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/* ld128 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 * XXX need to verify that nonzero integer multiples of pi/2 within the 35221234Skargl * range get no closer to a long double than 2**-140, or that 36221234Skargl * ilogb(x) + ilogb(min_delta) < 45 - -140. 37221234Skargl */ 38221234Skargl/* 39221234Skargl * invpio2: 113 bits of 2/pi 40221234Skargl * pio2_1: first 68 bits of pi/2 412116Sjkh * pio2_1t: pi/2 - pio2_1 42221234Skargl * pio2_2: second 68 bits of pi/2 432116Sjkh * pio2_2t: pi/2 - (pio2_1+pio2_2) 44221234Skargl * pio2_3: third 68 bits of pi/2 452116Sjkh * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) 462116Sjkh */ 472116Sjkh 488870Srgrimesstatic const double 492116Sjkhzero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 50221234Skargltwo24 = 1.67772160000000000000e+07; /* 0x41700000, 0x00000000 */ 512116Sjkh 52221234Skarglstatic const long double 53221234Skarglinvpio2 = 6.3661977236758134307553505349005747e-01L, /* 0x145f306dc9c882a53f84eafa3ea6a.0p-113 */ 54221234Skarglpio2_1 = 1.5707963267948966192292994253909555e+00L, /* 0x1921fb54442d18469800000000000.0p-112 */ 55221234Skarglpio2_1t = 2.0222662487959507323996846200947577e-21L, /* 0x13198a2e03707344a4093822299f3.0p-181 */ 56221234Skarglpio2_2 = 2.0222662487959507323994779168837751e-21L, /* 0x13198a2e03707344a400000000000.0p-181 */ 57221234Skarglpio2_2t = 2.0670321098263988236496903051604844e-43L, /* 0x127044533e63a0105df531d89cd91.0p-254 */ 58221234Skarglpio2_3 = 2.0670321098263988236499468110329591e-43L, /* 0x127044533e63a0105e00000000000.0p-254 */ 59221234Skarglpio2_3t = -2.5650587247459238361625433492959285e-65L; /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */ 60221234Skargl 61222508Skarglstatic inline __always_inline int 62221234Skargl__ieee754_rem_pio2l(long double x, long double *y) 632116Sjkh{ 64221234Skargl union IEEEl2bits u,u1; 65221234Skargl long double z,w,t,r,fn; 66221234Skargl double tx[5],ty[3]; 67221234Skargl int64_t n; 68221234Skargl int e0,ex,i,j,nx; 69221234Skargl int16_t expsign; 702116Sjkh 71221234Skargl u.e = x; 72221234Skargl expsign = u.xbits.expsign; 73221234Skargl ex = expsign & 0x7fff; 74221234Skargl if (ex < BIAS + 45 || ex == BIAS + 45 && 75221234Skargl u.bits.manh < 0x921fb54442d1LL) { 76221234Skargl /* |x| ~< 2^45*(pi/2), medium size */ 77176465Sbde /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 78221234Skargl fn = x*invpio2+0x1.8p112; 79221234Skargl fn = fn-0x1.8p112; 80221234Skargl#ifdef HAVE_EFFICIENT_I64RINT 81221234Skargl n = i64rint(fn); 82176465Sbde#else 83221234Skargl n = fn; 84176465Sbde#endif 85176476Sbde r = x-fn*pio2_1; 86221234Skargl w = fn*pio2_1t; /* 1st round good to 180 bit */ 87176466Sbde { 88221234Skargl union IEEEl2bits u2; 89221234Skargl int ex1; 90221234Skargl j = ex; 91141296Sdas y[0] = r-w; 92221234Skargl u2.e = y[0]; 93221234Skargl ex1 = u2.xbits.expsign & 0x7fff; 94221234Skargl i = j-ex1; 95221234Skargl if(i>51) { /* 2nd iteration needed, good to 248 */ 962116Sjkh t = r; 97141296Sdas w = fn*pio2_2; 982116Sjkh r = t-w; 99141296Sdas w = fn*pio2_2t-((t-r)-w); 1002116Sjkh y[0] = r-w; 101221234Skargl u2.e = y[0]; 102221234Skargl ex1 = u2.xbits.expsign & 0x7fff; 103221234Skargl i = j-ex1; 104221234Skargl if(i>119) { /* 3rd iteration need, 316 bits acc */ 1052116Sjkh t = r; /* will cover all possible cases */ 106141296Sdas w = fn*pio2_3; 1072116Sjkh r = t-w; 108141296Sdas w = fn*pio2_3t-((t-r)-w); 1092116Sjkh y[0] = r-w; 1102116Sjkh } 1112116Sjkh } 1122116Sjkh } 1132116Sjkh y[1] = (r-y[0])-w; 114176476Sbde return n; 1152116Sjkh } 116141296Sdas /* 1172116Sjkh * all other (large) arguments 1182116Sjkh */ 119221234Skargl if(ex==0x7fff) { /* x is inf or NaN */ 1202116Sjkh y[0]=y[1]=x-x; return 0; 1212116Sjkh } 1222116Sjkh /* set z = scalbn(|x|,ilogb(x)-23) */ 123221234Skargl u1.e = x; 124221234Skargl e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */ 125221234Skargl u1.xbits.expsign = ex - e0; 126221234Skargl z = u1.e; 127221234Skargl for(i=0;i<4;i++) { 1282116Sjkh tx[i] = (double)((int32_t)(z)); 1292116Sjkh z = (z-tx[i])*two24; 1302116Sjkh } 131221234Skargl tx[4] = z; 132221234Skargl nx = 5; 1332116Sjkh while(tx[nx-1]==zero) nx--; /* skip zero term */ 134221234Skargl n = __kernel_rem_pio2(tx,ty,e0,nx,3); 135221234Skargl t = (long double)ty[2] + ty[1]; 136221234Skargl r = t + ty[0]; 137221234Skargl w = ty[0] - (r - t); 138221234Skargl if(expsign<0) {y[0] = -r; y[1] = -w; return -n;} 139221234Skargl y[0] = r; y[1] = w; return n; 1402116Sjkh} 141