1 2 /* @(#)k_rem_pio2.c 5.1 93/09/24 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunPro, 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 __ieee754_remainder(double x,double p)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 = ilogb(z)-23 39 * z = scalbn(z,-e0) 40 * for i = 0,1,2 41 * x[i] = 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 scalbn(), 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 #ifndef _DOUBLE_IS_32BITS 133 134 #ifdef __STDC__ 135 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ 136 #else 137 static int init_jk[] = {2,3,4,6}; 138 #endif 139 140 #ifdef __STDC__ 141 static const double PIo2[] = { 142 #else 143 static double PIo2[] = { 144 #endif 145 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ 146 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ 147 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ 148 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ 149 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ 150 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ 151 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ 152 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ 153 }; 154 155 #ifdef __STDC__ 156 static const double 157 #else 158 static double 159 #endif 160 zero = 0.0, 161 one = 1.0, 162 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 163 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ 164 165 #ifdef __STDC__ 166 int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2) 167 #else 168 int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) 169 double x[], y[]; int e0,nx,prec; int32_t ipio2[]; 170 #endif 171 { 172 int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; 173 double z,fw,f[20],fq[20],q[20]; 174 175 /* initialize jk*/ 176 jk = init_jk[prec]; 177 jp = jk; 178 179 /* determine jx,jv,q0, note that 3>q0 */ 180 jx = nx-1; 181 jv = (e0-3)/24; if(jv<0) jv=0; 182 q0 = e0-24*(jv+1); 183 184 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ 185 j = jv-jx; m = jx+jk; 186 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; 187 188 /* compute q[0],q[1],...q[jk] */ 189 for (i=0;i<=jk;i++) { 190 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; 191 } 192 193 jz = jk; 194 recompute: 195 /* distill q[] into iq[] reversingly */ 196 for(i=0,j=jz,z=q[jz];j>0;i++,j--) { 197 fw = (double)((int32_t)(twon24* z)); 198 iq[i] = (int32_t)(z-two24*fw); 199 z = q[j-1]+fw; 200 } 201 202 /* compute n */ 203 z = scalbn(z,(int)q0); /* actual value of z */ 204 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ 205 n = (int32_t) z; 206 z -= (double)n; 207 ih = 0; 208 if(q0>0) { /* need iq[jz-1] to determine n */ 209 i = (iq[jz-1]>>(24-q0)); n += i; 210 iq[jz-1] -= i<<(24-q0); 211 ih = iq[jz-1]>>(23-q0); 212 } 213 else if(q0==0) ih = iq[jz-1]>>23; 214 else if(z>=0.5) ih=2; 215 216 if(ih>0) { /* q > 0.5 */ 217 n += 1; carry = 0; 218 for(i=0;i<jz ;i++) { /* compute 1-q */ 219 j = iq[i]; 220 if(carry==0) { 221 if(j!=0) { 222 carry = 1; iq[i] = 0x1000000- j; 223 } 224 } else iq[i] = 0xffffff - j; 225 } 226 if(q0>0) { /* rare case: chance is 1 in 12 */ 227 switch(q0) { 228 case 1: 229 iq[jz-1] &= 0x7fffff; break; 230 case 2: 231 iq[jz-1] &= 0x3fffff; break; 232 } 233 } 234 if(ih==2) { 235 z = one - z; 236 if(carry!=0) z -= scalbn(one,(int)q0); 237 } 238 } 239 240 /* check if recomputation is needed */ 241 if(z==zero) { 242 j = 0; 243 for (i=jz-1;i>=jk;i--) j |= iq[i]; 244 if(j==0) { /* need recomputation */ 245 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ 246 247 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ 248 f[jx+i] = (double) ipio2[jv+i]; 249 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; 250 q[i] = fw; 251 } 252 jz += k; 253 goto recompute; 254 } 255 } 256 257 /* chop off zero terms */ 258 if(z==0.0) { 259 jz -= 1; q0 -= 24; 260 while(iq[jz]==0) { jz--; q0-=24;} 261 } else { /* break z into 24-bit if necessary */ 262 z = scalbn(z,-(int)q0); 263 if(z>=two24) { 264 fw = (double)((int32_t)(twon24*z)); 265 iq[jz] = (int32_t)(z-two24*fw); 266 jz += 1; q0 += 24; 267 iq[jz] = (int32_t) fw; 268 } else iq[jz] = (int32_t) z ; 269 } 270 271 /* convert integer "bit" chunk to floating-point value */ 272 fw = scalbn(one,(int)q0); 273 for(i=jz;i>=0;i--) { 274 q[i] = fw*(double)iq[i]; fw*=twon24; 275 } 276 277 /* compute PIo2[0,...,jp]*q[jz,...,0] */ 278 for(i=jz;i>=0;i--) { 279 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; 280 fq[jz-i] = fw; 281 } 282 283 /* compress fq[] into y[] */ 284 switch(prec) { 285 case 0: 286 fw = 0.0; 287 for (i=jz;i>=0;i--) fw += fq[i]; 288 y[0] = (ih==0)? fw: -fw; 289 break; 290 case 1: 291 case 2: 292 fw = 0.0; 293 for (i=jz;i>=0;i--) fw += fq[i]; 294 y[0] = (ih==0)? fw: -fw; 295 fw = fq[0]-fw; 296 for (i=1;i<=jz;i++) fw += fq[i]; 297 y[1] = (ih==0)? fw: -fw; 298 break; 299 case 3: /* painful */ 300 for (i=jz;i>0;i--) { 301 fw = fq[i-1]+fq[i]; 302 fq[i] += fq[i-1]-fw; 303 fq[i-1] = fw; 304 } 305 for (i=jz;i>1;i--) { 306 fw = fq[i-1]+fq[i]; 307 fq[i] += fq[i-1]-fw; 308 fq[i-1] = fw; 309 } 310 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; 311 if(ih==0) { 312 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; 313 } else { 314 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; 315 } 316 } 317 return n&7; 318 } 319 320 #endif /* defined(_DOUBLE_IS_32BITS) */ 321