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 /* Use a specialized rint() to get fn. Assume round-to-nearest. */ 78 fn = x*invpio2+0x1.8p112; 79 fn = fn-0x1.8p112; 80 #ifdef HAVE_EFFICIENT_I64RINT 81 n = i64rint(fn); 82 #else 83 n = fn; 84 #endif 85 r = x-fn*pio2_1; 86 w = fn*pio2_1t; /* 1st round good to 180 bit */ 87 { 88 union IEEEl2bits u2; 89 int ex1; 90 j = ex; 91 y[0] = r-w; 92 u2.e = y[0]; 93 ex1 = u2.xbits.expsign & 0x7fff; 94 i = j-ex1; 95 if(i>51) { /* 2nd iteration needed, good to 248 */ 96 t = r; 97 w = fn*pio2_2; 98 r = t-w; 99 w = fn*pio2_2t-((t-r)-w); 100 y[0] = r-w; 101 u2.e = y[0]; 102 ex1 = u2.xbits.expsign & 0x7fff; 103 i = j-ex1; 104 if(i>119) { /* 3rd iteration need, 316 bits acc */ 105 t = r; /* will cover all possible cases */ 106 w = fn*pio2_3; 107 r = t-w; 108 w = fn*pio2_3t-((t-r)-w); 109 y[0] = r-w; 110 } 111 } 112 } 113 y[1] = (r-y[0])-w; 114 return n; 115 } 116 /* 117 * all other (large) arguments 118 */ 119 if(ex==0x7fff) { /* x is inf or NaN */ 120 y[0]=y[1]=x-x; return 0; 121 } 122 /* set z = scalbn(|x|,ilogb(x)-23) */ 123 u1.e = x; 124 e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */ 125 u1.xbits.expsign = ex - e0; 126 z = u1.e; 127 for(i=0;i<4;i++) { 128 tx[i] = (double)((int32_t)(z)); 129 z = (z-tx[i])*two24; 130 } 131 tx[4] = z; 132 nx = 5; 133 while(tx[nx-1]==zero) nx--; /* skip zero term */ 134 n = __kernel_rem_pio2(tx,ty,e0,nx,3); 135 t = (long double)ty[2] + ty[1]; 136 r = t + ty[0]; 137 w = ty[0] - (r - t); 138 if(expsign<0) {y[0] = -r; y[1] = -w; return -n;} 139 y[0] = r; y[1] = w; return n; 140 } 141