1] = fq[1]; |
| y[2] = fw; |
| } else { |
| y[0] = -fq[0]; |
| y[1] = -fq[1]; |
| y[2] = -fw; |
| } |
| } |
| return n & 7; |
| } |
