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