Atlas - k_rem_pio2.c
Home / ext / SDL / src / libm Lines: 1 | Size: 8603 bytes [Download] [Show on GitHub] [Search similar files] [Raw] [Raw (proxy)][FILE BEGIN]1#include "SDL_internal.h" 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/* 14 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) 15 * double x[],y[]; int e0,nx,prec; int ipio2[]; 16 * 17 * __kernel_rem_pio2 return the last three digits of N with 18 * y = x - N*pi/2 19 * so that |y| < pi/2. 20 * 21 * The method is to compute the integer (mod 8) and fraction parts of 22 * (2/pi)*x without doing the full multiplication. In general we 23 * skip the part of the product that are known to be a huge integer ( 24 * more accurately, = 0 mod 8 ). Thus the number of operations are 25 * independent of the exponent of the input. 26 * 27 * (2/pi) is represented by an array of 24-bit integers in ipio2[]. 28 * 29 * Input parameters: 30 * x[] The input value (must be positive) is broken into nx 31 * pieces of 24-bit integers in double precision format. 32 * x[i] will be the i-th 24 bit of x. The scaled exponent 33 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 34 * match x's up to 24 bits. 35 * 36 * Example of breaking a double positive z into x[0]+x[1]+x[2]: 37 * e0 = ilogb(z)-23 38 * z = scalbn(z,-e0) 39 * for i = 0,1,2 40 * x[i] = floor(z) 41 * z = (z-x[i])*2**24 42 * 43 * 44 * y[] ouput result in an array of double precision numbers. 45 * The dimension of y[] is: 46 * 24-bit precision 1 47 * 53-bit precision 2 48 * 64-bit precision 2 49 * 113-bit precision 3 50 * The actual value is the sum of them. Thus for 113-bit 51 * precision, one may have to do something like: 52 * 53 * long double t,w,r_head, r_tail; 54 * t = (long double)y[2] + (long double)y[1]; 55 * w = (long double)y[0]; 56 * r_head = t+w; 57 * r_tail = w - (r_head - t); 58 * 59 * e0 The exponent of x[0] 60 * 61 * nx dimension of x[] 62 * 63 * prec an integer indicating the precision: 64 * 0 24 bits (single) 65 * 1 53 bits (double) 66 * 2 64 bits (extended) 67 * 3 113 bits (quad) 68 * 69 * ipio2[] 70 * integer array, contains the (24*i)-th to (24*i+23)-th 71 * bit of 2/pi after binary point. The corresponding 72 * floating value is 73 * 74 * ipio2[i] * 2^(-24(i+1)). 75 * 76 * External function: 77 * double scalbn(), floor(); 78 * 79 * 80 * Here is the description of some local variables: 81 * 82 * jk jk+1 is the initial number of terms of ipio2[] needed 83 * in the computation. The recommended value is 2,3,4, 84 * 6 for single, double, extended,and quad. 85 * 86 * jz local integer variable indicating the number of 87 * terms of ipio2[] used. 88 * 89 * jx nx - 1 90 * 91 * jv index for pointing to the suitable ipio2[] for the 92 * computation. In general, we want 93 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 94 * is an integer. Thus 95 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv 96 * Hence jv = max(0,(e0-3)/24). 97 * 98 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. 99 * 100 * q[] double array with integral value, representing the 101 * 24-bits chunk of the product of x and 2/pi. 102 * 103 * q0 the corresponding exponent of q[0]. Note that the 104 * exponent for q[i] would be q0-24*i. 105 * 106 * PIo2[] double precision array, obtained by cutting pi/2 107 * into 24 bits chunks. 108 * 109 * f[] ipio2[] in floating point 110 * 111 * iq[] integer array by breaking up q[] in 24-bits chunk. 112 * 113 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] 114 * 115 * ih integer. If >0 it indicates q[] is >= 0.5, hence 116 * it also indicates the *sign* of the result. 117 * 118 */ 119 120 121/* 122 * Constants: 123 * The hexadecimal values are the intended ones for the following 124 * constants. The decimal values may be used, provided that the 125 * compiler will convert from decimal to binary accurately enough 126 * to produce the hexadecimal values shown. 127 */ 128 129#include "math_libm.h" 130#include "math_private.h" 131 132 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(const 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 (iq) - (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 SDL_zero(fq); 275 for(i=jz;i>=0;i--) { 276 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; 277 fq[jz-i] = fw; 278 } 279 280 /* compress fq[] into y[] */ 281 switch(prec) { 282 case 0: 283 fw = 0.0; 284 for (i=jz;i>=0;i--) fw += fq[i]; 285 y[0] = (ih==0)? fw: -fw; 286 break; 287 case 1: 288 case 2: 289 fw = 0.0; 290 for (i=jz;i>=0;i--) fw += fq[i]; 291 y[0] = (ih==0)? fw: -fw; 292 fw = fq[0]-fw; 293 for (i=1;i<=jz;i++) fw += fq[i]; 294 y[1] = (ih==0)? fw: -fw; 295 break; 296 case 3: /* painful */ 297 for (i=jz;i>0;i--) { 298 fw = fq[i-1]+fq[i]; 299 fq[i] += fq[i-1]-fw; 300 fq[i-1] = fw; 301 } 302 for (i=jz;i>1;i--) { 303 fw = fq[i-1]+fq[i]; 304 fq[i] += fq[i-1]-fw; 305 fq[i-1] = fw; 306 } 307 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; 308 if(ih==0) { 309 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; 310 } else { 311 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; 312 } 313 } 314 return n&7; 315} 316[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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