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| Rev | Author | Line No. | Line |
|---|---|---|---|
| 2 | pj | 1 | /* e_j1f.c -- float version of e_j1.c. |
| 2 | * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. |
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| 3 | */ |
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| 4 | |||
| 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 | |||
| 16 | #ifndef lint |
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| 17 | static char rcsid[] = "$\Id: e_j1f.c,v 1.2 1995/05/30 05:48:23 rgrimes Exp $"; |
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| 18 | #endif |
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| 19 | |||
| 20 | #include "math.h" |
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| 21 | #include "math_private.h" |
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| 22 | |||
| 23 | #ifdef __STDC__ |
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| 24 | static float ponef(float), qonef(float); |
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| 25 | #else |
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| 26 | static float ponef(), qonef(); |
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| 27 | #endif |
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| 28 | |||
| 29 | #ifdef __STDC__ |
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| 30 | static const float |
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| 31 | #else |
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| 32 | static float |
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| 33 | #endif |
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| 34 | huge = 1e30, |
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| 35 | one = 1.0, |
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| 36 | invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ |
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| 37 | tpi = 6.3661974669e-01, /* 0x3f22f983 */ |
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| 38 | /* R0/S0 on [0,2] */ |
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| 39 | r00 = -6.2500000000e-02, /* 0xbd800000 */ |
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| 40 | r01 = 1.4070566976e-03, /* 0x3ab86cfd */ |
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| 41 | r02 = -1.5995563444e-05, /* 0xb7862e36 */ |
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| 42 | r03 = 4.9672799207e-08, /* 0x335557d2 */ |
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| 43 | s01 = 1.9153760746e-02, /* 0x3c9ce859 */ |
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| 44 | s02 = 1.8594678841e-04, /* 0x3942fab6 */ |
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| 45 | s03 = 1.1771846857e-06, /* 0x359dffc2 */ |
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| 46 | s04 = 5.0463624390e-09, /* 0x31ad6446 */ |
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| 47 | s05 = 1.2354227016e-11; /* 0x2d59567e */ |
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| 48 | |||
| 49 | #ifdef __STDC__ |
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| 50 | static const float zero = 0.0; |
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| 51 | #else |
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| 52 | static float zero = 0.0; |
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| 53 | #endif |
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| 54 | |||
| 55 | #ifdef __STDC__ |
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| 56 | float __ieee754_j1f(float x) |
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| 57 | #else |
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| 58 | float __ieee754_j1f(x) |
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| 59 | float x; |
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| 60 | #endif |
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| 61 | { |
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| 62 | float z, s,c,ss,cc,r,u,v,y; |
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| 63 | int32_t hx,ix; |
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| 64 | |||
| 65 | GET_FLOAT_WORD(hx,x); |
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| 66 | ix = hx&0x7fffffff; |
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| 67 | if(ix>=0x7f800000) return one/x; |
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| 68 | y = fabsf(x); |
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| 69 | if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
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| 70 | s = sinf(y); |
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| 71 | c = cosf(y); |
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| 72 | ss = -s-c; |
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| 73 | cc = s-c; |
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| 74 | if(ix<0x7f000000) { /* make sure y+y not overflow */ |
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| 75 | z = cosf(y+y); |
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| 76 | if ((s*c)>zero) cc = z/ss; |
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| 77 | else ss = z/cc; |
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| 78 | } |
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| 79 | /* |
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| 80 | * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) |
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| 81 | * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) |
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| 82 | */ |
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| 83 | if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(y); |
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| 84 | else { |
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| 85 | u = ponef(y); v = qonef(y); |
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| 86 | z = invsqrtpi*(u*cc-v*ss)/sqrtf(y); |
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| 87 | } |
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| 88 | if(hx<0) return -z; |
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| 89 | else return z; |
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| 90 | } |
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| 91 | if(ix<0x32000000) { /* |x|<2**-27 */ |
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| 92 | if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */ |
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| 93 | } |
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| 94 | z = x*x; |
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| 95 | r = z*(r00+z*(r01+z*(r02+z*r03))); |
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| 96 | s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); |
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| 97 | r *= x; |
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| 98 | return(x*(float)0.5+r/s); |
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| 99 | } |
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| 100 | |||
| 101 | #ifdef __STDC__ |
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| 102 | static const float U0[5] = { |
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| 103 | #else |
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| 104 | static float U0[5] = { |
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| 105 | #endif |
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| 106 | -1.9605709612e-01, /* 0xbe48c331 */ |
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| 107 | 5.0443872809e-02, /* 0x3d4e9e3c */ |
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| 108 | -1.9125689287e-03, /* 0xbafaaf2a */ |
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| 109 | 2.3525259166e-05, /* 0x37c5581c */ |
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| 110 | -9.1909917899e-08, /* 0xb3c56003 */ |
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| 111 | }; |
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| 112 | #ifdef __STDC__ |
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| 113 | static const float V0[5] = { |
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| 114 | #else |
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| 115 | static float V0[5] = { |
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| 116 | #endif |
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| 117 | 1.9916731864e-02, /* 0x3ca3286a */ |
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| 118 | 2.0255257550e-04, /* 0x3954644b */ |
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| 119 | 1.3560879779e-06, /* 0x35b602d4 */ |
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| 120 | 6.2274145840e-09, /* 0x31d5f8eb */ |
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| 121 | 1.6655924903e-11, /* 0x2d9281cf */ |
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| 122 | }; |
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| 123 | |||
| 124 | #ifdef __STDC__ |
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| 125 | float __ieee754_y1f(float x) |
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| 126 | #else |
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| 127 | float __ieee754_y1f(x) |
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| 128 | float x; |
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| 129 | #endif |
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| 130 | { |
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| 131 | float z, s,c,ss,cc,u,v; |
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| 132 | int32_t hx,ix; |
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| 133 | |||
| 134 | GET_FLOAT_WORD(hx,x); |
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| 135 | ix = 0x7fffffff&hx; |
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| 136 | /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ |
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| 137 | if(ix>=0x7f800000) return one/(x+x*x); |
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| 138 | if(ix==0) return -one/zero; |
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| 139 | if(hx<0) return zero/zero; |
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| 140 | if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
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| 141 | s = sinf(x); |
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| 142 | c = cosf(x); |
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| 143 | ss = -s-c; |
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| 144 | cc = s-c; |
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| 145 | if(ix<0x7f000000) { /* make sure x+x not overflow */ |
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| 146 | z = cosf(x+x); |
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| 147 | if ((s*c)>zero) cc = z/ss; |
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| 148 | else ss = z/cc; |
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| 149 | } |
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| 150 | /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) |
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| 151 | * where x0 = x-3pi/4 |
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| 152 | * Better formula: |
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| 153 | * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) |
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| 154 | * = 1/sqrt(2) * (sin(x) - cos(x)) |
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| 155 | * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) |
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| 156 | * = -1/sqrt(2) * (cos(x) + sin(x)) |
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| 157 | * To avoid cancellation, use |
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| 158 | * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) |
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| 159 | * to compute the worse one. |
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| 160 | */ |
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| 161 | if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x); |
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| 162 | else { |
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| 163 | u = ponef(x); v = qonef(x); |
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| 164 | z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); |
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| 165 | } |
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| 166 | return z; |
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| 167 | } |
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| 168 | if(ix<=0x24800000) { /* x < 2**-54 */ |
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| 169 | return(-tpi/x); |
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| 170 | } |
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| 171 | z = x*x; |
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| 172 | u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4]))); |
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| 173 | v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4])))); |
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| 174 | return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x)); |
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| 175 | } |
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| 176 | |||
| 177 | /* For x >= 8, the asymptotic expansions of pone is |
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| 178 | * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. |
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| 179 | * We approximate pone by |
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| 180 | * pone(x) = 1 + (R/S) |
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| 181 | * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10 |
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| 182 | * S = 1 + ps0*s^2 + ... + ps4*s^10 |
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| 183 | * and |
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| 184 | * | pone(x)-1-R/S | <= 2 ** ( -60.06) |
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| 185 | */ |
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| 186 | |||
| 187 | #ifdef __STDC__ |
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| 188 | static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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| 189 | #else |
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| 190 | static float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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| 191 | #endif |
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| 192 | 0.0000000000e+00, /* 0x00000000 */ |
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| 193 | 1.1718750000e-01, /* 0x3df00000 */ |
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| 194 | 1.3239480972e+01, /* 0x4153d4ea */ |
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| 195 | 4.1205184937e+02, /* 0x43ce06a3 */ |
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| 196 | 3.8747453613e+03, /* 0x45722bed */ |
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| 197 | 7.9144794922e+03, /* 0x45f753d6 */ |
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| 198 | }; |
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| 199 | #ifdef __STDC__ |
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| 200 | static const float ps8[5] = { |
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| 201 | #else |
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| 202 | static float ps8[5] = { |
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| 203 | #endif |
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| 204 | 1.1420736694e+02, /* 0x42e46a2c */ |
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| 205 | 3.6509309082e+03, /* 0x45642ee5 */ |
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| 206 | 3.6956207031e+04, /* 0x47105c35 */ |
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| 207 | 9.7602796875e+04, /* 0x47bea166 */ |
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| 208 | 3.0804271484e+04, /* 0x46f0a88b */ |
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| 209 | }; |
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| 210 | |||
| 211 | #ifdef __STDC__ |
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| 212 | static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
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| 213 | #else |
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| 214 | static float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
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| 215 | #endif |
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| 216 | 1.3199052094e-11, /* 0x2d68333f */ |
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| 217 | 1.1718749255e-01, /* 0x3defffff */ |
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| 218 | 6.8027510643e+00, /* 0x40d9b023 */ |
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| 219 | 1.0830818176e+02, /* 0x42d89dca */ |
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| 220 | 5.1763616943e+02, /* 0x440168b7 */ |
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| 221 | 5.2871520996e+02, /* 0x44042dc6 */ |
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| 222 | }; |
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| 223 | #ifdef __STDC__ |
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| 224 | static const float ps5[5] = { |
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| 225 | #else |
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| 226 | static float ps5[5] = { |
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| 227 | #endif |
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| 228 | 5.9280597687e+01, /* 0x426d1f55 */ |
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| 229 | 9.9140142822e+02, /* 0x4477d9b1 */ |
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| 230 | 5.3532670898e+03, /* 0x45a74a23 */ |
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| 231 | 7.8446904297e+03, /* 0x45f52586 */ |
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| 232 | 1.5040468750e+03, /* 0x44bc0180 */ |
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| 233 | }; |
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| 234 | |||
| 235 | #ifdef __STDC__ |
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| 236 | static const float pr3[6] = { |
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| 237 | #else |
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| 238 | static float pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
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| 239 | #endif |
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| 240 | 3.0250391081e-09, /* 0x314fe10d */ |
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| 241 | 1.1718686670e-01, /* 0x3defffab */ |
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| 242 | 3.9329774380e+00, /* 0x407bb5e7 */ |
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| 243 | 3.5119403839e+01, /* 0x420c7a45 */ |
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| 244 | 9.1055007935e+01, /* 0x42b61c2a */ |
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| 245 | 4.8559066772e+01, /* 0x42423c7c */ |
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| 246 | }; |
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| 247 | #ifdef __STDC__ |
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| 248 | static const float ps3[5] = { |
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| 249 | #else |
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| 250 | static float ps3[5] = { |
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| 251 | #endif |
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| 252 | 3.4791309357e+01, /* 0x420b2a4d */ |
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| 253 | 3.3676245117e+02, /* 0x43a86198 */ |
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| 254 | 1.0468714600e+03, /* 0x4482dbe3 */ |
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| 255 | 8.9081134033e+02, /* 0x445eb3ed */ |
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| 256 | 1.0378793335e+02, /* 0x42cf936c */ |
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| 257 | }; |
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| 258 | |||
| 259 | #ifdef __STDC__ |
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| 260 | static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
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| 261 | #else |
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| 262 | static float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
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| 263 | #endif |
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| 264 | 1.0771083225e-07, /* 0x33e74ea8 */ |
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| 265 | 1.1717621982e-01, /* 0x3deffa16 */ |
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| 266 | 2.3685150146e+00, /* 0x401795c0 */ |
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| 267 | 1.2242610931e+01, /* 0x4143e1bc */ |
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| 268 | 1.7693971634e+01, /* 0x418d8d41 */ |
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| 269 | 5.0735230446e+00, /* 0x40a25a4d */ |
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| 270 | }; |
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| 271 | #ifdef __STDC__ |
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| 272 | static const float ps2[5] = { |
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| 273 | #else |
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| 274 | static float ps2[5] = { |
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| 275 | #endif |
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| 276 | 2.1436485291e+01, /* 0x41ab7dec */ |
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| 277 | 1.2529022980e+02, /* 0x42fa9499 */ |
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| 278 | 2.3227647400e+02, /* 0x436846c7 */ |
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| 279 | 1.1767937469e+02, /* 0x42eb5bd7 */ |
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| 280 | 8.3646392822e+00, /* 0x4105d590 */ |
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| 281 | }; |
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| 282 | |||
| 283 | #ifdef __STDC__ |
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| 284 | static float ponef(float x) |
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| 285 | #else |
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| 286 | static float ponef(x) |
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| 287 | float x; |
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| 288 | #endif |
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| 289 | { |
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| 290 | #ifdef __STDC__ |
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| 291 | const float *p,*q; |
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| 292 | #else |
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| 293 | float *p,*q; |
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| 294 | #endif |
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| 295 | float z,r,s; |
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| 296 | int32_t ix; |
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| 297 | GET_FLOAT_WORD(ix,x); |
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| 298 | ix &= 0x7fffffff; |
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| 299 | if(ix>=0x41000000) {p = pr8; q= ps8;} |
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| 300 | else if(ix>=0x40f71c58){p = pr5; q= ps5;} |
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| 301 | else if(ix>=0x4036db68){p = pr3; q= ps3;} |
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| 302 | else if(ix>=0x40000000){p = pr2; q= ps2;} |
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| 303 | z = one/(x*x); |
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| 304 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
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| 305 | s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); |
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| 306 | return one+ r/s; |
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| 307 | } |
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| 308 | |||
| 309 | |||
| 310 | /* For x >= 8, the asymptotic expansions of qone is |
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| 311 | * 3/8 s - 105/1024 s^3 - ..., where s = 1/x. |
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| 312 | * We approximate pone by |
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| 313 | * qone(x) = s*(0.375 + (R/S)) |
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| 314 | * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 |
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| 315 | * S = 1 + qs1*s^2 + ... + qs6*s^12 |
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| 316 | * and |
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| 317 | * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) |
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| 318 | */ |
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| 319 | |||
| 320 | #ifdef __STDC__ |
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| 321 | static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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| 322 | #else |
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| 323 | static float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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| 324 | #endif |
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| 325 | 0.0000000000e+00, /* 0x00000000 */ |
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| 326 | -1.0253906250e-01, /* 0xbdd20000 */ |
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| 327 | -1.6271753311e+01, /* 0xc1822c8d */ |
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| 328 | -7.5960174561e+02, /* 0xc43de683 */ |
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| 329 | -1.1849806641e+04, /* 0xc639273a */ |
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| 330 | -4.8438511719e+04, /* 0xc73d3683 */ |
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| 331 | }; |
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| 332 | #ifdef __STDC__ |
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| 333 | static const float qs8[6] = { |
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| 334 | #else |
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| 335 | static float qs8[6] = { |
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| 336 | #endif |
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| 337 | 1.6139537048e+02, /* 0x43216537 */ |
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| 338 | 7.8253862305e+03, /* 0x45f48b17 */ |
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| 339 | 1.3387534375e+05, /* 0x4802bcd6 */ |
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| 340 | 7.1965775000e+05, /* 0x492fb29c */ |
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| 341 | 6.6660125000e+05, /* 0x4922be94 */ |
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| 342 | -2.9449025000e+05, /* 0xc88fcb48 */ |
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| 343 | }; |
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| 344 | |||
| 345 | #ifdef __STDC__ |
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| 346 | static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
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| 347 | #else |
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| 348 | static float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
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| 349 | #endif |
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| 350 | -2.0897993405e-11, /* 0xadb7d219 */ |
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| 351 | -1.0253904760e-01, /* 0xbdd1fffe */ |
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| 352 | -8.0564479828e+00, /* 0xc100e736 */ |
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| 353 | -1.8366960144e+02, /* 0xc337ab6b */ |
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| 354 | -1.3731937256e+03, /* 0xc4aba633 */ |
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| 355 | -2.6124443359e+03, /* 0xc523471c */ |
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| 356 | }; |
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| 357 | #ifdef __STDC__ |
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| 358 | static const float qs5[6] = { |
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| 359 | #else |
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| 360 | static float qs5[6] = { |
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| 361 | #endif |
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| 362 | 8.1276550293e+01, /* 0x42a28d98 */ |
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| 363 | 1.9917987061e+03, /* 0x44f8f98f */ |
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| 364 | 1.7468484375e+04, /* 0x468878f8 */ |
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| 365 | 4.9851425781e+04, /* 0x4742bb6d */ |
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| 366 | 2.7948074219e+04, /* 0x46da5826 */ |
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| 367 | -4.7191835938e+03, /* 0xc5937978 */ |
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| 368 | }; |
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| 369 | |||
| 370 | #ifdef __STDC__ |
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| 371 | static const float qr3[6] = { |
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| 372 | #else |
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| 373 | static float qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
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| 374 | #endif |
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| 375 | -5.0783124372e-09, /* 0xb1ae7d4f */ |
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| 376 | -1.0253783315e-01, /* 0xbdd1ff5b */ |
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| 377 | -4.6101160049e+00, /* 0xc0938612 */ |
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| 378 | -5.7847221375e+01, /* 0xc267638e */ |
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| 379 | -2.2824453735e+02, /* 0xc3643e9a */ |
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| 380 | -2.1921012878e+02, /* 0xc35b35cb */ |
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| 381 | }; |
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| 382 | #ifdef __STDC__ |
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| 383 | static const float qs3[6] = { |
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| 384 | #else |
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| 385 | static float qs3[6] = { |
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| 386 | #endif |
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| 387 | 4.7665153503e+01, /* 0x423ea91e */ |
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| 388 | 6.7386511230e+02, /* 0x4428775e */ |
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| 389 | 3.3801528320e+03, /* 0x45534272 */ |
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| 390 | 5.5477290039e+03, /* 0x45ad5dd5 */ |
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| 391 | 1.9031191406e+03, /* 0x44ede3d0 */ |
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| 392 | -1.3520118713e+02, /* 0xc3073381 */ |
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| 393 | }; |
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| 394 | |||
| 395 | #ifdef __STDC__ |
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| 396 | static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
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| 397 | #else |
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| 398 | static float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
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| 399 | #endif |
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| 400 | -1.7838172539e-07, /* 0xb43f8932 */ |
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| 401 | -1.0251704603e-01, /* 0xbdd1f475 */ |
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| 402 | -2.7522056103e+00, /* 0xc0302423 */ |
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| 403 | -1.9663616180e+01, /* 0xc19d4f16 */ |
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| 404 | -4.2325313568e+01, /* 0xc2294d1f */ |
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| 405 | -2.1371921539e+01, /* 0xc1aaf9b2 */ |
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| 406 | }; |
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| 407 | #ifdef __STDC__ |
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| 408 | static const float qs2[6] = { |
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| 409 | #else |
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| 410 | static float qs2[6] = { |
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| 411 | #endif |
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| 412 | 2.9533363342e+01, /* 0x41ec4454 */ |
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| 413 | 2.5298155212e+02, /* 0x437cfb47 */ |
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| 414 | 7.5750280762e+02, /* 0x443d602e */ |
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| 415 | 7.3939318848e+02, /* 0x4438d92a */ |
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| 416 | 1.5594900513e+02, /* 0x431bf2f2 */ |
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| 417 | -4.9594988823e+00, /* 0xc09eb437 */ |
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| 418 | }; |
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| 419 | |||
| 420 | #ifdef __STDC__ |
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| 421 | static float qonef(float x) |
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| 422 | #else |
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| 423 | static float qonef(x) |
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| 424 | float x; |
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| 425 | #endif |
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| 426 | { |
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| 427 | #ifdef __STDC__ |
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| 428 | const float *p,*q; |
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| 429 | #else |
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| 430 | float *p,*q; |
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| 431 | #endif |
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| 432 | float s,r,z; |
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| 433 | int32_t ix; |
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| 434 | GET_FLOAT_WORD(ix,x); |
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| 435 | ix &= 0x7fffffff; |
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| 436 | if(ix>=0x40200000) {p = qr8; q= qs8;} |
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| 437 | else if(ix>=0x40f71c58){p = qr5; q= qs5;} |
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| 438 | else if(ix>=0x4036db68){p = qr3; q= qs3;} |
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| 439 | else if(ix>=0x40000000){p = qr2; q= qs2;} |
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| 440 | z = one/(x*x); |
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| 441 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
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| 442 | s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); |
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| 443 | return ((float).375 + r/s)/x; |
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| 444 | } |