k_rem_pio2.c revision 15122:b211a52a7439
1/* 2 * Copyright (c) 1998, 2013, Oracle and/or its affiliates. All rights reserved. 3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. 4 * 5 * This code is free software; you can redistribute it and/or modify it 6 * under the terms of the GNU General Public License version 2 only, as 7 * published by the Free Software Foundation. Oracle designates this 8 * particular file as subject to the "Classpath" exception as provided 9 * by Oracle in the LICENSE file that accompanied this code. 10 * 11 * This code is distributed in the hope that it will be useful, but WITHOUT 12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 14 * version 2 for more details (a copy is included in the LICENSE file that 15 * accompanied this code). 16 * 17 * You should have received a copy of the GNU General Public License version 18 * 2 along with this work; if not, write to the Free Software Foundation, 19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 20 * 21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 22 * or visit www.oracle.com if you need additional information or have any 23 * questions. 24 */ 25 26/* 27 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) 28 * double x[],y[]; int e0,nx,prec; int ipio2[]; 29 * 30 * __kernel_rem_pio2 return the last three digits of N with 31 * y = x - N*pi/2 32 * so that |y| < pi/2. 33 * 34 * The method is to compute the integer (mod 8) and fraction parts of 35 * (2/pi)*x without doing the full multiplication. In general we 36 * skip the part of the product that are known to be a huge integer ( 37 * more accurately, = 0 mod 8 ). Thus the number of operations are 38 * independent of the exponent of the input. 39 * 40 * (2/pi) is represented by an array of 24-bit integers in ipio2[]. 41 * 42 * Input parameters: 43 * x[] The input value (must be positive) is broken into nx 44 * pieces of 24-bit integers in double precision format. 45 * x[i] will be the i-th 24 bit of x. The scaled exponent 46 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 47 * match x's up to 24 bits. 48 * 49 * Example of breaking a double positive z into x[0]+x[1]+x[2]: 50 * e0 = ilogb(z)-23 51 * z = scalbn(z,-e0) 52 * for i = 0,1,2 53 * x[i] = floor(z) 54 * z = (z-x[i])*2**24 55 * 56 * 57 * y[] output result in an array of double precision numbers. 58 * The dimension of y[] is: 59 * 24-bit precision 1 60 * 53-bit precision 2 61 * 64-bit precision 2 62 * 113-bit precision 3 63 * The actual value is the sum of them. Thus for 113-bit 64 * precison, one may have to do something like: 65 * 66 * long double t,w,r_head, r_tail; 67 * t = (long double)y[2] + (long double)y[1]; 68 * w = (long double)y[0]; 69 * r_head = t+w; 70 * r_tail = w - (r_head - t); 71 * 72 * e0 The exponent of x[0] 73 * 74 * nx dimension of x[] 75 * 76 * prec an integer indicating the precision: 77 * 0 24 bits (single) 78 * 1 53 bits (double) 79 * 2 64 bits (extended) 80 * 3 113 bits (quad) 81 * 82 * ipio2[] 83 * integer array, contains the (24*i)-th to (24*i+23)-th 84 * bit of 2/pi after binary point. The corresponding 85 * floating value is 86 * 87 * ipio2[i] * 2^(-24(i+1)). 88 * 89 * External function: 90 * double scalbn(), floor(); 91 * 92 * 93 * Here is the description of some local variables: 94 * 95 * jk jk+1 is the initial number of terms of ipio2[] needed 96 * in the computation. The recommended value is 2,3,4, 97 * 6 for single, double, extended,and quad. 98 * 99 * jz local integer variable indicating the number of 100 * terms of ipio2[] used. 101 * 102 * jx nx - 1 103 * 104 * jv index for pointing to the suitable ipio2[] for the 105 * computation. In general, we want 106 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 107 * is an integer. Thus 108 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv 109 * Hence jv = max(0,(e0-3)/24). 110 * 111 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. 112 * 113 * q[] double array with integral value, representing the 114 * 24-bits chunk of the product of x and 2/pi. 115 * 116 * q0 the corresponding exponent of q[0]. Note that the 117 * exponent for q[i] would be q0-24*i. 118 * 119 * PIo2[] double precision array, obtained by cutting pi/2 120 * into 24 bits chunks. 121 * 122 * f[] ipio2[] in floating point 123 * 124 * iq[] integer array by breaking up q[] in 24-bits chunk. 125 * 126 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] 127 * 128 * ih integer. If >0 it indicates q[] is >= 0.5, hence 129 * it also indicates the *sign* of the result. 130 * 131 */ 132 133 134/* 135 * Constants: 136 * The hexadecimal values are the intended ones for the following 137 * constants. The decimal values may be used, provided that the 138 * compiler will convert from decimal to binary accurately enough 139 * to produce the hexadecimal values shown. 140 */ 141 142#include "fdlibm.h" 143 144#ifdef __STDC__ 145static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ 146#else 147static int init_jk[] = {2,3,4,6}; 148#endif 149 150#ifdef __STDC__ 151static const double PIo2[] = { 152#else 153static double PIo2[] = { 154#endif 155 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ 156 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ 157 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ 158 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ 159 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ 160 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ 161 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ 162 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ 163}; 164 165#ifdef __STDC__ 166static const double 167#else 168static double 169#endif 170zero = 0.0, 171one = 1.0, 172two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 173twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ 174 175#ifdef __STDC__ 176 int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) 177#else 178 int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) 179 double x[], y[]; int e0,nx,prec; int ipio2[]; 180#endif 181{ 182 int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; 183 double z,fw,f[20],fq[20],q[20]; 184 185 /* initialize jk*/ 186 jk = init_jk[prec]; 187 jp = jk; 188 189 /* determine jx,jv,q0, note that 3>q0 */ 190 jx = nx-1; 191 jv = (e0-3)/24; if(jv<0) 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; m = jx+jk; 196 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; 197 198 /* compute q[0],q[1],...q[jk] */ 199 for (i=0;i<=jk;i++) { 200 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; 201 } 202 203 jz = jk; 204recompute: 205 /* distill q[] into iq[] reversingly */ 206 for(i=0,j=jz,z=q[jz];j>0;i++,j--) { 207 fw = (double)((int)(twon24* z)); 208 iq[i] = (int)(z-two24*fw); 209 z = q[j-1]+fw; 210 } 211 212 /* compute n */ 213 z = scalbn(z,q0); /* actual value of z */ 214 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ 215 n = (int) z; 216 z -= (double)n; 217 ih = 0; 218 if(q0>0) { /* need iq[jz-1] to determine n */ 219 i = (iq[jz-1]>>(24-q0)); n += i; 220 iq[jz-1] -= i<<(24-q0); 221 ih = iq[jz-1]>>(23-q0); 222 } 223 else if(q0==0) ih = iq[jz-1]>>23; 224 else if(z>=0.5) ih=2; 225 226 if(ih>0) { /* q > 0.5 */ 227 n += 1; carry = 0; 228 for(i=0;i<jz ;i++) { /* compute 1-q */ 229 j = iq[i]; 230 if(carry==0) { 231 if(j!=0) { 232 carry = 1; iq[i] = 0x1000000- j; 233 } 234 } else iq[i] = 0xffffff - j; 235 } 236 if(q0>0) { /* rare case: chance is 1 in 12 */ 237 switch(q0) { 238 case 1: 239 iq[jz-1] &= 0x7fffff; break; 240 case 2: 241 iq[jz-1] &= 0x3fffff; break; 242 } 243 } 244 if(ih==2) { 245 z = one - z; 246 if(carry!=0) z -= scalbn(one,q0); 247 } 248 } 249 250 /* check if recomputation is needed */ 251 if(z==zero) { 252 j = 0; 253 for (i=jz-1;i>=jk;i--) j |= iq[i]; 254 if(j==0) { /* need recomputation */ 255 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ 256 257 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ 258 f[jx+i] = (double) ipio2[jv+i]; 259 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; 260 q[i] = fw; 261 } 262 jz += k; 263 goto recompute; 264 } 265 } 266 267 /* chop off zero terms */ 268 if(z==0.0) { 269 jz -= 1; q0 -= 24; 270 while(iq[jz]==0) { jz--; q0-=24;} 271 } else { /* break z into 24-bit if necessary */ 272 z = scalbn(z,-q0); 273 if(z>=two24) { 274 fw = (double)((int)(twon24*z)); 275 iq[jz] = (int)(z-two24*fw); 276 jz += 1; q0 += 24; 277 iq[jz] = (int) fw; 278 } else iq[jz] = (int) z ; 279 } 280 281 /* convert integer "bit" chunk to floating-point value */ 282 fw = scalbn(one,q0); 283 for(i=jz;i>=0;i--) { 284 q[i] = fw*(double)iq[i]; fw*=twon24; 285 } 286 287 /* compute PIo2[0,...,jp]*q[jz,...,0] */ 288 for(i=jz;i>=0;i--) { 289 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; 290 fq[jz-i] = fw; 291 } 292 293 /* compress fq[] into y[] */ 294 switch(prec) { 295 case 0: 296 fw = 0.0; 297 for (i=jz;i>=0;i--) fw += fq[i]; 298 y[0] = (ih==0)? fw: -fw; 299 break; 300 case 1: 301 case 2: 302 fw = 0.0; 303 for (i=jz;i>=0;i--) fw += fq[i]; 304 y[0] = (ih==0)? fw: -fw; 305 fw = fq[0]-fw; 306 for (i=1;i<=jz;i++) fw += fq[i]; 307 y[1] = (ih==0)? fw: -fw; 308 break; 309 case 3: /* painful */ 310 for (i=jz;i>0;i--) { 311 fw = fq[i-1]+fq[i]; 312 fq[i] += fq[i-1]-fw; 313 fq[i-1] = fw; 314 } 315 for (i=jz;i>1;i--) { 316 fw = fq[i-1]+fq[i]; 317 fq[i] += fq[i-1]-fw; 318 fq[i-1] = fw; 319 } 320 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; 321 if(ih==0) { 322 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; 323 } else { 324 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; 325 } 326 } 327 return n&7; 328} 329