1 2 /* @(#)k_rem_pio2.c 1.3 95/01/18 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunSoft, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 */ 13 14 /* 15 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) 16 * double x[],y[]; int e0,nx,prec; int ipio2[]; 17 * 18 * __kernel_rem_pio2 return the last three digits of N with 19 * y = x - N*pi/2 20 * so that |y| < pi/2. 21 * 22 * The method is to compute the integer (mod 8) and fraction parts of 23 * (2/pi)*x without doing the full multiplication. In general we 24 * skip the part of the product that are known to be a huge integer ( 25 * more accurately, = 0 mod 8 ). Thus the number of operations are 26 * independent of the exponent of the input. 27 * 28 * (2/pi) is represented by an array of 24-bit integers in ipio2[]. 29 * 30 * Input parameters: 31 * x[] The input value (must be positive) is broken into nx 32 * pieces of 24-bit integers in double precision format. 33 * x[i] will be the i-th 24 bit of x. The scaled exponent 34 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 35 * match x's up to 24 bits. 36 * 37 * Example of breaking a double positive z into x[0]+x[1]+x[2]: 38 * e0 = ieee_ilogb(z)-23 39 * z = ieee_scalbn(z,-e0) 40 * for i = 0,1,2 41 * x[i] = ieee_floor(z) 42 * z = (z-x[i])*2**24 43 * 44 * 45 * y[] ouput result in an array of double precision numbers. 46 * The dimension of y[] is: 47 * 24-bit precision 1 48 * 53-bit precision 2 49 * 64-bit precision 2 50 * 113-bit precision 3 51 * The actual value is the sum of them. Thus for 113-bit 52 * precison, one may have to do something like: 53 * 54 * long double t,w,r_head, r_tail; 55 * t = (long double)y[2] + (long double)y[1]; 56 * w = (long double)y[0]; 57 * r_head = t+w; 58 * r_tail = w - (r_head - t); 59 * 60 * e0 The exponent of x[0] 61 * 62 * nx dimension of x[] 63 * 64 * prec an integer indicating the precision: 65 * 0 24 bits (single) 66 * 1 53 bits (double) 67 * 2 64 bits (extended) 68 * 3 113 bits (quad) 69 * 70 * ipio2[] 71 * integer array, contains the (24*i)-th to (24*i+23)-th 72 * bit of 2/pi after binary point. The corresponding 73 * floating value is 74 * 75 * ipio2[i] * 2^(-24(i+1)). 76 * 77 * External function: 78 * double ieee_scalbn(), ieee_floor(); 79 * 80 * 81 * Here is the description of some local variables: 82 * 83 * jk jk+1 is the initial number of terms of ipio2[] needed 84 * in the computation. The recommended value is 2,3,4, 85 * 6 for single, double, extended,and quad. 86 * 87 * jz local integer variable indicating the number of 88 * terms of ipio2[] used. 89 * 90 * jx nx - 1 91 * 92 * jv index for pointing to the suitable ipio2[] for the 93 * computation. In general, we want 94 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 95 * is an integer. Thus 96 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv 97 * Hence jv = max(0,(e0-3)/24). 98 * 99 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. 100 * 101 * q[] double array with integral value, representing the 102 * 24-bits chunk of the product of x and 2/pi. 103 * 104 * q0 the corresponding exponent of q[0]. Note that the 105 * exponent for q[i] would be q0-24*i. 106 * 107 * PIo2[] double precision array, obtained by cutting pi/2 108 * into 24 bits chunks. 109 * 110 * f[] ipio2[] in floating point 111 * 112 * iq[] integer array by breaking up q[] in 24-bits chunk. 113 * 114 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] 115 * 116 * ih integer. If >0 it indicates q[] is >= 0.5, hence 117 * it also indicates the *sign* of the result. 118 * 119 */ 120 121 122 /* 123 * Constants: 124 * The hexadecimal values are the intended ones for the following 125 * constants. The decimal values may be used, provided that the 126 * compiler will convert from decimal to binary accurately enough 127 * to produce the hexadecimal values shown. 128 */ 129 130 #include "fdlibm.h" 131 132 #ifdef __STDC__ 133 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ 134 #else 135 static int init_jk[] = {2,3,4,6}; 136 #endif 137 138 #ifdef __STDC__ 139 static const double PIo2[] = { 140 #else 141 static double PIo2[] = { 142 #endif 143 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ 144 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ 145 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ 146 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ 147 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ 148 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ 149 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ 150 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ 151 }; 152 153 #ifdef __STDC__ 154 static const double 155 #else 156 static double 157 #endif 158 zero = 0.0, 159 one = 1.0, 160 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 161 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ 162 163 #ifdef __STDC__ 164 int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) 165 #else 166 int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) 167 double x[], y[]; int e0,nx,prec; int ipio2[]; 168 #endif 169 { 170 int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; 171 double z,fw,f[20],fq[20],q[20]; 172 173 /* initialize jk*/ 174 jk = init_jk[prec]; 175 jp = jk; 176 177 /* determine jx,jv,q0, note that 3>q0 */ 178 jx = nx-1; 179 jv = (e0-3)/24; if(jv<0) jv=0; 180 q0 = e0-24*(jv+1); 181 182 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ 183 j = jv-jx; m = jx+jk; 184 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; 185 186 /* compute q[0],q[1],...q[jk] */ 187 for (i=0;i<=jk;i++) { 188 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; 189 } 190 191 jz = jk; 192 recompute: 193 /* distill q[] into iq[] reversingly */ 194 for(i=0,j=jz,z=q[jz];j>0;i++,j--) { 195 fw = (double)((int)(twon24* z)); 196 iq[i] = (int)(z-two24*fw); 197 z = q[j-1]+fw; 198 } 199 200 /* compute n */ 201 z = ieee_scalbn(z,q0); /* actual value of z */ 202 z -= 8.0*ieee_floor(z*0.125); /* trim off integer >= 8 */ 203 n = (int) z; 204 z -= (double)n; 205 ih = 0; 206 if(q0>0) { /* need iq[jz-1] to determine n */ 207 i = (iq[jz-1]>>(24-q0)); n += i; 208 iq[jz-1] -= i<<(24-q0); 209 ih = iq[jz-1]>>(23-q0); 210 } 211 else if(q0==0) ih = iq[jz-1]>>23; 212 else if(z>=0.5) ih=2; 213 214 if(ih>0) { /* q > 0.5 */ 215 n += 1; carry = 0; 216 for(i=0;i<jz ;i++) { /* compute 1-q */ 217 j = iq[i]; 218 if(carry==0) { 219 if(j!=0) { 220 carry = 1; iq[i] = 0x1000000- j; 221 } 222 } else iq[i] = 0xffffff - j; 223 } 224 if(q0>0) { /* rare case: chance is 1 in 12 */ 225 switch(q0) { 226 case 1: 227 iq[jz-1] &= 0x7fffff; break; 228 case 2: 229 iq[jz-1] &= 0x3fffff; break; 230 } 231 } 232 if(ih==2) { 233 z = one - z; 234 if(carry!=0) z -= ieee_scalbn(one,q0); 235 } 236 } 237 238 /* check if recomputation is needed */ 239 if(z==zero) { 240 j = 0; 241 for (i=jz-1;i>=jk;i--) j |= iq[i]; 242 if(j==0) { /* need recomputation */ 243 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ 244 245 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ 246 f[jx+i] = (double) ipio2[jv+i]; 247 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; 248 q[i] = fw; 249 } 250 jz += k; 251 goto recompute; 252 } 253 } 254 255 /* chop off zero terms */ 256 if(z==0.0) { 257 jz -= 1; q0 -= 24; 258 while(iq[jz]==0) { jz--; q0-=24;} 259 } else { /* break z into 24-bit if necessary */ 260 z = ieee_scalbn(z,-q0); 261 if(z>=two24) { 262 fw = (double)((int)(twon24*z)); 263 iq[jz] = (int)(z-two24*fw); 264 jz += 1; q0 += 24; 265 iq[jz] = (int) fw; 266 } else iq[jz] = (int) z ; 267 } 268 269 /* convert integer "bit" chunk to floating-point value */ 270 fw = ieee_scalbn(one,q0); 271 for(i=jz;i>=0;i--) { 272 q[i] = fw*(double)iq[i]; fw*=twon24; 273 } 274 275 /* compute PIo2[0,...,jp]*q[jz,...,0] */ 276 for(i=jz;i>=0;i--) { 277 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; 278 fq[jz-i] = fw; 279 } 280 281 /* compress fq[] into y[] */ 282 switch(prec) { 283 case 0: 284 fw = 0.0; 285 for (i=jz;i>=0;i--) fw += fq[i]; 286 y[0] = (ih==0)? fw: -fw; 287 break; 288 case 1: 289 case 2: 290 fw = 0.0; 291 for (i=jz;i>=0;i--) fw += fq[i]; 292 y[0] = (ih==0)? fw: -fw; 293 fw = fq[0]-fw; 294 for (i=1;i<=jz;i++) fw += fq[i]; 295 y[1] = (ih==0)? fw: -fw; 296 break; 297 case 3: /* painful */ 298 for (i=jz;i>0;i--) { 299 fw = fq[i-1]+fq[i]; 300 fq[i] += fq[i-1]-fw; 301 fq[i-1] = fw; 302 } 303 for (i=jz;i>1;i--) { 304 fw = fq[i-1]+fq[i]; 305 fq[i] += fq[i-1]-fw; 306 fq[i-1] = fw; 307 } 308 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; 309 if(ih==0) { 310 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; 311 } else { 312 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; 313 } 314 } 315 return n&7; 316 } 317
