1 /* From: @(#)e_rem_pio2.c 1.4 95/01/18 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. 6 * 7 * Developed at SunSoft, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 * 13 * Optimized by Bruce D. Evans. 14 */ 15 16 #include <sys/cdefs.h> 17 __FBSDID("$FreeBSD$"); 18 19 /* ld128 version of __ieee754_rem_pio2l(x,y) 20 * 21 * return the remainder of x rem pi/2 in y[0]+y[1] 22 * use __kernel_rem_pio2() 23 */ 24 25 #include <float.h> 26 27 #include "math.h" 28 #include "math_private.h" 29 #include "fpmath.h" 30 31 #define BIAS (LDBL_MAX_EXP - 1) 32 33 /* 34 * XXX need to verify that nonzero integer multiples of pi/2 within the 35 * range get no closer to a long double than 2**-140, or that 36 * ilogb(x) + ilogb(min_delta) < 45 - -140. 37 */ 38 /* 39 * invpio2: 113 bits of 2/pi 40 * pio2_1: first 68 bits of pi/2 41 * pio2_1t: pi/2 - pio2_1 42 * pio2_2: second 68 bits of pi/2 43 * pio2_2t: pi/2 - (pio2_1+pio2_2) 44 * pio2_3: third 68 bits of pi/2 45 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) 46 */ 47 48 static const double 49 zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ 50 two24 = 1.67772160000000000000e+07; /* 0x41700000, 0x00000000 */ 51 52 static const long double 53 invpio2 = 6.3661977236758134307553505349005747e-01L, /* 0x145f306dc9c882a53f84eafa3ea6a.0p-113 */ 54 pio2_1 = 1.5707963267948966192292994253909555e+00L, /* 0x1921fb54442d18469800000000000.0p-112 */ 55 pio2_1t = 2.0222662487959507323996846200947577e-21L, /* 0x13198a2e03707344a4093822299f3.0p-181 */ 56 pio2_2 = 2.0222662487959507323994779168837751e-21L, /* 0x13198a2e03707344a400000000000.0p-181 */ 57 pio2_2t = 2.0670321098263988236496903051604844e-43L, /* 0x127044533e63a0105df531d89cd91.0p-254 */ 58 pio2_3 = 2.0670321098263988236499468110329591e-43L, /* 0x127044533e63a0105e00000000000.0p-254 */ 59 pio2_3t = -2.5650587247459238361625433492959285e-65L; /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */ 60 61 static inline __always_inline int 62 __ieee754_rem_pio2l(long double x, long double *y) 63 { 64 union IEEEl2bits u,u1; 65 long double z,w,t,r,fn; 66 double tx[5],ty[3]; 67 int64_t n; 68 int e0,ex,i,j,nx; 69 int16_t expsign; 70 71 u.e = x; 72 expsign = u.xbits.expsign; 73 ex = expsign & 0x7fff; 74 if (ex < BIAS + 45 || ex == BIAS + 45 && 75 u.bits.manh < 0x921fb54442d1LL) { 76 /* |x| ~< 2^45*(pi/2), medium size */ 77 /* TODO: use only double precision for fn, as in expl(). */ 78 fn = rnintl(x * invpio2); 79 n = i64rint(fn); 80 r = x-fn*pio2_1; 81 w = fn*pio2_1t; /* 1st round good to 180 bit */ 82 { 83 union IEEEl2bits u2; 84 int ex1; 85 j = ex; 86 y[0] = r-w; 87 u2.e = y[0]; 88 ex1 = u2.xbits.expsign & 0x7fff; 89 i = j-ex1; 90 if(i>51) { /* 2nd iteration needed, good to 248 */ 91 t = r; 92 w = fn*pio2_2; 93 r = t-w; 94 w = fn*pio2_2t-((t-r)-w); 95 y[0] = r-w; 96 u2.e = y[0]; 97 ex1 = u2.xbits.expsign & 0x7fff; 98 i = j-ex1; 99 if(i>119) { /* 3rd iteration need, 316 bits acc */ 100 t = r; /* will cover all possible cases */ 101 w = fn*pio2_3; 102 r = t-w; 103 w = fn*pio2_3t-((t-r)-w); 104 y[0] = r-w; 105 } 106 } 107 } 108 y[1] = (r-y[0])-w; 109 return n; 110 } 111 /* 112 * all other (large) arguments 113 */ 114 if(ex==0x7fff) { /* x is inf or NaN */ 115 y[0]=y[1]=x-x; return 0; 116 } 117 /* set z = scalbn(|x|,ilogb(x)-23) */ 118 u1.e = x; 119 e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */ 120 u1.xbits.expsign = ex - e0; 121 z = u1.e; 122 for(i=0;i<4;i++) { 123 tx[i] = (double)((int32_t)(z)); 124 z = (z-tx[i])*two24; 125 } 126 tx[4] = z; 127 nx = 5; 128 while(tx[nx-1]==zero) nx--; /* skip zero term */ 129 n = __kernel_rem_pio2(tx,ty,e0,nx,3); 130 t = (long double)ty[2] + ty[1]; 131 r = t + ty[0]; 132 w = ty[0] - (r - t); 133 if(expsign<0) {y[0] = -r; y[1] = -w; return -n;} 134 y[0] = r; y[1] = w; return n; 135 } 136