Atlas - k_rem_pio2.c
Home / ext / SDL2 / src / libm Lines: 1 | Size: 8705 bytes [Download] [Show on GitHub] [Search similar files] [Raw] [Raw (proxy)][FILE BEGIN]1/* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunPro, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 12/* 13 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) 14 * double x[],y[]; int e0,nx,prec; int ipio2[]; 15 * 16 * __kernel_rem_pio2 return the last three digits of N with 17 * y = x - N*pi/2 18 * so that |y| < pi/2. 19 * 20 * The method is to compute the integer (mod 8) and fraction parts of 21 * (2/pi)*x without doing the full multiplication. In general we 22 * skip the part of the product that are known to be a huge integer ( 23 * more accurately, = 0 mod 8 ). Thus the number of operations are 24 * independent of the exponent of the input. 25 * 26 * (2/pi) is represented by an array of 24-bit integers in ipio2[]. 27 * 28 * Input parameters: 29 * x[] The input value (must be positive) is broken into nx 30 * pieces of 24-bit integers in double precision format. 31 * x[i] will be the i-th 24 bit of x. The scaled exponent 32 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 33 * match x's up to 24 bits. 34 * 35 * Example of breaking a double positive z into x[0]+x[1]+x[2]: 36 * e0 = ilogb(z)-23 37 * z = scalbn(z,-e0) 38 * for i = 0,1,2 39 * x[i] = floor(z) 40 * z = (z-x[i])*2**24 41 * 42 * 43 * y[] ouput result in an array of double precision numbers. 44 * The dimension of y[] is: 45 * 24-bit precision 1 46 * 53-bit precision 2 47 * 64-bit precision 2 48 * 113-bit precision 3 49 * The actual value is the sum of them. Thus for 113-bit 50 * precison, one may have to do something like: 51 * 52 * long double t,w,r_head, r_tail; 53 * t = (long double)y[2] + (long double)y[1]; 54 * w = (long double)y[0]; 55 * r_head = t+w; 56 * r_tail = w - (r_head - t); 57 * 58 * e0 The exponent of x[0] 59 * 60 * nx dimension of x[] 61 * 62 * prec an integer indicating the precision: 63 * 0 24 bits (single) 64 * 1 53 bits (double) 65 * 2 64 bits (extended) 66 * 3 113 bits (quad) 67 * 68 * ipio2[] 69 * integer array, contains the (24*i)-th to (24*i+23)-th 70 * bit of 2/pi after binary point. The corresponding 71 * floating value is 72 * 73 * ipio2[i] * 2^(-24(i+1)). 74 * 75 * External function: 76 * double scalbn(), floor(); 77 * 78 * 79 * Here is the description of some local variables: 80 * 81 * jk jk+1 is the initial number of terms of ipio2[] needed 82 * in the computation. The recommended value is 2,3,4, 83 * 6 for single, double, extended,and quad. 84 * 85 * jz local integer variable indicating the number of 86 * terms of ipio2[] used. 87 * 88 * jx nx - 1 89 * 90 * jv index for pointing to the suitable ipio2[] for the 91 * computation. In general, we want 92 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 93 * is an integer. Thus 94 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv 95 * Hence jv = max(0,(e0-3)/24). 96 * 97 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. 98 * 99 * q[] double array with integral value, representing the 100 * 24-bits chunk of the product of x and 2/pi. 101 * 102 * q0 the corresponding exponent of q[0]. Note that the 103 * exponent for q[i] would be q0-24*i. 104 * 105 * PIo2[] double precision array, obtained by cutting pi/2 106 * into 24 bits chunks. 107 * 108 * f[] ipio2[] in floating point 109 * 110 * iq[] integer array by breaking up q[] in 24-bits chunk. 111 * 112 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] 113 * 114 * ih integer. If >0 it indicates q[] is >= 0.5, hence 115 * it also indicates the *sign* of the result. 116 * 117 */ 118 119 120/* 121 * Constants: 122 * The hexadecimal values are the intended ones for the following 123 * constants. The decimal values may be used, provided that the 124 * compiler will convert from decimal to binary accurately enough 125 * to produce the hexadecimal values shown. 126 */ 127 128#include "math_libm.h" 129#include "math_private.h" 130 131#include "SDL_assert.h" 132 133static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ 134 135static const double PIo2[] = { 136 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ 137 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ 138 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ 139 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ 140 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ 141 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ 142 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ 143 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ 144}; 145 146static const double 147zero = 0.0, 148one = 1.0, 149two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 150twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ 151 152int32_t attribute_hidden __kernel_rem_pio2(double *x, double *y, int e0, int nx, const unsigned int prec, const int32_t *ipio2) 153{ 154 int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; 155 double z,fw,f[20],fq[20],q[20]; 156 157 if (nx < 1) { 158 return 0; 159 } 160 161 /* initialize jk*/ 162 SDL_assert(prec < SDL_arraysize(init_jk)); 163 jk = init_jk[prec]; 164 SDL_assert(jk > 0); 165 jp = jk; 166 167 /* determine jx,jv,q0, note that 3>q0 */ 168 jx = nx-1; 169 jv = (e0-3)/24; if(jv<0) jv=0; 170 q0 = e0-24*(jv+1); 171 172 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ 173 j = jv-jx; m = jx+jk; 174 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; 175 if ((m+1) < SDL_arraysize(f)) { 176 SDL_memset(&f[m+1], 0, sizeof (f) - ((m+1) * sizeof (f[0]))); 177 } 178 179 /* compute q[0],q[1],...q[jk] */ 180 for (i=0;i<=jk;i++) { 181 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; 182 q[i] = fw; 183 } 184 185 jz = jk; 186recompute: 187 /* distill q[] into iq[] reversingly */ 188 for(i=0,j=jz,z=q[jz];j>0;i++,j--) { 189 fw = (double)((int32_t)(twon24* z)); 190 iq[i] = (int32_t)(z-two24*fw); 191 z = q[j-1]+fw; 192 } 193 if (jz < SDL_arraysize(iq)) { 194 SDL_memset(&iq[jz], 0, sizeof (q) - (jz * sizeof (iq[0]))); 195 } 196 197 /* compute n */ 198 z = scalbn(z,q0); /* actual value of z */ 199 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ 200 n = (int32_t) z; 201 z -= (double)n; 202 ih = 0; 203 if(q0>0) { /* need iq[jz-1] to determine n */ 204 i = (iq[jz-1]>>(24-q0)); n += i; 205 iq[jz-1] -= i<<(24-q0); 206 ih = iq[jz-1]>>(23-q0); 207 } 208 else if(q0==0) ih = iq[jz-1]>>23; 209 else if(z>=0.5) ih=2; 210 211 if(ih>0) { /* q > 0.5 */ 212 n += 1; carry = 0; 213 for(i=0;i<jz ;i++) { /* compute 1-q */ 214 j = iq[i]; 215 if(carry==0) { 216 if(j!=0) { 217 carry = 1; iq[i] = 0x1000000- j; 218 } 219 } else iq[i] = 0xffffff - j; 220 } 221 if(q0>0) { /* rare case: chance is 1 in 12 */ 222 switch(q0) { 223 case 1: 224 iq[jz-1] &= 0x7fffff; break; 225 case 2: 226 iq[jz-1] &= 0x3fffff; break; 227 } 228 } 229 if(ih==2) { 230 z = one - z; 231 if(carry!=0) z -= scalbn(one,q0); 232 } 233 } 234 235 /* check if recomputation is needed */ 236 if(z==zero) { 237 j = 0; 238 for (i=jz-1;i>=jk;i--) j |= iq[i]; 239 if(j==0) { /* need recomputation */ 240 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ 241 242 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ 243 f[jx+i] = (double) ipio2[jv+i]; 244 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; 245 q[i] = fw; 246 } 247 jz += k; 248 goto recompute; 249 } 250 } 251 252 /* chop off zero terms */ 253 if(z==0.0) { 254 jz -= 1; q0 -= 24; 255 SDL_assert(jz >= 0); 256 while(iq[jz]==0) { jz--; SDL_assert(jz >= 0); q0-=24;} 257 } else { /* break z into 24-bit if necessary */ 258 z = scalbn(z,-q0); 259 if(z>=two24) { 260 fw = (double)((int32_t)(twon24*z)); 261 iq[jz] = (int32_t)(z-two24*fw); 262 jz += 1; q0 += 24; 263 iq[jz] = (int32_t) fw; 264 } else iq[jz] = (int32_t) z ; 265 } 266 267 /* convert integer "bit" chunk to floating-point value */ 268 fw = scalbn(one,q0); 269 for(i=jz;i>=0;i--) { 270 q[i] = fw*(double)iq[i]; fw*=twon24; 271 } 272 273 /* compute PIo2[0,...,jp]*q[jz,...,0] */ 274 for(i=jz;i>=0;i--) { 275 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; 276 fq[jz-i] = fw; 277 } 278 if ((jz+1) < SDL_arraysize(f)) { 279 SDL_memset(&fq[jz+1], 0, sizeof (fq) - ((jz+1) * sizeof (fq[0]))); 280 } 281 282 /* compress fq[] into y[] */ 283 switch(prec) { 284 case 0: 285 fw = 0.0; 286 for (i=jz;i>=0;i--) fw += fq[i]; 287 y[0] = (ih==0)? fw: -fw; 288 break; 289 case 1: 290 case 2: 291 fw = 0.0; 292 for (i=jz;i>=0;i--) fw += fq[i]; 293 y[0] = (ih==0)? fw: -fw; 294 fw = fq[0]-fw; 295 for (i=1;i<=jz;i++) fw += fq[i]; 296 y[1] = (ih==0)? fw: -fw; 297 break; 298 case 3: /* painful */ 299 for (i=jz;i>0;i--) { 300 fw = fq[i-1]+fq[i]; 301 fq[i] += fq[i-1]-fw; 302 fq[i-1] = fw; 303 } 304 for (i=jz;i>1;i--) { 305 fw = fq[i-1]+fq[i]; 306 fq[i] += fq[i-1]-fw; 307 fq[i-1] = fw; 308 } 309 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; 310 if(ih==0) { 311 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; 312 } else { 313 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; 314 } 315 } 316 return n&7; 317} 318[FILE END](C) 2025 0x4248 (C) 2025 4248 Media and 4248 Systems, All part of 0x4248 See LICENCE files for more information. 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