[352] | 1 | /* origin: FreeBSD /usr/src/lib/msun/ld80/e_rem_pio2.c */
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| 2 | /*
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| 3 | * ====================================================
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| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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| 5 | * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
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| 6 | *
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| 7 | * Developed at SunSoft, a Sun Microsystems, Inc. business.
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| 8 | * Permission to use, copy, modify, and distribute this
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| 9 | * software is freely granted, provided that this notice
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| 10 | * is preserved.
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| 11 | * ====================================================
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| 12 | *
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| 13 | * Optimized by Bruce D. Evans.
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| 14 | */
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| 15 | #include "libm.h"
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| 16 | #if (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
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| 17 | /* ld80 and ld128 version of __rem_pio2(x,y)
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| 18 | *
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| 19 | * return the remainder of x rem pi/2 in y[0]+y[1]
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| 20 | * use __rem_pio2_large() for large x
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| 21 | */
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| 22 |
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| 23 | static const long double toint = 1.5/LDBL_EPSILON;
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| 24 |
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| 25 | #if LDBL_MANT_DIG == 64
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| 26 | /* u ~< 0x1p25*pi/2 */
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| 27 | #define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.m>>48) < ((0x3fff + 25)<<16 | 0x921f>>1 | 0x8000))
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| 28 | #define QUOBITS(x) ((uint32_t)(int32_t)x & 0x7fffffff)
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| 29 | #define ROUND1 22
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| 30 | #define ROUND2 61
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| 31 | #define NX 3
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| 32 | #define NY 2
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| 33 | /*
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| 34 | * invpio2: 64 bits of 2/pi
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| 35 | * pio2_1: first 39 bits of pi/2
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| 36 | * pio2_1t: pi/2 - pio2_1
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| 37 | * pio2_2: second 39 bits of pi/2
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| 38 | * pio2_2t: pi/2 - (pio2_1+pio2_2)
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| 39 | * pio2_3: third 39 bits of pi/2
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| 40 | * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
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| 41 | */
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| 42 | static const double
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| 43 | pio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */
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| 44 | pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */
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| 45 | pio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */
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| 46 | static const long double
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| 47 | invpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */
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| 48 | pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */
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| 49 | pio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */
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| 50 | pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */
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| 51 | #elif LDBL_MANT_DIG == 113
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| 52 | /* u ~< 0x1p45*pi/2 */
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| 53 | #define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.top) < ((0x3fff + 45)<<16 | 0x921f))
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| 54 | #define QUOBITS(x) ((uint32_t)(int64_t)x & 0x7fffffff)
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| 55 | #define ROUND1 51
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| 56 | #define ROUND2 119
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| 57 | #define NX 5
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| 58 | #define NY 3
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| 59 | static const long double
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| 60 | invpio2 = 6.3661977236758134307553505349005747e-01L, /* 0x145f306dc9c882a53f84eafa3ea6a.0p-113 */
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| 61 | pio2_1 = 1.5707963267948966192292994253909555e+00L, /* 0x1921fb54442d18469800000000000.0p-112 */
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| 62 | pio2_1t = 2.0222662487959507323996846200947577e-21L, /* 0x13198a2e03707344a4093822299f3.0p-181 */
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| 63 | pio2_2 = 2.0222662487959507323994779168837751e-21L, /* 0x13198a2e03707344a400000000000.0p-181 */
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| 64 | pio2_2t = 2.0670321098263988236496903051604844e-43L, /* 0x127044533e63a0105df531d89cd91.0p-254 */
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| 65 | pio2_3 = 2.0670321098263988236499468110329591e-43L, /* 0x127044533e63a0105e00000000000.0p-254 */
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| 66 | pio2_3t = -2.5650587247459238361625433492959285e-65L; /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */
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| 67 | #endif
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| 68 |
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| 69 | int __rem_pio2l(long double x, long double *y)
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| 70 | {
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| 71 | union ldshape u,uz;
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| 72 | long double z,w,t,r,fn;
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| 73 | double tx[NX],ty[NY];
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| 74 | int ex,ey,n,i;
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| 75 |
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| 76 | u.f = x;
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| 77 | ex = u.i.se & 0x7fff;
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| 78 | if (SMALL(u)) {
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| 79 | /* rint(x/(pi/2)), Assume round-to-nearest. */
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| 80 | fn = x*invpio2 + toint - toint;
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| 81 | n = QUOBITS(fn);
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| 82 | r = x-fn*pio2_1;
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| 83 | w = fn*pio2_1t; /* 1st round good to 102/180 bits (ld80/ld128) */
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| 84 | y[0] = r-w;
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| 85 | u.f = y[0];
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| 86 | ey = u.i.se & 0x7fff;
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| 87 | if (ex - ey > ROUND1) { /* 2nd iteration needed, good to 141/248 (ld80/ld128) */
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| 88 | t = r;
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| 89 | w = fn*pio2_2;
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| 90 | r = t-w;
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| 91 | w = fn*pio2_2t-((t-r)-w);
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| 92 | y[0] = r-w;
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| 93 | u.f = y[0];
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| 94 | ey = u.i.se & 0x7fff;
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| 95 | if (ex - ey > ROUND2) { /* 3rd iteration, good to 180/316 bits */
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| 96 | t = r; /* will cover all possible cases (not verified for ld128) */
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| 97 | w = fn*pio2_3;
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| 98 | r = t-w;
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| 99 | w = fn*pio2_3t-((t-r)-w);
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| 100 | y[0] = r-w;
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| 101 | }
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| 102 | }
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| 103 | y[1] = (r - y[0]) - w;
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| 104 | return n;
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| 105 | }
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| 106 | /*
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| 107 | * all other (large) arguments
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| 108 | */
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| 109 | if (ex == 0x7fff) { /* x is inf or NaN */
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| 110 | y[0] = y[1] = x - x;
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| 111 | return 0;
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| 112 | }
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| 113 | /* set z = scalbn(|x|,-ilogb(x)+23) */
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| 114 | uz.f = x;
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| 115 | uz.i.se = 0x3fff + 23;
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| 116 | z = uz.f;
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| 117 | for (i=0; i < NX - 1; i++) {
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| 118 | tx[i] = (double)(int32_t)z;
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| 119 | z = (z-tx[i])*0x1p24;
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| 120 | }
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| 121 | tx[i] = z;
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| 122 | while (tx[i] == 0)
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| 123 | i--;
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| 124 | n = __rem_pio2_large(tx, ty, ex-0x3fff-23, i+1, NY);
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| 125 | w = ty[1];
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| 126 | if (NY == 3)
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| 127 | w += ty[2];
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| 128 | r = ty[0] + w;
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| 129 | /* TODO: for ld128 this does not follow the recommendation of the
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| 130 | comments of __rem_pio2_large which seem wrong if |ty[0]| > |ty[1]+ty[2]| */
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| 131 | w -= r - ty[0];
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| 132 | if (u.i.se >> 15) {
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| 133 | y[0] = -r;
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| 134 | y[1] = -w;
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| 135 | return -n;
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| 136 | }
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| 137 | y[0] = r;
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| 138 | y[1] = w;
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| 139 | return n;
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| 140 | }
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| 141 | #endif
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