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 #include <sys/cdefs.h> 14 #if defined(LIBM_SCCS) && !defined(lint) 15 __RCSID("$NetBSD: k_rem_pio2.c,v 1.10 2002/05/26 22:01:53 wiz Exp $"); 16 #endif 17 18 /* 19 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) 20 * double x[],y[]; int e0,nx,prec; int ipio2[]; 21 * 22 * __kernel_rem_pio2 return the last three digits of N with 23 * y = x - N*pi/2 24 * so that |y| < pi/2. 25 * 26 * The method is to compute the integer (mod 8) and fraction parts of 27 * (2/pi)*x without doing the full multiplication. In general we 28 * skip the part of the product that are known to be a huge integer ( 29 * more accurately, = 0 mod 8 ). Thus the number of operations are 30 * independent of the exponent of the input. 31 * 32 * (2/pi) is represented by an array of 24-bit integers in ipio2[]. 33 * 34 * Input parameters: 35 * x[] The input value (must be positive) is broken into nx 36 * pieces of 24-bit integers in double precision format. 37 * x[i] will be the i-th 24 bit of x. The scaled exponent 38 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 39 * match x's up to 24 bits. 40 * 41 * Example of breaking a double positive z into x[0]+x[1]+x[2]: 42 * e0 = ilogb(z)-23 43 * z = scalbn(z,-e0) 44 * for i = 0,1,2 45 * x[i] = floor(z) 46 * z = (z-x[i])*2**24 47 * 48 * 49 * y[] ouput result in an array of double precision numbers. 50 * The dimension of y[] is: 51 * 24-bit precision 1 52 * 53-bit precision 2 53 * 64-bit precision 2 54 * 113-bit precision 3 55 * The actual value is the sum of them. Thus for 113-bit 56 * precison, one may have to do something like: 57 * 58 * long double t,w,r_head, r_tail; 59 * t = (long double)y[2] + (long double)y[1]; 60 * w = (long double)y[0]; 61 * r_head = t+w; 62 * r_tail = w - (r_head - t); 63 * 64 * e0 The exponent of x[0] 65 * 66 * nx dimension of x[] 67 * 68 * prec an integer indicating the precision: 69 * 0 24 bits (single) 70 * 1 53 bits (double) 71 * 2 64 bits (extended) 72 * 3 113 bits (quad) 73 * 74 * ipio2[] 75 * integer array, contains the (24*i)-th to (24*i+23)-th 76 * bit of 2/pi after binary point. The corresponding 77 * floating value is 78 * 79 * ipio2[i] * 2^(-24(i+1)). 80 * 81 * External function: 82 * double scalbn(), floor(); 83 * 84 * 85 * Here is the description of some local variables: 86 * 87 * jk jk+1 is the initial number of terms of ipio2[] needed 88 * in the computation. The recommended value is 2,3,4, 89 * 6 for single, double, extended,and quad. 90 * 91 * jz local integer variable indicating the number of 92 * terms of ipio2[] used. 93 * 94 * jx nx - 1 95 * 96 * jv index for pointing to the suitable ipio2[] for the 97 * computation. In general, we want 98 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 99 * is an integer. Thus 100 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv 101 * Hence jv = max(0,(e0-3)/24). 102 * 103 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. 104 * 105 * q[] double array with integral value, representing the 106 * 24-bits chunk of the product of x and 2/pi. 107 * 108 * q0 the corresponding exponent of q[0]. Note that the 109 * exponent for q[i] would be q0-24*i. 110 * 111 * PIo2[] double precision array, obtained by cutting pi/2 112 * into 24 bits chunks. 113 * 114 * f[] ipio2[] in floating point 115 * 116 * iq[] integer array by breaking up q[] in 24-bits chunk. 117 * 118 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] 119 * 120 * ih integer. If >0 it indicates q[] is >= 0.5, hence 121 * it also indicates the *sign* of the result. 122 * 123 */ 124 125 126 /* 127 * Constants: 128 * The hexadecimal values are the intended ones for the following 129 * constants. The decimal values may be used, provided that the 130 * compiler will convert from decimal to binary accurately enough 131 * to produce the hexadecimal values shown. 132 */ 133 134 #include "math.h" 135 #include "math_private.h" 136 137 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ 138 139 static const double PIo2[] = { 140 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ 141 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ 142 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ 143 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ 144 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ 145 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ 146 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ 147 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ 148 }; 149 150 static const double 151 zero = 0.0, 152 one = 1.0, 153 two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 154 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ 155 156 int 157 __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2) 158 { 159 int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; 160 double z,fw,f[20],fq[20],q[20]; 161 162 /* initialize jk*/ 163 jk = init_jk[prec]; 164 jp = jk; 165 166 /* determine jx,jv,q0, note that 3>q0 */ 167 jx = nx-1; 168 jv = (e0-3)/24; if(jv<0) jv=0; 169 q0 = e0-24*(jv+1); 170 171 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ 172 j = jv-jx; m = jx+jk; 173 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; 174 175 /* compute q[0],q[1],...q[jk] */ 176 for (i=0;i<=jk;i++) { 177 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; 178 } 179 180 jz = jk; 181 recompute: 182 /* distill q[] into iq[] reversingly */ 183 for(i=0,j=jz,z=q[jz];j>0;i++,j--) { 184 fw = (double)((int32_t)(twon24* z)); 185 iq[i] = (int32_t)(z-two24*fw); 186 z = q[j-1]+fw; 187 } 188 189 /* compute n */ 190 z = scalbn(z,q0); /* actual value of z */ 191 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ 192 n = (int32_t) z; 193 z -= (double)n; 194 ih = 0; 195 if(q0>0) { /* need iq[jz-1] to determine n */ 196 i = (iq[jz-1]>>(24-q0)); n += i; 197 iq[jz-1] -= i<<(24-q0); 198 ih = iq[jz-1]>>(23-q0); 199 } 200 else if(q0==0) ih = iq[jz-1]>>23; 201 else if(z>=0.5) ih=2; 202 203 if(ih>0) { /* q > 0.5 */ 204 n += 1; carry = 0; 205 for(i=0;i<jz ;i++) { /* compute 1-q */ 206 j = iq[i]; 207 if(carry==0) { 208 if(j!=0) { 209 carry = 1; iq[i] = 0x1000000- j; 210 } 211 } else iq[i] = 0xffffff - j; 212 } 213 if(q0>0) { /* rare case: chance is 1 in 12 */ 214 switch(q0) { 215 case 1: 216 iq[jz-1] &= 0x7fffff; break; 217 case 2: 218 iq[jz-1] &= 0x3fffff; break; 219 } 220 } 221 if(ih==2) { 222 z = one - z; 223 if(carry!=0) z -= scalbn(one,q0); 224 } 225 } 226 227 /* check if recomputation is needed */ 228 if(z==zero) { 229 j = 0; 230 for (i=jz-1;i>=jk;i--) j |= iq[i]; 231 if(j==0) { /* need recomputation */ 232 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ 233 234 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ 235 f[jx+i] = (double) ipio2[jv+i]; 236 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; 237 q[i] = fw; 238 } 239 jz += k; 240 goto recompute; 241 } 242 } 243 244 /* chop off zero terms */ 245 if(z==0.0) { 246 jz -= 1; q0 -= 24; 247 while(iq[jz]==0) { jz--; q0-=24;} 248 } else { /* break z into 24-bit if necessary */ 249 z = scalbn(z,-q0); 250 if(z>=two24) { 251 fw = (double)((int32_t)(twon24*z)); 252 iq[jz] = (int32_t)(z-two24*fw); 253 jz += 1; q0 += 24; 254 iq[jz] = (int32_t) fw; 255 } else iq[jz] = (int32_t) z ; 256 } 257 258 /* convert integer "bit" chunk to floating-point value */ 259 fw = scalbn(one,q0); 260 for(i=jz;i>=0;i--) { 261 q[i] = fw*(double)iq[i]; fw*=twon24; 262 } 263 264 /* compute PIo2[0,...,jp]*q[jz,...,0] */ 265 for(i=jz;i>=0;i--) { 266 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; 267 fq[jz-i] = fw; 268 } 269 270 /* compress fq[] into y[] */ 271 switch(prec) { 272 case 0: 273 fw = 0.0; 274 for (i=jz;i>=0;i--) fw += fq[i]; 275 y[0] = (ih==0)? fw: -fw; 276 break; 277 case 1: 278 case 2: 279 fw = 0.0; 280 for (i=jz;i>=0;i--) fw += fq[i]; 281 y[0] = (ih==0)? fw: -fw; 282 fw = fq[0]-fw; 283 for (i=1;i<=jz;i++) fw += fq[i]; 284 y[1] = (ih==0)? fw: -fw; 285 break; 286 case 3: /* painful */ 287 for (i=jz;i>0;i--) { 288 fw = fq[i-1]+fq[i]; 289 fq[i] += fq[i-1]-fw; 290 fq[i-1] = fw; 291 } 292 for (i=jz;i>1;i--) { 293 fw = fq[i-1]+fq[i]; 294 fq[i] += fq[i-1]-fw; 295 fq[i-1] = fw; 296 } 297 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; 298 if(ih==0) { 299 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; 300 } else { 301 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; 302 } 303 } 304 return n&7; 305 } 306