1169691Skan/* From: @(#)e_rem_pio2.c 1.4 95/01/18 */ 2169691Skan/* 3169691Skan * ==================================================== 4169691Skan * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5169691Skan * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. 6169691Skan * 7169691Skan * Developed at SunSoft, a Sun Microsystems, Inc. business. 8169691Skan * Permission to use, copy, modify, and distribute this 9169691Skan * software is freely granted, provided that this notice 10169691Skan * is preserved. 11169691Skan * ==================================================== 12169691Skan * 13169691Skan * Optimized by Bruce D. Evans. 14169691Skan */ 15169691Skan 16169691Skan#include <sys/cdefs.h> 17169691Skan__FBSDID("$FreeBSD: head/lib/msun/ld128/e_rem_pio2l.h 336545 2018-07-20 12:42:24Z bde $"); 18169691Skan 19169691Skan/* ld128 version of __ieee754_rem_pio2l(x,y) 20169691Skan * 21169691Skan * return the remainder of x rem pi/2 in y[0]+y[1] 22169691Skan * use __kernel_rem_pio2() 23169691Skan */ 24169691Skan 25169691Skan#include <float.h> 26169691Skan 27169691Skan#include "math.h" 28169691Skan#include "math_private.h" 29169691Skan#include "fpmath.h" 30169691Skan 31169691Skan#define BIAS (LDBL_MAX_EXP - 1) 32169691Skan 33169691Skan/* 34169691Skan * XXX need to verify that nonzero integer multiples of pi/2 within the 35169691Skan * range get no closer to a long double than 2**-140, or that 36169691Skan * ilogb(x) + ilogb(min_delta) < 45 - -140. 37169691Skan */ 38169691Skan/* 39169691Skan * invpio2: 113 bits of 2/pi 40169691Skan * pio2_1: first 68 bits of pi/2 41169691Skan * pio2_1t: pi/2 - pio2_1 42169691Skan * pio2_2: second 68 bits of pi/2 43169691Skan * pio2_2t: pi/2 - (pio2_1+pio2_2) 44169691Skan * pio2_3: third 68 bits of pi/2 45169691Skan * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) 46169691Skan */ 47169691Skan 48169691Skanstatic const double 49169691Skanzero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 50169691Skantwo24 = 1.67772160000000000000e+07; /* 0x41700000, 0x00000000 */ 51169691Skan 52169691Skanstatic const long double 53169691Skaninvpio2 = 6.3661977236758134307553505349005747e-01L, /* 0x145f306dc9c882a53f84eafa3ea6a.0p-113 */ 54169691Skanpio2_1 = 1.5707963267948966192292994253909555e+00L, /* 0x1921fb54442d18469800000000000.0p-112 */ 55169691Skanpio2_1t = 2.0222662487959507323996846200947577e-21L, /* 0x13198a2e03707344a4093822299f3.0p-181 */ 56169691Skanpio2_2 = 2.0222662487959507323994779168837751e-21L, /* 0x13198a2e03707344a400000000000.0p-181 */ 57169691Skanpio2_2t = 2.0670321098263988236496903051604844e-43L, /* 0x127044533e63a0105df531d89cd91.0p-254 */ 58169691Skanpio2_3 = 2.0670321098263988236499468110329591e-43L, /* 0x127044533e63a0105e00000000000.0p-254 */ 59169691Skanpio2_3t = -2.5650587247459238361625433492959285e-65L; /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */ 60169691Skan 61169691Skanstatic inline __always_inline int 62169691Skan__ieee754_rem_pio2l(long double x, long double *y) 63169691Skan{ 64169691Skan union IEEEl2bits u,u1; 65169691Skan long double z,w,t,r,fn; 66169691Skan double tx[5],ty[3]; 67169691Skan int64_t n; 68169691Skan int e0,ex,i,j,nx; 69169691Skan int16_t expsign; 70169691Skan 71169691Skan u.e = x; 72169691Skan expsign = u.xbits.expsign; 73169691Skan ex = expsign & 0x7fff; 74169691Skan if (ex < BIAS + 45 || ex == BIAS + 45 && 75169691Skan u.bits.manh < 0x921fb54442d1LL) { 76169691Skan /* |x| ~< 2^45*(pi/2), medium size */ 77169691Skan /* TODO: use only double precision for fn, as in expl(). */ 78169691Skan fn = rnintl(x * invpio2); 79169691Skan n = i64rint(fn); 80169691Skan r = x-fn*pio2_1; 81169691Skan w = fn*pio2_1t; /* 1st round good to 180 bit */ 82169691Skan { 83169691Skan union IEEEl2bits u2; 84169691Skan int ex1; 85169691Skan j = ex; 86169691Skan y[0] = r-w; 87169691Skan u2.e = y[0]; 88169691Skan ex1 = u2.xbits.expsign & 0x7fff; 89169691Skan i = j-ex1; 90169691Skan if(i>51) { /* 2nd iteration needed, good to 248 */ 91169691Skan t = r; 92169691Skan w = fn*pio2_2; 93169691Skan r = t-w; 94169691Skan w = fn*pio2_2t-((t-r)-w); 95169691Skan y[0] = r-w; 96169691Skan u2.e = y[0]; 97169691Skan ex1 = u2.xbits.expsign & 0x7fff; 98169691Skan i = j-ex1; 99169691Skan if(i>119) { /* 3rd iteration need, 316 bits acc */ 100169691Skan t = r; /* will cover all possible cases */ 101169691Skan w = fn*pio2_3; 102169691Skan r = t-w; 103169691Skan w = fn*pio2_3t-((t-r)-w); 104169691Skan y[0] = r-w; 105169691Skan } 106169691Skan } 107169691Skan } 108169691Skan y[1] = (r-y[0])-w; 109169691Skan return n; 110169691Skan } 111169691Skan /* 112169691Skan * all other (large) arguments 113169691Skan */ 114169691Skan if(ex==0x7fff) { /* x is inf or NaN */ 115169691Skan y[0]=y[1]=x-x; return 0; 116169691Skan } 117169691Skan /* set z = scalbn(|x|,ilogb(x)-23) */ 118169691Skan u1.e = x; 119169691Skan e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */ 120169691Skan u1.xbits.expsign = ex - e0; 121169691Skan z = u1.e; 122169691Skan for(i=0;i<4;i++) { 123169691Skan tx[i] = (double)((int32_t)(z)); 124169691Skan z = (z-tx[i])*two24; 125169691Skan } 126169691Skan tx[4] = z; 127169691Skan nx = 5; 128169691Skan while(tx[nx-1]==zero) nx--; /* skip zero term */ 129169691Skan n = __kernel_rem_pio2(tx,ty,e0,nx,3); 130169691Skan t = (long double)ty[2] + ty[1]; 131169691Skan r = t + ty[0]; 132169691Skan w = ty[0] - (r - t); 133169691Skan if(expsign<0) {y[0] = -r; y[1] = -w; return -n;} 134169691Skan y[0] = r; y[1] = w; return n; 135169691Skan} 136169691Skan