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| Rev | Author | Line No. | Line |
|---|---|---|---|
| 2 | pj | 1 | /* e_j0f.c -- float version of e_j0.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_j0f.c,v 1.2 1995/05/30 05:48:19 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 pzerof(float), qzerof(float); |
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| 25 | #else |
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| 26 | static float pzerof(), qzerof(); |
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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, 2.00] */ |
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| 39 | R02 = 1.5625000000e-02, /* 0x3c800000 */ |
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| 40 | R03 = -1.8997929874e-04, /* 0xb947352e */ |
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| 41 | R04 = 1.8295404516e-06, /* 0x35f58e88 */ |
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| 42 | R05 = -4.6183270541e-09, /* 0xb19eaf3c */ |
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| 43 | S01 = 1.5619102865e-02, /* 0x3c7fe744 */ |
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| 44 | S02 = 1.1692678527e-04, /* 0x38f53697 */ |
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| 45 | S03 = 5.1354652442e-07, /* 0x3509daa6 */ |
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| 46 | S04 = 1.1661400734e-09; /* 0x30a045e8 */ |
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| 47 | |||
| 48 | #ifdef __STDC__ |
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| 49 | static const float zero = 0.0; |
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| 50 | #else |
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| 51 | static float zero = 0.0; |
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| 52 | #endif |
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| 53 | |||
| 54 | #ifdef __STDC__ |
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| 55 | float __ieee754_j0f(float x) |
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| 56 | #else |
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| 57 | float __ieee754_j0f(x) |
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| 58 | float x; |
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| 59 | #endif |
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| 60 | { |
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| 61 | float z, s,c,ss,cc,r,u,v; |
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| 62 | int32_t hx,ix; |
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| 63 | |||
| 64 | GET_FLOAT_WORD(hx,x); |
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| 65 | ix = hx&0x7fffffff; |
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| 66 | if(ix>=0x7f800000) return one/(x*x); |
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| 67 | x = fabsf(x); |
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| 68 | if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
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| 69 | s = sinf(x); |
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| 70 | c = cosf(x); |
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| 71 | ss = s-c; |
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| 72 | cc = s+c; |
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| 73 | if(ix<0x7f000000) { /* make sure x+x not overflow */ |
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| 74 | z = -cosf(x+x); |
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| 75 | if ((s*c)<zero) cc = z/ss; |
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| 76 | else ss = z/cc; |
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| 77 | } |
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| 78 | /* |
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| 79 | * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) |
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| 80 | * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) |
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| 81 | */ |
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| 82 | if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x); |
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| 83 | else { |
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| 84 | u = pzerof(x); v = qzerof(x); |
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| 85 | z = invsqrtpi*(u*cc-v*ss)/sqrtf(x); |
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| 86 | } |
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| 87 | return z; |
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| 88 | } |
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| 89 | if(ix<0x39000000) { /* |x| < 2**-13 */ |
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| 90 | if(huge+x>one) { /* raise inexact if x != 0 */ |
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| 91 | if(ix<0x32000000) return one; /* |x|<2**-27 */ |
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| 92 | else return one - (float)0.25*x*x; |
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| 93 | } |
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| 94 | } |
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| 95 | z = x*x; |
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| 96 | r = z*(R02+z*(R03+z*(R04+z*R05))); |
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| 97 | s = one+z*(S01+z*(S02+z*(S03+z*S04))); |
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| 98 | if(ix < 0x3F800000) { /* |x| < 1.00 */ |
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| 99 | return one + z*((float)-0.25+(r/s)); |
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| 100 | } else { |
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| 101 | u = (float)0.5*x; |
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| 102 | return((one+u)*(one-u)+z*(r/s)); |
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| 103 | } |
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| 104 | } |
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| 105 | |||
| 106 | #ifdef __STDC__ |
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| 107 | static const float |
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| 108 | #else |
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| 109 | static float |
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| 110 | #endif |
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| 111 | u00 = -7.3804296553e-02, /* 0xbd9726b5 */ |
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| 112 | u01 = 1.7666645348e-01, /* 0x3e34e80d */ |
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| 113 | u02 = -1.3818567619e-02, /* 0xbc626746 */ |
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| 114 | u03 = 3.4745343146e-04, /* 0x39b62a69 */ |
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| 115 | u04 = -3.8140706238e-06, /* 0xb67ff53c */ |
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| 116 | u05 = 1.9559013964e-08, /* 0x32a802ba */ |
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| 117 | u06 = -3.9820518410e-11, /* 0xae2f21eb */ |
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| 118 | v01 = 1.2730483897e-02, /* 0x3c509385 */ |
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| 119 | v02 = 7.6006865129e-05, /* 0x389f65e0 */ |
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| 120 | v03 = 2.5915085189e-07, /* 0x348b216c */ |
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| 121 | v04 = 4.4111031494e-10; /* 0x2ff280c2 */ |
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| 122 | |||
| 123 | #ifdef __STDC__ |
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| 124 | float __ieee754_y0f(float x) |
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| 125 | #else |
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| 126 | float __ieee754_y0f(x) |
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| 127 | float x; |
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| 128 | #endif |
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| 129 | { |
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| 130 | float z, s,c,ss,cc,u,v; |
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| 131 | int32_t hx,ix; |
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| 132 | |||
| 133 | GET_FLOAT_WORD(hx,x); |
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| 134 | ix = 0x7fffffff&hx; |
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| 135 | /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */ |
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| 136 | if(ix>=0x7f800000) return one/(x+x*x); |
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| 137 | if(ix==0) return -one/zero; |
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| 138 | if(hx<0) return zero/zero; |
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| 139 | if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
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| 140 | /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) |
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| 141 | * where x0 = x-pi/4 |
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| 142 | * Better formula: |
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| 143 | * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) |
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| 144 | * = 1/sqrt(2) * (sin(x) + cos(x)) |
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| 145 | * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) |
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| 146 | * = 1/sqrt(2) * (sin(x) - cos(x)) |
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| 147 | * To avoid cancellation, use |
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| 148 | * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) |
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| 149 | * to compute the worse one. |
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| 150 | */ |
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| 151 | s = sinf(x); |
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| 152 | c = cosf(x); |
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| 153 | ss = s-c; |
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| 154 | cc = s+c; |
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| 155 | /* |
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| 156 | * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) |
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| 157 | * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) |
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| 158 | */ |
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| 159 | if(ix<0x7f000000) { /* make sure x+x not overflow */ |
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| 160 | z = -cosf(x+x); |
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| 161 | if ((s*c)<zero) cc = z/ss; |
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| 162 | else ss = z/cc; |
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| 163 | } |
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| 164 | if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x); |
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| 165 | else { |
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| 166 | u = pzerof(x); v = qzerof(x); |
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| 167 | z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); |
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| 168 | } |
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| 169 | return z; |
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| 170 | } |
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| 171 | if(ix<=0x32000000) { /* x < 2**-27 */ |
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| 172 | return(u00 + tpi*__ieee754_logf(x)); |
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| 173 | } |
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| 174 | z = x*x; |
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| 175 | u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); |
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| 176 | v = one+z*(v01+z*(v02+z*(v03+z*v04))); |
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| 177 | return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x))); |
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| 178 | } |
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| 179 | |||
| 180 | /* The asymptotic expansions of pzero is |
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| 181 | * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. |
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| 182 | * For x >= 2, We approximate pzero by |
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| 183 | * pzero(x) = 1 + (R/S) |
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| 184 | * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 |
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| 185 | * S = 1 + pS0*s^2 + ... + pS4*s^10 |
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| 186 | * and |
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| 187 | * | pzero(x)-1-R/S | <= 2 ** ( -60.26) |
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| 188 | */ |
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| 189 | #ifdef __STDC__ |
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| 190 | static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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| 191 | #else |
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| 192 | static float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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| 193 | #endif |
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| 194 | 0.0000000000e+00, /* 0x00000000 */ |
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| 195 | -7.0312500000e-02, /* 0xbd900000 */ |
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| 196 | -8.0816707611e+00, /* 0xc1014e86 */ |
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| 197 | -2.5706311035e+02, /* 0xc3808814 */ |
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| 198 | -2.4852163086e+03, /* 0xc51b5376 */ |
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| 199 | -5.2530439453e+03, /* 0xc5a4285a */ |
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| 200 | }; |
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| 201 | #ifdef __STDC__ |
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| 202 | static const float pS8[5] = { |
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| 203 | #else |
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| 204 | static float pS8[5] = { |
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| 205 | #endif |
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| 206 | 1.1653436279e+02, /* 0x42e91198 */ |
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| 207 | 3.8337448730e+03, /* 0x456f9beb */ |
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| 208 | 4.0597855469e+04, /* 0x471e95db */ |
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| 209 | 1.1675296875e+05, /* 0x47e4087c */ |
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| 210 | 4.7627726562e+04, /* 0x473a0bba */ |
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| 211 | }; |
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| 212 | #ifdef __STDC__ |
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| 213 | static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
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| 214 | #else |
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| 215 | static float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
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| 216 | #endif |
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| 217 | -1.1412546255e-11, /* 0xad48c58a */ |
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| 218 | -7.0312492549e-02, /* 0xbd8fffff */ |
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| 219 | -4.1596107483e+00, /* 0xc0851b88 */ |
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| 220 | -6.7674766541e+01, /* 0xc287597b */ |
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| 221 | -3.3123129272e+02, /* 0xc3a59d9b */ |
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| 222 | -3.4643338013e+02, /* 0xc3ad3779 */ |
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| 223 | }; |
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| 224 | #ifdef __STDC__ |
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| 225 | static const float pS5[5] = { |
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| 226 | #else |
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| 227 | static float pS5[5] = { |
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| 228 | #endif |
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| 229 | 6.0753936768e+01, /* 0x42730408 */ |
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| 230 | 1.0512523193e+03, /* 0x44836813 */ |
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| 231 | 5.9789707031e+03, /* 0x45bad7c4 */ |
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| 232 | 9.6254453125e+03, /* 0x461665c8 */ |
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| 233 | 2.4060581055e+03, /* 0x451660ee */ |
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| 234 | }; |
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| 235 | |||
| 236 | #ifdef __STDC__ |
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| 237 | static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
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| 238 | #else |
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| 239 | static float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
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| 240 | #endif |
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| 241 | -2.5470459075e-09, /* 0xb12f081b */ |
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| 242 | -7.0311963558e-02, /* 0xbd8fffb8 */ |
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| 243 | -2.4090321064e+00, /* 0xc01a2d95 */ |
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| 244 | -2.1965976715e+01, /* 0xc1afba52 */ |
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| 245 | -5.8079170227e+01, /* 0xc2685112 */ |
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| 246 | -3.1447946548e+01, /* 0xc1fb9565 */ |
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| 247 | }; |
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| 248 | #ifdef __STDC__ |
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| 249 | static const float pS3[5] = { |
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| 250 | #else |
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| 251 | static float pS3[5] = { |
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| 252 | #endif |
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| 253 | 3.5856033325e+01, /* 0x420f6c94 */ |
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| 254 | 3.6151397705e+02, /* 0x43b4c1ca */ |
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| 255 | 1.1936077881e+03, /* 0x44953373 */ |
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| 256 | 1.1279968262e+03, /* 0x448cffe6 */ |
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| 257 | 1.7358093262e+02, /* 0x432d94b8 */ |
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| 258 | }; |
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| 259 | |||
| 260 | #ifdef __STDC__ |
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| 261 | static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
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| 262 | #else |
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| 263 | static float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
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| 264 | #endif |
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| 265 | -8.8753431271e-08, /* 0xb3be98b7 */ |
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| 266 | -7.0303097367e-02, /* 0xbd8ffb12 */ |
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| 267 | -1.4507384300e+00, /* 0xbfb9b1cc */ |
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| 268 | -7.6356959343e+00, /* 0xc0f4579f */ |
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| 269 | -1.1193166733e+01, /* 0xc1331736 */ |
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| 270 | -3.2336456776e+00, /* 0xc04ef40d */ |
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| 271 | }; |
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| 272 | #ifdef __STDC__ |
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| 273 | static const float pS2[5] = { |
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| 274 | #else |
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| 275 | static float pS2[5] = { |
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| 276 | #endif |
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| 277 | 2.2220300674e+01, /* 0x41b1c32d */ |
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| 278 | 1.3620678711e+02, /* 0x430834f0 */ |
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| 279 | 2.7047027588e+02, /* 0x43873c32 */ |
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| 280 | 1.5387539673e+02, /* 0x4319e01a */ |
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| 281 | 1.4657617569e+01, /* 0x416a859a */ |
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| 282 | }; |
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| 283 | |||
| 284 | #ifdef __STDC__ |
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| 285 | static float pzerof(float x) |
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| 286 | #else |
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| 287 | static float pzerof(x) |
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| 288 | float x; |
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| 289 | #endif |
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| 290 | { |
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| 291 | #ifdef __STDC__ |
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| 292 | const float *p,*q; |
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| 293 | #else |
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| 294 | float *p,*q; |
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| 295 | #endif |
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| 296 | float z,r,s; |
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| 297 | int32_t ix; |
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| 298 | GET_FLOAT_WORD(ix,x); |
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| 299 | ix &= 0x7fffffff; |
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| 300 | if(ix>=0x41000000) {p = pR8; q= pS8;} |
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| 301 | else if(ix>=0x40f71c58){p = pR5; q= pS5;} |
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| 302 | else if(ix>=0x4036db68){p = pR3; q= pS3;} |
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| 303 | else if(ix>=0x40000000){p = pR2; q= pS2;} |
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| 304 | z = one/(x*x); |
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| 305 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
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| 306 | s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); |
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| 307 | return one+ r/s; |
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| 308 | } |
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| 309 | |||
| 310 | |||
| 311 | /* For x >= 8, the asymptotic expansions of qzero is |
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| 312 | * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. |
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| 313 | * We approximate pzero by |
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| 314 | * qzero(x) = s*(-1.25 + (R/S)) |
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| 315 | * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 |
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| 316 | * S = 1 + qS0*s^2 + ... + qS5*s^12 |
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| 317 | * and |
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| 318 | * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) |
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| 319 | */ |
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| 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 | 7.3242187500e-02, /* 0x3d960000 */ |
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| 327 | 1.1768206596e+01, /* 0x413c4a93 */ |
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| 328 | 5.5767340088e+02, /* 0x440b6b19 */ |
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| 329 | 8.8591972656e+03, /* 0x460a6cca */ |
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| 330 | 3.7014625000e+04, /* 0x471096a0 */ |
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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 | 1.6377603149e+02, /* 0x4323c6aa */ |
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| 338 | 8.0983447266e+03, /* 0x45fd12c2 */ |
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| 339 | 1.4253829688e+05, /* 0x480b3293 */ |
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| 340 | 8.0330925000e+05, /* 0x49441ed4 */ |
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| 341 | 8.4050156250e+05, /* 0x494d3359 */ |
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| 342 | -3.4389928125e+05, /* 0xc8a7eb69 */ |
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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 | 1.8408595828e-11, /* 0x2da1ec79 */ |
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| 351 | 7.3242180049e-02, /* 0x3d95ffff */ |
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| 352 | 5.8356351852e+00, /* 0x40babd86 */ |
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| 353 | 1.3511157227e+02, /* 0x43071c90 */ |
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| 354 | 1.0272437744e+03, /* 0x448067cd */ |
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| 355 | 1.9899779053e+03, /* 0x44f8bf4b */ |
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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 | 8.2776611328e+01, /* 0x42a58da0 */ |
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| 363 | 2.0778142090e+03, /* 0x4501dd07 */ |
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| 364 | 1.8847289062e+04, /* 0x46933e94 */ |
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| 365 | 5.6751113281e+04, /* 0x475daf1d */ |
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| 366 | 3.5976753906e+04, /* 0x470c88c1 */ |
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| 367 | -5.3543427734e+03, /* 0xc5a752be */ |
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| 368 | }; |
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| 369 | |||
| 370 | #ifdef __STDC__ |
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| 371 | static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
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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 | 4.3774099900e-09, /* 0x3196681b */ |
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| 376 | 7.3241114616e-02, /* 0x3d95ff70 */ |
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| 377 | 3.3442313671e+00, /* 0x405607e3 */ |
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| 378 | 4.2621845245e+01, /* 0x422a7cc5 */ |
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| 379 | 1.7080809021e+02, /* 0x432acedf */ |
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| 380 | 1.6673394775e+02, /* 0x4326bbe4 */ |
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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 | 4.8758872986e+01, /* 0x42430916 */ |
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| 388 | 7.0968920898e+02, /* 0x44316c1c */ |
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| 389 | 3.7041481934e+03, /* 0x4567825f */ |
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| 390 | 6.4604252930e+03, /* 0x45c9e367 */ |
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| 391 | 2.5163337402e+03, /* 0x451d4557 */ |
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| 392 | -1.4924745178e+02, /* 0xc3153f59 */ |
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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 | 1.5044444979e-07, /* 0x342189db */ |
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| 401 | 7.3223426938e-02, /* 0x3d95f62a */ |
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| 402 | 1.9981917143e+00, /* 0x3fffc4bf */ |
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| 403 | 1.4495602608e+01, /* 0x4167edfd */ |
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| 404 | 3.1666231155e+01, /* 0x41fd5471 */ |
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| 405 | 1.6252708435e+01, /* 0x4182058c */ |
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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 | 3.0365585327e+01, /* 0x41f2ecb8 */ |
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| 413 | 2.6934811401e+02, /* 0x4386ac8f */ |
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| 414 | 8.4478375244e+02, /* 0x44533229 */ |
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| 415 | 8.8293585205e+02, /* 0x445cbbe5 */ |
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| 416 | 2.1266638184e+02, /* 0x4354aa98 */ |
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| 417 | -5.3109550476e+00, /* 0xc0a9f358 */ |
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| 418 | }; |
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| 419 | |||
| 420 | #ifdef __STDC__ |
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| 421 | static float qzerof(float x) |
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| 422 | #else |
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| 423 | static float qzerof(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>=0x41000000) {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).125 + r/s)/x; |
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| 444 | } |