2757
+ − 1 /* @(#)k_rem_pio2.c 5.1 93/09/24 */
+ − 2 /*
+ − 3 * ====================================================
+ − 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ − 5 *
+ − 6 * Developed at SunPro, a Sun Microsystems, Inc. business.
+ − 7 * Permission to use, copy, modify, and distribute this
+ − 8 * software is freely granted, provided that this notice
+ − 9 * is preserved.
+ − 10 * ====================================================
+ − 11 */
+ − 12
+ − 13 #if defined(LIBM_SCCS) && !defined(lint)
3162
+ − 14 static const char rcsid[] =
2757
+ − 15 "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $";
+ − 16 #endif
+ − 17
+ − 18 /*
+ − 19 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
+ − 20 * double x[],y[]; int e0,nx,prec; int ipio2[];
+ − 21 *
+ − 22 * __kernel_rem_pio2 return the last three digits of N with
+ − 23 * y = x - N*pi/2
+ − 24 * so that |y| < pi/2.
+ − 25 *
+ − 26 * The method is to compute the integer (mod 8) and fraction parts of
+ − 27 * (2/pi)*x without doing the full multiplication. In general we
+ − 28 * skip the part of the product that are known to be a huge integer (
+ − 29 * more accurately, = 0 mod 8 ). Thus the number of operations are
+ − 30 * independent of the exponent of the input.
+ − 31 *
+ − 32 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
+ − 33 *
+ − 34 * Input parameters:
+ − 35 * x[] The input value (must be positive) is broken into nx
+ − 36 * pieces of 24-bit integers in double precision format.
+ − 37 * x[i] will be the i-th 24 bit of x. The scaled exponent
+ − 38 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
+ − 39 * match x's up to 24 bits.
+ − 40 *
+ − 41 * Example of breaking a double positive z into x[0]+x[1]+x[2]:
+ − 42 * e0 = ilogb(z)-23
+ − 43 * z = scalbn(z,-e0)
+ − 44 * for i = 0,1,2
+ − 45 * x[i] = floor(z)
+ − 46 * z = (z-x[i])*2**24
+ − 47 *
+ − 48 *
+ − 49 * y[] ouput result in an array of double precision numbers.
+ − 50 * The dimension of y[] is:
+ − 51 * 24-bit precision 1
+ − 52 * 53-bit precision 2
+ − 53 * 64-bit precision 2
+ − 54 * 113-bit precision 3
+ − 55 * The actual value is the sum of them. Thus for 113-bit
+ − 56 * precison, one may have to do something like:
+ − 57 *
+ − 58 * long double t,w,r_head, r_tail;
+ − 59 * t = (long double)y[2] + (long double)y[1];
+ − 60 * w = (long double)y[0];
+ − 61 * r_head = t+w;
+ − 62 * r_tail = w - (r_head - t);
+ − 63 *
+ − 64 * e0 The exponent of x[0]
+ − 65 *
+ − 66 * nx dimension of x[]
+ − 67 *
+ − 68 * prec an integer indicating the precision:
+ − 69 * 0 24 bits (single)
+ − 70 * 1 53 bits (double)
+ − 71 * 2 64 bits (extended)
+ − 72 * 3 113 bits (quad)
+ − 73 *
+ − 74 * ipio2[]
+ − 75 * integer array, contains the (24*i)-th to (24*i+23)-th
+ − 76 * bit of 2/pi after binary point. The corresponding
+ − 77 * floating value is
+ − 78 *
+ − 79 * ipio2[i] * 2^(-24(i+1)).
+ − 80 *
+ − 81 * External function:
+ − 82 * double scalbn(), floor();
+ − 83 *
+ − 84 *
+ − 85 * Here is the description of some local variables:
+ − 86 *
+ − 87 * jk jk+1 is the initial number of terms of ipio2[] needed
+ − 88 * in the computation. The recommended value is 2,3,4,
+ − 89 * 6 for single, double, extended,and quad.
+ − 90 *
+ − 91 * jz local integer variable indicating the number of
+ − 92 * terms of ipio2[] used.
+ − 93 *
+ − 94 * jx nx - 1
+ − 95 *
+ − 96 * jv index for pointing to the suitable ipio2[] for the
+ − 97 * computation. In general, we want
+ − 98 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
+ − 99 * is an integer. Thus
+ − 100 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
+ − 101 * Hence jv = max(0,(e0-3)/24).
+ − 102 *
+ − 103 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
+ − 104 *
+ − 105 * q[] double array with integral value, representing the
+ − 106 * 24-bits chunk of the product of x and 2/pi.
+ − 107 *
+ − 108 * q0 the corresponding exponent of q[0]. Note that the
+ − 109 * exponent for q[i] would be q0-24*i.
+ − 110 *
+ − 111 * PIo2[] double precision array, obtained by cutting pi/2
+ − 112 * into 24 bits chunks.
+ − 113 *
+ − 114 * f[] ipio2[] in floating point
+ − 115 *
+ − 116 * iq[] integer array by breaking up q[] in 24-bits chunk.
+ − 117 *
+ − 118 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
+ − 119 *
+ − 120 * ih integer. If >0 it indicates q[] is >= 0.5, hence
+ − 121 * it also indicates the *sign* of the result.
+ − 122 *
+ − 123 */
+ − 124
+ − 125
+ − 126 /*
+ − 127 * Constants:
+ − 128 * The hexadecimal values are the intended ones for the following
+ − 129 * constants. The decimal values may be used, provided that the
+ − 130 * compiler will convert from decimal to binary accurately enough
+ − 131 * to produce the hexadecimal values shown.
+ − 132 */
+ − 133
+ − 134 #include "math.h"
+ − 135 #include "math_private.h"
+ − 136
+ − 137 libm_hidden_proto(scalbn)
+ − 138 libm_hidden_proto(floor)
+ − 139 #ifdef __STDC__
+ − 140 static const int init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */
+ − 141 #else
+ − 142 static int init_jk[] = { 2, 3, 4, 6 };
+ − 143 #endif
+ − 144
+ − 145 #ifdef __STDC__
+ − 146 static const double PIo2[] = {
+ − 147 #else
+ − 148 static double PIo2[] = {
+ − 149 #endif
+ − 150 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
+ − 151 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
+ − 152 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
+ − 153 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
+ − 154 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
+ − 155 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
+ − 156 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
+ − 157 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
+ − 158 };
+ − 159
+ − 160 #ifdef __STDC__
+ − 161 static const double
+ − 162 #else
+ − 163 static double
+ − 164 #endif
+ − 165 zero = 0.0, one = 1.0, two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
+ − 166 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
+ − 167
+ − 168 #ifdef __STDC__
+ − 169 int attribute_hidden
+ − 170 __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
+ − 171 const int32_t * ipio2)
+ − 172 #else
+ − 173 int attribute_hidden
+ − 174 __kernel_rem_pio2(x, y, e0, nx, prec, ipio2)
+ − 175 double x[], y[];
+ − 176 int e0, nx, prec;
+ − 177 int32_t ipio2[];
+ − 178 #endif
+ − 179 {
+ − 180 int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
+ − 181 double z, fw, f[20], fq[20], q[20];
+ − 182
+ − 183 /* initialize jk */
+ − 184 jk = init_jk[prec];
+ − 185 jp = jk;
+ − 186
+ − 187 /* determine jx,jv,q0, note that 3>q0 */
+ − 188 jx = nx - 1;
+ − 189 jv = (e0 - 3) / 24;
+ − 190 if (jv < 0)
+ − 191 jv = 0;
+ − 192 q0 = e0 - 24 * (jv + 1);
+ − 193
+ − 194 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
+ − 195 j = jv - jx;
+ − 196 m = jx + jk;
+ − 197 for (i = 0; i <= m; i++, j++)
+ − 198 f[i] = (j < 0) ? zero : (double) ipio2[j];
+ − 199
+ − 200 /* compute q[0],q[1],...q[jk] */
+ − 201 for (i = 0; i <= jk; i++) {
+ − 202 for (j = 0, fw = 0.0; j <= jx; j++)
+ − 203 fw += x[j] * f[jx + i - j];
+ − 204 q[i] = fw;
+ − 205 }
+ − 206
+ − 207 jz = jk;
+ − 208 recompute:
+ − 209 /* distill q[] into iq[] reversingly */
+ − 210 for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
+ − 211 fw = (double) ((int32_t) (twon24 * z));
+ − 212 iq[i] = (int32_t) (z - two24 * fw);
+ − 213 z = q[j - 1] + fw;
+ − 214 }
+ − 215
+ − 216 /* compute n */
+ − 217 z = scalbn(z, q0); /* actual value of z */
+ − 218 z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */
+ − 219 n = (int32_t) z;
+ − 220 z -= (double) n;
+ − 221 ih = 0;
+ − 222 if (q0 > 0) { /* need iq[jz-1] to determine n */
+ − 223 i = (iq[jz - 1] >> (24 - q0));
+ − 224 n += i;
+ − 225 iq[jz - 1] -= i << (24 - q0);
+ − 226 ih = iq[jz - 1] >> (23 - q0);
+ − 227 } else if (q0 == 0)
+ − 228 ih = iq[jz - 1] >> 23;
+ − 229 else if (z >= 0.5)
+ − 230 ih = 2;
+ − 231
+ − 232 if (ih > 0) { /* q > 0.5 */
+ − 233 n += 1;
+ − 234 carry = 0;
+ − 235 for (i = 0; i < jz; i++) { /* compute 1-q */
+ − 236 j = iq[i];
+ − 237 if (carry == 0) {
+ − 238 if (j != 0) {
+ − 239 carry = 1;
+ − 240 iq[i] = 0x1000000 - j;
+ − 241 }
+ − 242 } else
+ − 243 iq[i] = 0xffffff - j;
+ − 244 }
+ − 245 if (q0 > 0) { /* rare case: chance is 1 in 12 */
+ − 246 switch (q0) {
+ − 247 case 1:
+ − 248 iq[jz - 1] &= 0x7fffff;
+ − 249 break;
+ − 250 case 2:
+ − 251 iq[jz - 1] &= 0x3fffff;
+ − 252 break;
+ − 253 }
+ − 254 }
+ − 255 if (ih == 2) {
+ − 256 z = one - z;
+ − 257 if (carry != 0)
+ − 258 z -= scalbn(one, q0);
+ − 259 }
+ − 260 }
+ − 261
+ − 262 /* check if recomputation is needed */
+ − 263 if (z == zero) {
+ − 264 j = 0;
+ − 265 for (i = jz - 1; i >= jk; i--)
+ − 266 j |= iq[i];
+ − 267 if (j == 0) { /* need recomputation */
+ − 268 for (k = 1; iq[jk - k] == 0; k++); /* k = no. of terms needed */
+ − 269
+ − 270 for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */
+ − 271 f[jx + i] = (double) ipio2[jv + i];
+ − 272 for (j = 0, fw = 0.0; j <= jx; j++)
+ − 273 fw += x[j] * f[jx + i - j];
+ − 274 q[i] = fw;
+ − 275 }
+ − 276 jz += k;
+ − 277 goto recompute;
+ − 278 }
+ − 279 }
+ − 280
+ − 281 /* chop off zero terms */
+ − 282 if (z == 0.0) {
+ − 283 jz -= 1;
+ − 284 q0 -= 24;
+ − 285 while (iq[jz] == 0) {
+ − 286 jz--;
+ − 287 q0 -= 24;
+ − 288 }
+ − 289 } else { /* break z into 24-bit if necessary */
+ − 290 z = scalbn(z, -q0);
+ − 291 if (z >= two24) {
+ − 292 fw = (double) ((int32_t) (twon24 * z));
+ − 293 iq[jz] = (int32_t) (z - two24 * fw);
+ − 294 jz += 1;
+ − 295 q0 += 24;
+ − 296 iq[jz] = (int32_t) fw;
+ − 297 } else
+ − 298 iq[jz] = (int32_t) z;
+ − 299 }
+ − 300
+ − 301 /* convert integer "bit" chunk to floating-point value */
+ − 302 fw = scalbn(one, q0);
+ − 303 for (i = jz; i >= 0; i--) {
+ − 304 q[i] = fw * (double) iq[i];
+ − 305 fw *= twon24;
+ − 306 }
+ − 307
+ − 308 /* compute PIo2[0,...,jp]*q[jz,...,0] */
+ − 309 for (i = jz; i >= 0; i--) {
+ − 310 for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
+ − 311 fw += PIo2[k] * q[i + k];
+ − 312 fq[jz - i] = fw;
+ − 313 }
+ − 314
+ − 315 /* compress fq[] into y[] */
+ − 316 switch (prec) {
+ − 317 case 0:
+ − 318 fw = 0.0;
+ − 319 for (i = jz; i >= 0; i--)
+ − 320 fw += fq[i];
+ − 321 y[0] = (ih == 0) ? fw : -fw;
+ − 322 break;
+ − 323 case 1:
+ − 324 case 2:
+ − 325 fw = 0.0;
+ − 326 for (i = jz; i >= 0; i--)
+ − 327 fw += fq[i];
+ − 328 y[0] = (ih == 0) ? fw : -fw;
+ − 329 fw = fq[0] - fw;
+ − 330 for (i = 1; i <= jz; i++)
+ − 331 fw += fq[i];
+ − 332 y[1] = (ih == 0) ? fw : -fw;
+ − 333 break;
+ − 334 case 3: /* painful */
+ − 335 for (i = jz; i > 0; i--) {
+ − 336 fw = fq[i - 1] + fq[i];
+ − 337 fq[i] += fq[i - 1] - fw;
+ − 338 fq[i - 1] = fw;
+ − 339 }
+ − 340 for (i = jz; i > 1; i--) {
+ − 341 fw = fq[i - 1] + fq[i];
+ − 342 fq[i] += fq[i - 1] - fw;
+ − 343 fq[i - 1] = fw;
+ − 344 }
+ − 345 for (fw = 0.0, i = jz; i >= 2; i--)
+ − 346 fw += fq[i];
+ − 347 if (ih == 0) {
+ − 348 y[0] = fq[0];
+ − 349 y[1] = fq[1];
+ − 350 y[2] = fw;
+ − 351 } else {
+ − 352 y[0] = -fq[0];
+ − 353 y[1] = -fq[1];
+ − 354 y[2] = -fw;
+ − 355 }
+ − 356 }
+ − 357 return n & 7;
+ − 358 }
