1221234Skargl/* From: @(#)e_rem_pio2.c 1.4 95/01/18 */ 22116Sjkh/* 32116Sjkh * ==================================================== 42116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5221234Skargl * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. 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 * ==================================================== 12141296Sdas * 13176476Sbde * Optimized by Bruce D. Evans. 142116Sjkh */ 152116Sjkh 16176385Sbde#include <sys/cdefs.h> 17176385Sbde__FBSDID("$FreeBSD$"); 182116Sjkh 19221234Skargl/* ld80 version of __ieee754_rem_pio2l(x,y) 20141296Sdas * 21141296Sdas * return the remainder of x rem pi/2 in y[0]+y[1] 222116Sjkh * use __kernel_rem_pio2() 232116Sjkh */ 242116Sjkh 25176465Sbde#include <float.h> 26176465Sbde 272116Sjkh#include "math.h" 282116Sjkh#include "math_private.h" 29221234Skargl#include "fpmath.h" 302116Sjkh 31221234Skargl#define BIAS (LDBL_MAX_EXP - 1) 32221234Skargl 332116Sjkh/* 34221234Skargl * invpio2: 64 bits of 2/pi 35221234Skargl * pio2_1: first 39 bits of pi/2 362116Sjkh * pio2_1t: pi/2 - pio2_1 37221234Skargl * pio2_2: second 39 bits of pi/2 382116Sjkh * pio2_2t: pi/2 - (pio2_1+pio2_2) 39221234Skargl * pio2_3: third 39 bits of pi/2 402116Sjkh * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) 412116Sjkh */ 422116Sjkh 438870Srgrimesstatic const double 442116Sjkhzero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 452116Sjkhtwo24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ 46221234Skarglpio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */ 47221234Skarglpio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */ 48221234Skarglpio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */ 492116Sjkh 50221234Skargl#if defined(__amd64__) || defined(__i386__) 51221234Skargl/* Long double constants are slow on these arches, and broken on i386. */ 52221234Skarglstatic const volatile double 53221234Skarglinvpio2hi = 6.3661977236758138e-01, /* 0x145f306dc9c883.0p-53 */ 54221234Skarglinvpio2lo = -3.9356538861223811e-17, /* -0x16b00000000000.0p-107 */ 55221234Skarglpio2_1thi = -1.0746346554971943e-12, /* -0x12e7b9676733af.0p-92 */ 56221234Skarglpio2_1tlo = 8.8451028997905949e-29, /* 0x1c080000000000.0p-146 */ 57221234Skarglpio2_2thi = 6.3683171635109499e-25, /* 0x18a2e03707344a.0p-133 */ 58221234Skarglpio2_2tlo = 2.3183081793789774e-41, /* 0x10280000000000.0p-187 */ 59221234Skarglpio2_3thi = -2.7529965190440717e-37, /* -0x176b7ed8fbbacc.0p-174 */ 60221234Skarglpio2_3tlo = -4.2006647512740502e-54; /* -0x19c00000000000.0p-230 */ 61221234Skargl#define invpio2 ((long double)invpio2hi + invpio2lo) 62221234Skargl#define pio2_1t ((long double)pio2_1thi + pio2_1tlo) 63221234Skargl#define pio2_2t ((long double)pio2_2thi + pio2_2tlo) 64221234Skargl#define pio2_3t ((long double)pio2_3thi + pio2_3tlo) 65221234Skargl#else 66221234Skarglstatic const long double 67221234Skarglinvpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */ 68221234Skarglpio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */ 69221234Skarglpio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */ 70221234Skarglpio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */ 71176385Sbde#endif 72221234Skargl 73222508Skarglstatic inline __always_inline int 74221234Skargl__ieee754_rem_pio2l(long double x, long double *y) 752116Sjkh{ 76221234Skargl union IEEEl2bits u,u1; 77221234Skargl long double z,w,t,r,fn; 78176558Sbde double tx[3],ty[2]; 79221234Skargl int e0,ex,i,j,nx,n; 80221234Skargl int16_t expsign; 812116Sjkh 82221234Skargl u.e = x; 83221234Skargl expsign = u.xbits.expsign; 84221234Skargl ex = expsign & 0x7fff; 85223262Sbenl if (ex < BIAS + 25 || (ex == BIAS + 25 && u.bits.manh < 0xc90fdaa2)) { 86221234Skargl /* |x| ~< 2^25*(pi/2), medium size */ 87176465Sbde /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 88221234Skargl fn = x*invpio2+0x1.8p63; 89221234Skargl fn = fn-0x1.8p63; 90176467Sbde#ifdef HAVE_EFFICIENT_IRINT 91176465Sbde n = irint(fn); 92176465Sbde#else 93221234Skargl n = fn; 94176465Sbde#endif 95176476Sbde r = x-fn*pio2_1; 96221234Skargl w = fn*pio2_1t; /* 1st round good to 102 bit */ 97176466Sbde { 98221234Skargl union IEEEl2bits u2; 99221234Skargl int ex1; 100221234Skargl j = ex; 101141296Sdas y[0] = r-w; 102221234Skargl u2.e = y[0]; 103221234Skargl ex1 = u2.xbits.expsign & 0x7fff; 104221234Skargl i = j-ex1; 105221234Skargl if(i>22) { /* 2nd iteration needed, good to 141 */ 1062116Sjkh t = r; 107141296Sdas w = fn*pio2_2; 1082116Sjkh r = t-w; 109141296Sdas w = fn*pio2_2t-((t-r)-w); 1102116Sjkh y[0] = r-w; 111221234Skargl u2.e = y[0]; 112221234Skargl ex1 = u2.xbits.expsign & 0x7fff; 113221234Skargl i = j-ex1; 114221234Skargl if(i>61) { /* 3rd iteration need, 180 bits acc */ 1152116Sjkh t = r; /* will cover all possible cases */ 116141296Sdas w = fn*pio2_3; 1172116Sjkh r = t-w; 118141296Sdas w = fn*pio2_3t-((t-r)-w); 1192116Sjkh y[0] = r-w; 1202116Sjkh } 1212116Sjkh } 1222116Sjkh } 1232116Sjkh y[1] = (r-y[0])-w; 124176476Sbde return n; 1252116Sjkh } 126141296Sdas /* 1272116Sjkh * all other (large) arguments 1282116Sjkh */ 129221234Skargl if(ex==0x7fff) { /* x is inf or NaN */ 1302116Sjkh y[0]=y[1]=x-x; return 0; 1312116Sjkh } 1322116Sjkh /* set z = scalbn(|x|,ilogb(x)-23) */ 133221234Skargl u1.e = x; 134221234Skargl e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */ 135221234Skargl u1.xbits.expsign = ex - e0; 136221234Skargl z = u1.e; 1372116Sjkh for(i=0;i<2;i++) { 1382116Sjkh tx[i] = (double)((int32_t)(z)); 1392116Sjkh z = (z-tx[i])*two24; 1402116Sjkh } 1412116Sjkh tx[2] = z; 1422116Sjkh nx = 3; 1432116Sjkh while(tx[nx-1]==zero) nx--; /* skip zero term */ 144221234Skargl n = __kernel_rem_pio2(tx,ty,e0,nx,2); 145221234Skargl r = (long double)ty[0] + ty[1]; 146221234Skargl w = ty[1] - (r - ty[0]); 147221234Skargl if(expsign<0) {y[0] = -r; y[1] = -w; return -n;} 148221234Skargl y[0] = r; y[1] = w; return n; 1492116Sjkh} 150