[388] | 1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_j0f.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 pzerof(float), qzerof(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 y0)
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| 26 | {
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| 27 | float z,s,c,ss,cc;
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| 28 | /*
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| 29 | * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
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| 30 | * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
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| 31 | */
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| 32 | s = sinf(x);
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| 33 | c = cosf(x);
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| 34 | if (y0)
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| 35 | c = -c;
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| 36 | cc = s+c;
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| 37 | if (ix < 0x7f000000) {
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| 38 | ss = s-c;
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| 39 | z = -cosf(2*x);
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| 40 | if (s*c < 0)
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| 41 | cc = z/ss;
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| 42 | else
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| 43 | ss = z/cc;
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| 44 | if (ix < 0x58800000) {
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| 45 | if (y0)
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| 46 | ss = -ss;
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| 47 | cc = pzerof(x)*cc-qzerof(x)*ss;
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| 48 | }
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| 49 | }
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| 50 | return invsqrtpi*cc/sqrtf(x);
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| 51 | }
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| 52 |
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| 53 | /* R0/S0 on [0, 2.00] */
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| 54 | static const float
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| 55 | R02 = 1.5625000000e-02, /* 0x3c800000 */
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| 56 | R03 = -1.8997929874e-04, /* 0xb947352e */
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| 57 | R04 = 1.8295404516e-06, /* 0x35f58e88 */
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| 58 | R05 = -4.6183270541e-09, /* 0xb19eaf3c */
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| 59 | S01 = 1.5619102865e-02, /* 0x3c7fe744 */
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| 60 | S02 = 1.1692678527e-04, /* 0x38f53697 */
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| 61 | S03 = 5.1354652442e-07, /* 0x3509daa6 */
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| 62 | S04 = 1.1661400734e-09; /* 0x30a045e8 */
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| 63 |
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| 64 | float j0f(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 |
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| 69 | GET_FLOAT_WORD(ix, x);
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| 70 | ix &= 0x7fffffff;
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| 71 | if (ix >= 0x7f800000)
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| 72 | return 1/(x*x);
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| 73 | x = fabsf(x);
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| 74 |
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| 75 | if (ix >= 0x40000000) { /* |x| >= 2 */
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| 76 | /* large ulp error near zeros */
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| 77 | return common(ix, x, 0);
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| 78 | }
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| 79 | if (ix >= 0x3a000000) { /* |x| >= 2**-11 */
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| 80 | /* up to 4ulp error near 2 */
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| 81 | z = x*x;
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| 82 | r = z*(R02+z*(R03+z*(R04+z*R05)));
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| 83 | s = 1+z*(S01+z*(S02+z*(S03+z*S04)));
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| 84 | return (1+x/2)*(1-x/2) + z*(r/s);
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| 85 | }
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| 86 | if (ix >= 0x21800000) /* |x| >= 2**-60 */
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| 87 | x = 0.25f*x*x;
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| 88 | return 1 - x;
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| 89 | }
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| 90 |
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| 91 | static const float
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| 92 | u00 = -7.3804296553e-02, /* 0xbd9726b5 */
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| 93 | u01 = 1.7666645348e-01, /* 0x3e34e80d */
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| 94 | u02 = -1.3818567619e-02, /* 0xbc626746 */
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| 95 | u03 = 3.4745343146e-04, /* 0x39b62a69 */
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| 96 | u04 = -3.8140706238e-06, /* 0xb67ff53c */
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| 97 | u05 = 1.9559013964e-08, /* 0x32a802ba */
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| 98 | u06 = -3.9820518410e-11, /* 0xae2f21eb */
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| 99 | v01 = 1.2730483897e-02, /* 0x3c509385 */
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| 100 | v02 = 7.6006865129e-05, /* 0x389f65e0 */
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| 101 | v03 = 2.5915085189e-07, /* 0x348b216c */
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| 102 | v04 = 4.4111031494e-10; /* 0x2ff280c2 */
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| 103 |
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| 104 | float y0f(float x)
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| 105 | {
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| 106 | float z,u,v;
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| 107 | uint32_t ix;
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| 108 |
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| 109 | GET_FLOAT_WORD(ix, x);
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| 110 | if ((ix & 0x7fffffff) == 0)
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| 111 | return -1/0.0f;
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| 112 | if (ix>>31)
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| 113 | return 0/0.0f;
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| 114 | if (ix >= 0x7f800000)
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| 115 | return 1/x;
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| 116 | if (ix >= 0x40000000) { /* |x| >= 2.0 */
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| 117 | /* large ulp error near zeros */
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| 118 | return common(ix,x,1);
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| 119 | }
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| 120 | if (ix >= 0x39000000) { /* x >= 2**-13 */
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| 121 | /* large ulp error at x ~= 0.89 */
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| 122 | z = x*x;
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| 123 | u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
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| 124 | v = 1+z*(v01+z*(v02+z*(v03+z*v04)));
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| 125 | return u/v + tpi*(j0f(x)*logf(x));
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| 126 | }
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| 127 | return u00 + tpi*logf(x);
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| 128 | }
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| 129 |
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| 130 | /* The asymptotic expansions of pzero is
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| 131 | * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
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| 132 | * For x >= 2, We approximate pzero by
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| 133 | * pzero(x) = 1 + (R/S)
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| 134 | * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
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| 135 | * S = 1 + pS0*s^2 + ... + pS4*s^10
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| 136 | * and
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| 137 | * | pzero(x)-1-R/S | <= 2 ** ( -60.26)
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| 138 | */
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| 139 | static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
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| 140 | 0.0000000000e+00, /* 0x00000000 */
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| 141 | -7.0312500000e-02, /* 0xbd900000 */
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| 142 | -8.0816707611e+00, /* 0xc1014e86 */
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| 143 | -2.5706311035e+02, /* 0xc3808814 */
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| 144 | -2.4852163086e+03, /* 0xc51b5376 */
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| 145 | -5.2530439453e+03, /* 0xc5a4285a */
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| 146 | };
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| 147 | static const float pS8[5] = {
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| 148 | 1.1653436279e+02, /* 0x42e91198 */
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| 149 | 3.8337448730e+03, /* 0x456f9beb */
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| 150 | 4.0597855469e+04, /* 0x471e95db */
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| 151 | 1.1675296875e+05, /* 0x47e4087c */
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| 152 | 4.7627726562e+04, /* 0x473a0bba */
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| 153 | };
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| 154 | static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
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| 155 | -1.1412546255e-11, /* 0xad48c58a */
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| 156 | -7.0312492549e-02, /* 0xbd8fffff */
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| 157 | -4.1596107483e+00, /* 0xc0851b88 */
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| 158 | -6.7674766541e+01, /* 0xc287597b */
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| 159 | -3.3123129272e+02, /* 0xc3a59d9b */
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| 160 | -3.4643338013e+02, /* 0xc3ad3779 */
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| 161 | };
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| 162 | static const float pS5[5] = {
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| 163 | 6.0753936768e+01, /* 0x42730408 */
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| 164 | 1.0512523193e+03, /* 0x44836813 */
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| 165 | 5.9789707031e+03, /* 0x45bad7c4 */
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| 166 | 9.6254453125e+03, /* 0x461665c8 */
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| 167 | 2.4060581055e+03, /* 0x451660ee */
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| 168 | };
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| 169 |
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| 170 | static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
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| 171 | -2.5470459075e-09, /* 0xb12f081b */
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| 172 | -7.0311963558e-02, /* 0xbd8fffb8 */
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| 173 | -2.4090321064e+00, /* 0xc01a2d95 */
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| 174 | -2.1965976715e+01, /* 0xc1afba52 */
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| 175 | -5.8079170227e+01, /* 0xc2685112 */
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| 176 | -3.1447946548e+01, /* 0xc1fb9565 */
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| 177 | };
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| 178 | static const float pS3[5] = {
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| 179 | 3.5856033325e+01, /* 0x420f6c94 */
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| 180 | 3.6151397705e+02, /* 0x43b4c1ca */
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| 181 | 1.1936077881e+03, /* 0x44953373 */
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| 182 | 1.1279968262e+03, /* 0x448cffe6 */
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| 183 | 1.7358093262e+02, /* 0x432d94b8 */
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| 184 | };
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| 185 |
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| 186 | static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
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| 187 | -8.8753431271e-08, /* 0xb3be98b7 */
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| 188 | -7.0303097367e-02, /* 0xbd8ffb12 */
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| 189 | -1.4507384300e+00, /* 0xbfb9b1cc */
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| 190 | -7.6356959343e+00, /* 0xc0f4579f */
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| 191 | -1.1193166733e+01, /* 0xc1331736 */
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| 192 | -3.2336456776e+00, /* 0xc04ef40d */
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| 193 | };
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| 194 | static const float pS2[5] = {
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| 195 | 2.2220300674e+01, /* 0x41b1c32d */
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| 196 | 1.3620678711e+02, /* 0x430834f0 */
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| 197 | 2.7047027588e+02, /* 0x43873c32 */
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| 198 | 1.5387539673e+02, /* 0x4319e01a */
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| 199 | 1.4657617569e+01, /* 0x416a859a */
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| 200 | };
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| 201 |
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| 202 | static float pzerof(float x)
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| 203 | {
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| 204 | const float *p,*q;
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| 205 | float_t z,r,s;
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| 206 | uint32_t ix;
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| 207 |
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| 208 | GET_FLOAT_WORD(ix, x);
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| 209 | ix &= 0x7fffffff;
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| 210 | if (ix >= 0x41000000){p = pR8; q = pS8;}
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| 211 | else if (ix >= 0x409173eb){p = pR5; q = pS5;}
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| 212 | else if (ix >= 0x4036d917){p = pR3; q = pS3;}
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| 213 | else /*ix >= 0x40000000*/ {p = pR2; q = pS2;}
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| 214 | z = 1.0f/(x*x);
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| 215 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
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| 216 | s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
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| 217 | return 1.0f + r/s;
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| 218 | }
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| 219 |
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| 220 |
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| 221 | /* For x >= 8, the asymptotic expansions of qzero is
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| 222 | * -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
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| 223 | * We approximate pzero by
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| 224 | * qzero(x) = s*(-1.25 + (R/S))
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| 225 | * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
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| 226 | * S = 1 + qS0*s^2 + ... + qS5*s^12
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| 227 | * and
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| 228 | * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
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| 229 | */
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| 230 | static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
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| 231 | 0.0000000000e+00, /* 0x00000000 */
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| 232 | 7.3242187500e-02, /* 0x3d960000 */
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| 233 | 1.1768206596e+01, /* 0x413c4a93 */
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| 234 | 5.5767340088e+02, /* 0x440b6b19 */
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| 235 | 8.8591972656e+03, /* 0x460a6cca */
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| 236 | 3.7014625000e+04, /* 0x471096a0 */
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| 237 | };
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| 238 | static const float qS8[6] = {
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| 239 | 1.6377603149e+02, /* 0x4323c6aa */
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| 240 | 8.0983447266e+03, /* 0x45fd12c2 */
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| 241 | 1.4253829688e+05, /* 0x480b3293 */
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| 242 | 8.0330925000e+05, /* 0x49441ed4 */
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| 243 | 8.4050156250e+05, /* 0x494d3359 */
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| 244 | -3.4389928125e+05, /* 0xc8a7eb69 */
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| 245 | };
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| 246 |
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| 247 | static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
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| 248 | 1.8408595828e-11, /* 0x2da1ec79 */
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| 249 | 7.3242180049e-02, /* 0x3d95ffff */
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| 250 | 5.8356351852e+00, /* 0x40babd86 */
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| 251 | 1.3511157227e+02, /* 0x43071c90 */
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| 252 | 1.0272437744e+03, /* 0x448067cd */
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| 253 | 1.9899779053e+03, /* 0x44f8bf4b */
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| 254 | };
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| 255 | static const float qS5[6] = {
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| 256 | 8.2776611328e+01, /* 0x42a58da0 */
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| 257 | 2.0778142090e+03, /* 0x4501dd07 */
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| 258 | 1.8847289062e+04, /* 0x46933e94 */
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| 259 | 5.6751113281e+04, /* 0x475daf1d */
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| 260 | 3.5976753906e+04, /* 0x470c88c1 */
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| 261 | -5.3543427734e+03, /* 0xc5a752be */
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| 262 | };
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| 263 |
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| 264 | static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
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| 265 | 4.3774099900e-09, /* 0x3196681b */
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| 266 | 7.3241114616e-02, /* 0x3d95ff70 */
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| 267 | 3.3442313671e+00, /* 0x405607e3 */
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| 268 | 4.2621845245e+01, /* 0x422a7cc5 */
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| 269 | 1.7080809021e+02, /* 0x432acedf */
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| 270 | 1.6673394775e+02, /* 0x4326bbe4 */
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| 271 | };
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| 272 | static const float qS3[6] = {
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| 273 | 4.8758872986e+01, /* 0x42430916 */
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| 274 | 7.0968920898e+02, /* 0x44316c1c */
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| 275 | 3.7041481934e+03, /* 0x4567825f */
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| 276 | 6.4604252930e+03, /* 0x45c9e367 */
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| 277 | 2.5163337402e+03, /* 0x451d4557 */
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| 278 | -1.4924745178e+02, /* 0xc3153f59 */
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| 279 | };
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| 280 |
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| 281 | static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
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| 282 | 1.5044444979e-07, /* 0x342189db */
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| 283 | 7.3223426938e-02, /* 0x3d95f62a */
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| 284 | 1.9981917143e+00, /* 0x3fffc4bf */
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| 285 | 1.4495602608e+01, /* 0x4167edfd */
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| 286 | 3.1666231155e+01, /* 0x41fd5471 */
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| 287 | 1.6252708435e+01, /* 0x4182058c */
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| 288 | };
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| 289 | static const float qS2[6] = {
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| 290 | 3.0365585327e+01, /* 0x41f2ecb8 */
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| 291 | 2.6934811401e+02, /* 0x4386ac8f */
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| 292 | 8.4478375244e+02, /* 0x44533229 */
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| 293 | 8.8293585205e+02, /* 0x445cbbe5 */
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| 294 | 2.1266638184e+02, /* 0x4354aa98 */
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| 295 | -5.3109550476e+00, /* 0xc0a9f358 */
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| 296 | };
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| 297 |
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| 298 | static float qzerof(float x)
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| 299 | {
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| 300 | const float *p,*q;
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| 301 | float_t s,r,z;
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| 302 | uint32_t ix;
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| 303 |
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| 304 | GET_FLOAT_WORD(ix, x);
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| 305 | ix &= 0x7fffffff;
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| 306 | if (ix >= 0x41000000){p = qR8; q = qS8;}
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| 307 | else if (ix >= 0x409173eb){p = qR5; q = qS5;}
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| 308 | else if (ix >= 0x4036d917){p = qR3; q = qS3;}
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| 309 | else /*ix >= 0x40000000*/ {p = qR2; q = qS2;}
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| 310 | z = 1.0f/(x*x);
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| 311 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
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| 312 | 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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| 313 | return (-.125f + r/s)/x;
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| 314 | }
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