[352] | 1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_j1f.c */
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| 2 | /*
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| 3 | * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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| 4 | */
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| 5 | /*
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| 6 | * ====================================================
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| 7 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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| 8 | *
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| 9 | * Developed at SunPro, a Sun Microsystems, Inc. business.
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| 10 | * Permission to use, copy, modify, and distribute this
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| 11 | * software is freely granted, provided that this notice
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| 12 | * is preserved.
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| 13 | * ====================================================
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| 14 | */
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| 15 |
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| 16 | #define _GNU_SOURCE
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| 17 | #include "libm.h"
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| 18 |
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| 19 | static float ponef(float), qonef(float);
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| 20 |
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| 21 | static const float
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| 22 | invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */
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| 23 | tpi = 6.3661974669e-01; /* 0x3f22f983 */
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| 24 |
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| 25 | static float common(uint32_t ix, float x, int y1, int sign)
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| 26 | {
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| 27 | double z,s,c,ss,cc;
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| 28 |
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| 29 | s = sinf(x);
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| 30 | if (y1)
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| 31 | s = -s;
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| 32 | c = cosf(x);
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| 33 | cc = s-c;
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| 34 | if (ix < 0x7f000000) {
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| 35 | ss = -s-c;
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| 36 | z = cosf(2*x);
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| 37 | if (s*c > 0)
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| 38 | cc = z/ss;
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| 39 | else
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| 40 | ss = z/cc;
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| 41 | if (ix < 0x58800000) {
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| 42 | if (y1)
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| 43 | ss = -ss;
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| 44 | cc = ponef(x)*cc-qonef(x)*ss;
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| 45 | }
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| 46 | }
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| 47 | if (sign)
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| 48 | cc = -cc;
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| 49 | return invsqrtpi*cc/sqrtf(x);
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| 50 | }
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| 51 |
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| 52 | /* R0/S0 on [0,2] */
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| 53 | static const float
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| 54 | r00 = -6.2500000000e-02, /* 0xbd800000 */
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| 55 | r01 = 1.4070566976e-03, /* 0x3ab86cfd */
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| 56 | r02 = -1.5995563444e-05, /* 0xb7862e36 */
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| 57 | r03 = 4.9672799207e-08, /* 0x335557d2 */
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| 58 | s01 = 1.9153760746e-02, /* 0x3c9ce859 */
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| 59 | s02 = 1.8594678841e-04, /* 0x3942fab6 */
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| 60 | s03 = 1.1771846857e-06, /* 0x359dffc2 */
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| 61 | s04 = 5.0463624390e-09, /* 0x31ad6446 */
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| 62 | s05 = 1.2354227016e-11; /* 0x2d59567e */
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| 63 |
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| 64 | float j1f(float x)
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| 65 | {
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| 66 | float z,r,s;
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| 67 | uint32_t ix;
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| 68 | int sign;
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| 69 |
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| 70 | GET_FLOAT_WORD(ix, x);
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| 71 | sign = ix>>31;
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| 72 | ix &= 0x7fffffff;
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| 73 | if (ix >= 0x7f800000)
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| 74 | return 1/(x*x);
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| 75 | if (ix >= 0x40000000) /* |x| >= 2 */
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| 76 | return common(ix, fabsf(x), 0, sign);
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| 77 | if (ix >= 0x39000000) { /* |x| >= 2**-13 */
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| 78 | z = x*x;
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| 79 | r = z*(r00+z*(r01+z*(r02+z*r03)));
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| 80 | s = 1+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
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| 81 | z = 0.5f + r/s;
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| 82 | } else
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| 83 | z = 0.5f;
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| 84 | return z*x;
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| 85 | }
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| 86 |
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| 87 | static const float U0[5] = {
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| 88 | -1.9605709612e-01, /* 0xbe48c331 */
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| 89 | 5.0443872809e-02, /* 0x3d4e9e3c */
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| 90 | -1.9125689287e-03, /* 0xbafaaf2a */
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| 91 | 2.3525259166e-05, /* 0x37c5581c */
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| 92 | -9.1909917899e-08, /* 0xb3c56003 */
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| 93 | };
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| 94 | static const float V0[5] = {
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| 95 | 1.9916731864e-02, /* 0x3ca3286a */
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| 96 | 2.0255257550e-04, /* 0x3954644b */
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| 97 | 1.3560879779e-06, /* 0x35b602d4 */
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| 98 | 6.2274145840e-09, /* 0x31d5f8eb */
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| 99 | 1.6655924903e-11, /* 0x2d9281cf */
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| 100 | };
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| 101 |
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| 102 | float y1f(float x)
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| 103 | {
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| 104 | float z,u,v;
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| 105 | uint32_t ix;
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| 106 |
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| 107 | GET_FLOAT_WORD(ix, x);
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| 108 | if ((ix & 0x7fffffff) == 0)
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| 109 | return -1/0.0f;
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| 110 | if (ix>>31)
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| 111 | return 0/0.0f;
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| 112 | if (ix >= 0x7f800000)
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| 113 | return 1/x;
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| 114 | if (ix >= 0x40000000) /* |x| >= 2.0 */
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| 115 | return common(ix,x,1,0);
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| 116 | if (ix < 0x33000000) /* x < 2**-25 */
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| 117 | return -tpi/x;
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| 118 | z = x*x;
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| 119 | u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
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| 120 | v = 1.0f+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
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| 121 | return x*(u/v) + tpi*(j1f(x)*logf(x)-1.0f/x);
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| 122 | }
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| 123 |
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| 124 | /* For x >= 8, the asymptotic expansions of pone is
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| 125 | * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
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| 126 | * We approximate pone by
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| 127 | * pone(x) = 1 + (R/S)
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| 128 | * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
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| 129 | * S = 1 + ps0*s^2 + ... + ps4*s^10
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| 130 | * and
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| 131 | * | pone(x)-1-R/S | <= 2 ** ( -60.06)
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| 132 | */
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| 133 |
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| 134 | static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
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| 135 | 0.0000000000e+00, /* 0x00000000 */
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| 136 | 1.1718750000e-01, /* 0x3df00000 */
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| 137 | 1.3239480972e+01, /* 0x4153d4ea */
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| 138 | 4.1205184937e+02, /* 0x43ce06a3 */
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| 139 | 3.8747453613e+03, /* 0x45722bed */
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| 140 | 7.9144794922e+03, /* 0x45f753d6 */
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| 141 | };
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| 142 | static const float ps8[5] = {
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| 143 | 1.1420736694e+02, /* 0x42e46a2c */
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| 144 | 3.6509309082e+03, /* 0x45642ee5 */
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| 145 | 3.6956207031e+04, /* 0x47105c35 */
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| 146 | 9.7602796875e+04, /* 0x47bea166 */
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| 147 | 3.0804271484e+04, /* 0x46f0a88b */
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| 148 | };
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| 149 |
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| 150 | static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
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| 151 | 1.3199052094e-11, /* 0x2d68333f */
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| 152 | 1.1718749255e-01, /* 0x3defffff */
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| 153 | 6.8027510643e+00, /* 0x40d9b023 */
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| 154 | 1.0830818176e+02, /* 0x42d89dca */
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| 155 | 5.1763616943e+02, /* 0x440168b7 */
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| 156 | 5.2871520996e+02, /* 0x44042dc6 */
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| 157 | };
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| 158 | static const float ps5[5] = {
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| 159 | 5.9280597687e+01, /* 0x426d1f55 */
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| 160 | 9.9140142822e+02, /* 0x4477d9b1 */
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| 161 | 5.3532670898e+03, /* 0x45a74a23 */
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| 162 | 7.8446904297e+03, /* 0x45f52586 */
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| 163 | 1.5040468750e+03, /* 0x44bc0180 */
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| 164 | };
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| 165 |
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| 166 | static const float pr3[6] = {
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| 167 | 3.0250391081e-09, /* 0x314fe10d */
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| 168 | 1.1718686670e-01, /* 0x3defffab */
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| 169 | 3.9329774380e+00, /* 0x407bb5e7 */
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| 170 | 3.5119403839e+01, /* 0x420c7a45 */
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| 171 | 9.1055007935e+01, /* 0x42b61c2a */
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| 172 | 4.8559066772e+01, /* 0x42423c7c */
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| 173 | };
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| 174 | static const float ps3[5] = {
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| 175 | 3.4791309357e+01, /* 0x420b2a4d */
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| 176 | 3.3676245117e+02, /* 0x43a86198 */
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| 177 | 1.0468714600e+03, /* 0x4482dbe3 */
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| 178 | 8.9081134033e+02, /* 0x445eb3ed */
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| 179 | 1.0378793335e+02, /* 0x42cf936c */
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| 180 | };
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| 181 |
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| 182 | static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
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| 183 | 1.0771083225e-07, /* 0x33e74ea8 */
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| 184 | 1.1717621982e-01, /* 0x3deffa16 */
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| 185 | 2.3685150146e+00, /* 0x401795c0 */
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| 186 | 1.2242610931e+01, /* 0x4143e1bc */
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| 187 | 1.7693971634e+01, /* 0x418d8d41 */
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| 188 | 5.0735230446e+00, /* 0x40a25a4d */
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| 189 | };
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| 190 | static const float ps2[5] = {
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| 191 | 2.1436485291e+01, /* 0x41ab7dec */
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| 192 | 1.2529022980e+02, /* 0x42fa9499 */
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| 193 | 2.3227647400e+02, /* 0x436846c7 */
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| 194 | 1.1767937469e+02, /* 0x42eb5bd7 */
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| 195 | 8.3646392822e+00, /* 0x4105d590 */
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| 196 | };
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| 197 |
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| 198 | static float ponef(float x)
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| 199 | {
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| 200 | const float *p,*q;
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| 201 | float_t z,r,s;
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| 202 | uint32_t ix;
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| 203 |
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| 204 | GET_FLOAT_WORD(ix, x);
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| 205 | ix &= 0x7fffffff;
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| 206 | if (ix >= 0x41000000){p = pr8; q = ps8;}
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| 207 | else if (ix >= 0x409173eb){p = pr5; q = ps5;}
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| 208 | else if (ix >= 0x4036d917){p = pr3; q = ps3;}
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| 209 | else /*ix >= 0x40000000*/ {p = pr2; q = ps2;}
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| 210 | z = 1.0f/(x*x);
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| 211 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
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| 212 | s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
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| 213 | return 1.0f + r/s;
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| 214 | }
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| 215 |
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| 216 | /* For x >= 8, the asymptotic expansions of qone is
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| 217 | * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
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| 218 | * We approximate pone by
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| 219 | * qone(x) = s*(0.375 + (R/S))
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| 220 | * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
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| 221 | * S = 1 + qs1*s^2 + ... + qs6*s^12
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| 222 | * and
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| 223 | * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
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| 224 | */
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| 225 |
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| 226 | static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
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| 227 | 0.0000000000e+00, /* 0x00000000 */
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| 228 | -1.0253906250e-01, /* 0xbdd20000 */
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| 229 | -1.6271753311e+01, /* 0xc1822c8d */
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| 230 | -7.5960174561e+02, /* 0xc43de683 */
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| 231 | -1.1849806641e+04, /* 0xc639273a */
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| 232 | -4.8438511719e+04, /* 0xc73d3683 */
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| 233 | };
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| 234 | static const float qs8[6] = {
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| 235 | 1.6139537048e+02, /* 0x43216537 */
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| 236 | 7.8253862305e+03, /* 0x45f48b17 */
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| 237 | 1.3387534375e+05, /* 0x4802bcd6 */
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| 238 | 7.1965775000e+05, /* 0x492fb29c */
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| 239 | 6.6660125000e+05, /* 0x4922be94 */
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| 240 | -2.9449025000e+05, /* 0xc88fcb48 */
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| 241 | };
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| 242 |
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| 243 | static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
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| 244 | -2.0897993405e-11, /* 0xadb7d219 */
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| 245 | -1.0253904760e-01, /* 0xbdd1fffe */
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| 246 | -8.0564479828e+00, /* 0xc100e736 */
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| 247 | -1.8366960144e+02, /* 0xc337ab6b */
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| 248 | -1.3731937256e+03, /* 0xc4aba633 */
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| 249 | -2.6124443359e+03, /* 0xc523471c */
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| 250 | };
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| 251 | static const float qs5[6] = {
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| 252 | 8.1276550293e+01, /* 0x42a28d98 */
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| 253 | 1.9917987061e+03, /* 0x44f8f98f */
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| 254 | 1.7468484375e+04, /* 0x468878f8 */
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| 255 | 4.9851425781e+04, /* 0x4742bb6d */
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| 256 | 2.7948074219e+04, /* 0x46da5826 */
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| 257 | -4.7191835938e+03, /* 0xc5937978 */
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| 258 | };
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| 259 |
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| 260 | static const float qr3[6] = {
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| 261 | -5.0783124372e-09, /* 0xb1ae7d4f */
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| 262 | -1.0253783315e-01, /* 0xbdd1ff5b */
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| 263 | -4.6101160049e+00, /* 0xc0938612 */
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| 264 | -5.7847221375e+01, /* 0xc267638e */
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| 265 | -2.2824453735e+02, /* 0xc3643e9a */
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| 266 | -2.1921012878e+02, /* 0xc35b35cb */
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| 267 | };
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| 268 | static const float qs3[6] = {
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| 269 | 4.7665153503e+01, /* 0x423ea91e */
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| 270 | 6.7386511230e+02, /* 0x4428775e */
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| 271 | 3.3801528320e+03, /* 0x45534272 */
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| 272 | 5.5477290039e+03, /* 0x45ad5dd5 */
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| 273 | 1.9031191406e+03, /* 0x44ede3d0 */
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| 274 | -1.3520118713e+02, /* 0xc3073381 */
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| 275 | };
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| 276 |
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| 277 | static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
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| 278 | -1.7838172539e-07, /* 0xb43f8932 */
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| 279 | -1.0251704603e-01, /* 0xbdd1f475 */
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| 280 | -2.7522056103e+00, /* 0xc0302423 */
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| 281 | -1.9663616180e+01, /* 0xc19d4f16 */
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| 282 | -4.2325313568e+01, /* 0xc2294d1f */
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| 283 | -2.1371921539e+01, /* 0xc1aaf9b2 */
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| 284 | };
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| 285 | static const float qs2[6] = {
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| 286 | 2.9533363342e+01, /* 0x41ec4454 */
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| 287 | 2.5298155212e+02, /* 0x437cfb47 */
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| 288 | 7.5750280762e+02, /* 0x443d602e */
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| 289 | 7.3939318848e+02, /* 0x4438d92a */
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| 290 | 1.5594900513e+02, /* 0x431bf2f2 */
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| 291 | -4.9594988823e+00, /* 0xc09eb437 */
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| 292 | };
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| 293 |
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| 294 | static float qonef(float x)
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| 295 | {
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| 296 | const float *p,*q;
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| 297 | float_t s,r,z;
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| 298 | uint32_t ix;
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| 299 |
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| 300 | GET_FLOAT_WORD(ix, x);
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| 301 | ix &= 0x7fffffff;
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| 302 | if (ix >= 0x41000000){p = qr8; q = qs8;}
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| 303 | else if (ix >= 0x409173eb){p = qr5; q = qs5;}
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| 304 | else if (ix >= 0x4036d917){p = qr3; q = qs3;}
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| 305 | else /*ix >= 0x40000000*/ {p = qr2; q = qs2;}
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| 306 | z = 1.0f/(x*x);
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| 307 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
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| 308 | s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
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| 309 | return (.375f + r/s)/x;
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| 310 | }
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