k_rem_pio2.c revision 175507
1141296Sdas 2141296Sdas/* @(#)k_rem_pio2.c 1.3 95/01/18 */ 32116Sjkh/* 42116Sjkh * ==================================================== 52116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 62116Sjkh * 7141296Sdas * Developed at SunSoft, a Sun Microsystems, Inc. business. 82116Sjkh * Permission to use, copy, modify, and distribute this 9141296Sdas * software is freely granted, provided that this notice 102116Sjkh * is preserved. 112116Sjkh * ==================================================== 122116Sjkh */ 132116Sjkh 14175499Sbde#include <sys/cdefs.h> 15175499Sbde__FBSDID("$FreeBSD: head/lib/msun/src/k_rem_pio2.c 175507 2008-01-20 04:09:44Z bde $"); 162116Sjkh 172116Sjkh/* 182116Sjkh * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) 192116Sjkh * double x[],y[]; int e0,nx,prec; int ipio2[]; 20141296Sdas * 21141296Sdas * __kernel_rem_pio2 return the last three digits of N with 222116Sjkh * y = x - N*pi/2 232116Sjkh * so that |y| < pi/2. 242116Sjkh * 25141296Sdas * The method is to compute the integer (mod 8) and fraction parts of 262116Sjkh * (2/pi)*x without doing the full multiplication. In general we 272116Sjkh * skip the part of the product that are known to be a huge integer ( 282116Sjkh * more accurately, = 0 mod 8 ). Thus the number of operations are 292116Sjkh * independent of the exponent of the input. 302116Sjkh * 312116Sjkh * (2/pi) is represented by an array of 24-bit integers in ipio2[]. 322116Sjkh * 332116Sjkh * Input parameters: 34141296Sdas * x[] The input value (must be positive) is broken into nx 352116Sjkh * pieces of 24-bit integers in double precision format. 36141296Sdas * x[i] will be the i-th 24 bit of x. The scaled exponent 37141296Sdas * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 382116Sjkh * match x's up to 24 bits. 392116Sjkh * 402116Sjkh * Example of breaking a double positive z into x[0]+x[1]+x[2]: 412116Sjkh * e0 = ilogb(z)-23 422116Sjkh * z = scalbn(z,-e0) 432116Sjkh * for i = 0,1,2 442116Sjkh * x[i] = floor(z) 452116Sjkh * z = (z-x[i])*2**24 462116Sjkh * 472116Sjkh * 482116Sjkh * y[] ouput result in an array of double precision numbers. 492116Sjkh * The dimension of y[] is: 502116Sjkh * 24-bit precision 1 512116Sjkh * 53-bit precision 2 522116Sjkh * 64-bit precision 2 532116Sjkh * 113-bit precision 3 542116Sjkh * The actual value is the sum of them. Thus for 113-bit 552116Sjkh * precison, one may have to do something like: 562116Sjkh * 572116Sjkh * long double t,w,r_head, r_tail; 582116Sjkh * t = (long double)y[2] + (long double)y[1]; 592116Sjkh * w = (long double)y[0]; 602116Sjkh * r_head = t+w; 612116Sjkh * r_tail = w - (r_head - t); 622116Sjkh * 632116Sjkh * e0 The exponent of x[0] 642116Sjkh * 652116Sjkh * nx dimension of x[] 662116Sjkh * 672116Sjkh * prec an integer indicating the precision: 682116Sjkh * 0 24 bits (single) 692116Sjkh * 1 53 bits (double) 702116Sjkh * 2 64 bits (extended) 712116Sjkh * 3 113 bits (quad) 722116Sjkh * 732116Sjkh * ipio2[] 74141296Sdas * integer array, contains the (24*i)-th to (24*i+23)-th 75141296Sdas * bit of 2/pi after binary point. The corresponding 762116Sjkh * floating value is 772116Sjkh * 782116Sjkh * ipio2[i] * 2^(-24(i+1)). 792116Sjkh * 802116Sjkh * External function: 812116Sjkh * double scalbn(), floor(); 822116Sjkh * 832116Sjkh * 842116Sjkh * Here is the description of some local variables: 852116Sjkh * 862116Sjkh * jk jk+1 is the initial number of terms of ipio2[] needed 872116Sjkh * in the computation. The recommended value is 2,3,4, 882116Sjkh * 6 for single, double, extended,and quad. 892116Sjkh * 90141296Sdas * jz local integer variable indicating the number of 91141296Sdas * terms of ipio2[] used. 922116Sjkh * 932116Sjkh * jx nx - 1 942116Sjkh * 952116Sjkh * jv index for pointing to the suitable ipio2[] for the 962116Sjkh * computation. In general, we want 972116Sjkh * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 982116Sjkh * is an integer. Thus 992116Sjkh * e0-3-24*jv >= 0 or (e0-3)/24 >= jv 1002116Sjkh * Hence jv = max(0,(e0-3)/24). 1012116Sjkh * 1022116Sjkh * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. 1032116Sjkh * 1042116Sjkh * q[] double array with integral value, representing the 1052116Sjkh * 24-bits chunk of the product of x and 2/pi. 1062116Sjkh * 1072116Sjkh * q0 the corresponding exponent of q[0]. Note that the 1082116Sjkh * exponent for q[i] would be q0-24*i. 1092116Sjkh * 1102116Sjkh * PIo2[] double precision array, obtained by cutting pi/2 111141296Sdas * into 24 bits chunks. 1122116Sjkh * 113141296Sdas * f[] ipio2[] in floating point 1142116Sjkh * 1152116Sjkh * iq[] integer array by breaking up q[] in 24-bits chunk. 1162116Sjkh * 1172116Sjkh * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] 1182116Sjkh * 1192116Sjkh * ih integer. If >0 it indicates q[] is >= 0.5, hence 1202116Sjkh * it also indicates the *sign* of the result. 1212116Sjkh * 1222116Sjkh */ 1232116Sjkh 1242116Sjkh 1252116Sjkh/* 1262116Sjkh * Constants: 127141296Sdas * The hexadecimal values are the intended ones for the following 128141296Sdas * constants. The decimal values may be used, provided that the 129141296Sdas * compiler will convert from decimal to binary accurately enough 1302116Sjkh * to produce the hexadecimal values shown. 1312116Sjkh */ 1322116Sjkh 133175499Sbde#include <float.h> 134175499Sbde 1352116Sjkh#include "math.h" 1362116Sjkh#include "math_private.h" 1372116Sjkh 1382116Sjkhstatic const int init_jk[] = {2,3,4,6}; /* initial value for jk */ 1392116Sjkh 1402116Sjkhstatic const double PIo2[] = { 1412116Sjkh 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ 1422116Sjkh 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ 1432116Sjkh 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ 1442116Sjkh 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ 1452116Sjkh 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ 1462116Sjkh 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ 1472116Sjkh 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ 1482116Sjkh 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ 1492116Sjkh}; 1502116Sjkh 151141296Sdasstatic const double 1522116Sjkhzero = 0.0, 1532116Sjkhone = 1.0, 1542116Sjkhtwo24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 1552116Sjkhtwon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ 1562116Sjkh 1578870Srgrimes int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2) 1582116Sjkh{ 1592116Sjkh int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; 1602116Sjkh double z,fw,f[20],fq[20],q[20]; 1612116Sjkh 1622116Sjkh /* initialize jk*/ 1632116Sjkh jk = init_jk[prec]; 1642116Sjkh jp = jk; 1652116Sjkh 1662116Sjkh /* determine jx,jv,q0, note that 3>q0 */ 1672116Sjkh jx = nx-1; 1682116Sjkh jv = (e0-3)/24; if(jv<0) jv=0; 1692116Sjkh q0 = e0-24*(jv+1); 1702116Sjkh 1712116Sjkh /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ 1722116Sjkh j = jv-jx; m = jx+jk; 1732116Sjkh for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; 1742116Sjkh 1752116Sjkh /* compute q[0],q[1],...q[jk] */ 1762116Sjkh for (i=0;i<=jk;i++) { 1772116Sjkh for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; 1782116Sjkh } 1792116Sjkh 1802116Sjkh jz = jk; 1812116Sjkhrecompute: 1822116Sjkh /* distill q[] into iq[] reversingly */ 1832116Sjkh for(i=0,j=jz,z=q[jz];j>0;i++,j--) { 1842116Sjkh fw = (double)((int32_t)(twon24* z)); 1852116Sjkh iq[i] = (int32_t)(z-two24*fw); 1862116Sjkh z = q[j-1]+fw; 1872116Sjkh } 1882116Sjkh 1892116Sjkh /* compute n */ 1902116Sjkh z = scalbn(z,q0); /* actual value of z */ 1912116Sjkh z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ 1922116Sjkh n = (int32_t) z; 1932116Sjkh z -= (double)n; 1942116Sjkh ih = 0; 1952116Sjkh if(q0>0) { /* need iq[jz-1] to determine n */ 1962116Sjkh i = (iq[jz-1]>>(24-q0)); n += i; 1972116Sjkh iq[jz-1] -= i<<(24-q0); 1982116Sjkh ih = iq[jz-1]>>(23-q0); 199141296Sdas } 2002116Sjkh else if(q0==0) ih = iq[jz-1]>>23; 2012116Sjkh else if(z>=0.5) ih=2; 2022116Sjkh 2032116Sjkh if(ih>0) { /* q > 0.5 */ 2042116Sjkh n += 1; carry = 0; 2052116Sjkh for(i=0;i<jz ;i++) { /* compute 1-q */ 2062116Sjkh j = iq[i]; 2072116Sjkh if(carry==0) { 2082116Sjkh if(j!=0) { 2092116Sjkh carry = 1; iq[i] = 0x1000000- j; 2102116Sjkh } 2112116Sjkh } else iq[i] = 0xffffff - j; 2122116Sjkh } 2132116Sjkh if(q0>0) { /* rare case: chance is 1 in 12 */ 2142116Sjkh switch(q0) { 2152116Sjkh case 1: 2162116Sjkh iq[jz-1] &= 0x7fffff; break; 2172116Sjkh case 2: 2182116Sjkh iq[jz-1] &= 0x3fffff; break; 2192116Sjkh } 2202116Sjkh } 2212116Sjkh if(ih==2) { 2222116Sjkh z = one - z; 2232116Sjkh if(carry!=0) z -= scalbn(one,q0); 2242116Sjkh } 2252116Sjkh } 2262116Sjkh 2272116Sjkh /* check if recomputation is needed */ 2282116Sjkh if(z==zero) { 2292116Sjkh j = 0; 2302116Sjkh for (i=jz-1;i>=jk;i--) j |= iq[i]; 2312116Sjkh if(j==0) { /* need recomputation */ 2322116Sjkh for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ 2332116Sjkh 2342116Sjkh for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ 2352116Sjkh f[jx+i] = (double) ipio2[jv+i]; 2362116Sjkh for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; 2372116Sjkh q[i] = fw; 2382116Sjkh } 2392116Sjkh jz += k; 2402116Sjkh goto recompute; 2412116Sjkh } 2422116Sjkh } 2432116Sjkh 2442116Sjkh /* chop off zero terms */ 2452116Sjkh if(z==0.0) { 2462116Sjkh jz -= 1; q0 -= 24; 2472116Sjkh while(iq[jz]==0) { jz--; q0-=24;} 2482116Sjkh } else { /* break z into 24-bit if necessary */ 2492116Sjkh z = scalbn(z,-q0); 250141296Sdas if(z>=two24) { 2512116Sjkh fw = (double)((int32_t)(twon24*z)); 2522116Sjkh iq[jz] = (int32_t)(z-two24*fw); 2532116Sjkh jz += 1; q0 += 24; 2542116Sjkh iq[jz] = (int32_t) fw; 2552116Sjkh } else iq[jz] = (int32_t) z ; 2562116Sjkh } 2572116Sjkh 2582116Sjkh /* convert integer "bit" chunk to floating-point value */ 2592116Sjkh fw = scalbn(one,q0); 2602116Sjkh for(i=jz;i>=0;i--) { 2612116Sjkh q[i] = fw*(double)iq[i]; fw*=twon24; 2622116Sjkh } 2632116Sjkh 2642116Sjkh /* compute PIo2[0,...,jp]*q[jz,...,0] */ 2652116Sjkh for(i=jz;i>=0;i--) { 2662116Sjkh for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; 2672116Sjkh fq[jz-i] = fw; 2682116Sjkh } 2692116Sjkh 2702116Sjkh /* compress fq[] into y[] */ 2712116Sjkh switch(prec) { 2722116Sjkh case 0: 2732116Sjkh fw = 0.0; 2742116Sjkh for (i=jz;i>=0;i--) fw += fq[i]; 275141296Sdas y[0] = (ih==0)? fw: -fw; 2762116Sjkh break; 2772116Sjkh case 1: 2782116Sjkh case 2: 2792116Sjkh fw = 0.0; 280141296Sdas for (i=jz;i>=0;i--) fw += fq[i]; 281175507Sbde STRICT_ASSIGN(double,fw,fw); 282141296Sdas y[0] = (ih==0)? fw: -fw; 2832116Sjkh fw = fq[0]-fw; 2842116Sjkh for (i=1;i<=jz;i++) fw += fq[i]; 285141296Sdas y[1] = (ih==0)? fw: -fw; 2862116Sjkh break; 2872116Sjkh case 3: /* painful */ 2882116Sjkh for (i=jz;i>0;i--) { 289141296Sdas fw = fq[i-1]+fq[i]; 2902116Sjkh fq[i] += fq[i-1]-fw; 2912116Sjkh fq[i-1] = fw; 2922116Sjkh } 2932116Sjkh for (i=jz;i>1;i--) { 294141296Sdas fw = fq[i-1]+fq[i]; 2952116Sjkh fq[i] += fq[i-1]-fw; 2962116Sjkh fq[i-1] = fw; 2972116Sjkh } 298141296Sdas for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; 2992116Sjkh if(ih==0) { 3002116Sjkh y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; 3012116Sjkh } else { 3022116Sjkh y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; 3032116Sjkh } 3042116Sjkh } 3052116Sjkh return n&7; 3062116Sjkh} 307