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