1 /* @(#)k_rem_pio2.c 5.1 93/09/24 */ 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 #ifndef lint 14 static char rcsid[] = "$Id: k_rem_pio2.c,v 1.3 1994/02/18 02:25:58 jtc Exp $"; 15 #endif 16 17 /* 18 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) 19 * double x[],y[]; int e0,nx,prec; int ipio2[]; 20 * 21 * __kernel_rem_pio2 return the last three digits of N with 22 * y = x - N*pi/2 23 * so that |y| < pi/2. 24 * 25 * The method is to compute the integer (mod 8) and fraction parts of 26 * (2/pi)*x without doing the full multiplication. In general we 27 * skip the part of the product that are known to be a huge integer ( 28 * more accurately, = 0 mod 8 ). Thus the number of operations are 29 * independent of the exponent of the input. 30 * 31 * (2/pi) is represented by an array of 24-bit integers in ipio2[]. 32 * 33 * Input parameters: 34 * x[] The input value (must be positive) is broken into nx 35 * pieces of 24-bit integers in double precision format. 36 * x[i] will be the i-th 24 bit of x. The scaled exponent 37 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 38 * match x's up to 24 bits. 39 * 40 * Example of breaking a double positive z into x[0]+x[1]+x[2]: 41 * e0 = ilogb(z)-23 42 * z = scalbn(z,-e0) 43 * for i = 0,1,2 44 * x[i] = floor(z) 45 * z = (z-x[i])*2**24 46 * 47 * 48 * y[] ouput result in an array of double precision numbers. 49 * The dimension of y[] is: 50 * 24-bit precision 1 51 * 53-bit precision 2 52 * 64-bit precision 2 53 * 113-bit precision 3 54 * The actual value is the sum of them. Thus for 113-bit 55 * precison, one may have to do something like: 56 * 57 * long double t,w,r_head, r_tail; 58 * t = (long double)y[2] + (long double)y[1]; 59 * w = (long double)y[0]; 60 * r_head = t+w; 61 * r_tail = w - (r_head - t); 62 * 63 * e0 The exponent of x[0] 64 * 65 * nx dimension of x[] 66 * 67 * prec an integer indicating the precision: 68 * 0 24 bits (single) 69 * 1 53 bits (double) 70 * 2 64 bits (extended) 71 * 3 113 bits (quad) 72 * 73 * ipio2[] 74 * integer array, contains the (24*i)-th to (24*i+23)-th 75 * bit of 2/pi after binary point. The corresponding 76 * floating value is 77 * 78 * ipio2[i] * 2^(-24(i+1)). 79 * 80 * External function: 81 * double scalbn(), floor(); 82 * 83 * 84 * Here is the description of some local variables: 85 * 86 * jk jk+1 is the initial number of terms of ipio2[] needed 87 * in the computation. The recommended value is 2,3,4, 88 * 6 for single, double, extended,and quad. 89 * 90 * jz local integer variable indicating the number of 91 * terms of ipio2[] used. 92 * 93 * jx nx - 1 94 * 95 * jv index for pointing to the suitable ipio2[] for the 96 * computation. In general, we want 97 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 98 * is an integer. Thus 99 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv 100 * Hence jv = max(0,(e0-3)/24). 101 * 102 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. 103 * 104 * q[] double array with integral value, representing the 105 * 24-bits chunk of the product of x and 2/pi. 106 * 107 * q0 the corresponding exponent of q[0]. Note that the 108 * exponent for q[i] would be q0-24*i. 109 * 110 * PIo2[] double precision array, obtained by cutting pi/2 111 * into 24 bits chunks. 112 * 113 * f[] ipio2[] in floating point 114 * 115 * iq[] integer array by breaking up q[] in 24-bits chunk. 116 * 117 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] 118 * 119 * ih integer. If >0 it indicates q[] is >= 0.5, hence 120 * it also indicates the *sign* of the result. 121 * 122 */ 123 124 125 /* 126 * Constants: 127 * The hexadecimal values are the intended ones for the following 128 * constants. The decimal values may be used, provided that the 129 * compiler will convert from decimal to binary accurately enough 130 * to produce the hexadecimal values shown. 131 */ 132 133 #include <math.h> 134 135 #ifdef __STDC__ 136 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ 137 #else 138 static int init_jk[] = {2,3,4,6}; 139 #endif 140 141 #ifdef __STDC__ 142 static const double PIo2[] = { 143 #else 144 static double PIo2[] = { 145 #endif 146 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ 147 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ 148 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ 149 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ 150 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ 151 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ 152 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ 153 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ 154 }; 155 156 #ifdef __STDC__ 157 static const double 158 #else 159 static double 160 #endif 161 zero = 0.0, 162 one = 1.0, 163 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 164 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ 165 166 #ifdef __STDC__ 167 int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) 168 #else 169 int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) 170 double x[], y[]; int e0,nx,prec; int ipio2[]; 171 #endif 172 { 173 int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; 174 double z,fw,f[20],fq[20],q[20]; 175 176 /* initialize jk*/ 177 jk = init_jk[prec]; 178 jp = jk; 179 180 /* determine jx,jv,q0, note that 3>q0 */ 181 jx = nx-1; 182 jv = (e0-3)/24; if(jv<0) jv=0; 183 q0 = e0-24*(jv+1); 184 185 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ 186 j = jv-jx; m = jx+jk; 187 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; 188 189 /* compute q[0],q[1],...q[jk] */ 190 for (i=0;i<=jk;i++) { 191 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; 192 } 193 194 jz = jk; 195 recompute: 196 /* distill q[] into iq[] reversingly */ 197 for(i=0,j=jz,z=q[jz];j>0;i++,j--) { 198 fw = (double)((int)(twon24* z)); 199 iq[i] = (int)(z-two24*fw); 200 z = q[j-1]+fw; 201 } 202 203 /* compute n */ 204 z = scalbn(z,q0); /* actual value of z */ 205 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ 206 n = (int) z; 207 z -= (double)n; 208 ih = 0; 209 if(q0>0) { /* need iq[jz-1] to determine n */ 210 i = (iq[jz-1]>>(24-q0)); n += i; 211 iq[jz-1] -= i<<(24-q0); 212 ih = iq[jz-1]>>(23-q0); 213 } 214 else if(q0==0) ih = iq[jz-1]>>23; 215 else if(z>=0.5) ih=2; 216 217 if(ih>0) { /* q > 0.5 */ 218 n += 1; carry = 0; 219 for(i=0;i<jz ;i++) { /* compute 1-q */ 220 j = iq[i]; 221 if(carry==0) { 222 if(j!=0) { 223 carry = 1; iq[i] = 0x1000000- j; 224 } 225 } else iq[i] = 0xffffff - j; 226 } 227 if(q0>0) { /* rare case: chance is 1 in 12 */ 228 switch(q0) { 229 case 1: 230 iq[jz-1] &= 0x7fffff; break; 231 case 2: 232 iq[jz-1] &= 0x3fffff; break; 233 } 234 } 235 if(ih==2) { 236 z = one - z; 237 if(carry!=0) z -= scalbn(one,q0); 238 } 239 } 240 241 /* check if recomputation is needed */ 242 if(z==zero) { 243 j = 0; 244 for (i=jz-1;i>=jk;i--) j |= iq[i]; 245 if(j==0) { /* need recomputation */ 246 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ 247 248 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ 249 f[jx+i] = (double) ipio2[jv+i]; 250 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; 251 q[i] = fw; 252 } 253 jz += k; 254 goto recompute; 255 } 256 } 257 258 /* chop off zero terms */ 259 if(z==0.0) { 260 jz -= 1; q0 -= 24; 261 while(iq[jz]==0) { jz--; q0-=24;} 262 } else { /* break z into 24-bit if necessary */ 263 z = scalbn(z,-q0); 264 if(z>=two24) { 265 fw = (double)((int)(twon24*z)); 266 iq[jz] = (int)(z-two24*fw); 267 jz += 1; q0 += 24; 268 iq[jz] = (int) fw; 269 } else iq[jz] = (int) z ; 270 } 271 272 /* convert integer "bit" chunk to floating-point value */ 273 fw = scalbn(one,q0); 274 for(i=jz;i>=0;i--) { 275 q[i] = fw*(double)iq[i]; fw*=twon24; 276 } 277 278 /* compute PIo2[0,...,jp]*q[jz,...,0] */ 279 for(i=jz;i>=0;i--) { 280 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; 281 fq[jz-i] = fw; 282 } 283 284 /* compress fq[] into y[] */ 285 switch(prec) { 286 case 0: 287 fw = 0.0; 288 for (i=jz;i>=0;i--) fw += fq[i]; 289 y[0] = (ih==0)? fw: -fw; 290 break; 291 case 1: 292 case 2: 293 fw = 0.0; 294 for (i=jz;i>=0;i--) fw += fq[i]; 295 y[0] = (ih==0)? fw: -fw; 296 fw = fq[0]-fw; 297 for (i=1;i<=jz;i++) fw += fq[i]; 298 y[1] = (ih==0)? fw: -fw; 299 break; 300 case 3: /* painful */ 301 for (i=jz;i>0;i--) { 302 fw = fq[i-1]+fq[i]; 303 fq[i] += fq[i-1]-fw; 304 fq[i-1] = fw; 305 } 306 for (i=jz;i>1;i--) { 307 fw = fq[i-1]+fq[i]; 308 fq[i] += fq[i-1]-fw; 309 fq[i-1] = fw; 310 } 311 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; 312 if(ih==0) { 313 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; 314 } else { 315 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; 316 } 317 } 318 return n&7; 319 } 320