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| Rev | Author | Line No. | Line |
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| 2 | pj | 1 | /* e_jnf.c -- float version of e_jn.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_jnf.c,v 1.3 1995/05/30 05:48:25 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 const float |
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| 25 | #else |
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| 26 | static float |
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| 27 | #endif |
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| 28 | invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ |
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| 29 | two = 2.0000000000e+00, /* 0x40000000 */ |
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| 30 | one = 1.0000000000e+00; /* 0x3F800000 */ |
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| 31 | |||
| 32 | #ifdef __STDC__ |
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| 33 | static const float zero = 0.0000000000e+00; |
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| 34 | #else |
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| 35 | static float zero = 0.0000000000e+00; |
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| 36 | #endif |
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| 37 | |||
| 38 | #ifdef __STDC__ |
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| 39 | float __ieee754_jnf(int n, float x) |
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| 40 | #else |
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| 41 | float __ieee754_jnf(n,x) |
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| 42 | int n; float x; |
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| 43 | #endif |
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| 44 | { |
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| 45 | int32_t i,hx,ix, sgn; |
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| 46 | float a, b, temp, di; |
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| 47 | float z, w; |
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| 48 | |||
| 49 | /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) |
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| 50 | * Thus, J(-n,x) = J(n,-x) |
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| 51 | */ |
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| 52 | GET_FLOAT_WORD(hx,x); |
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| 53 | ix = 0x7fffffff&hx; |
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| 54 | /* if J(n,NaN) is NaN */ |
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| 55 | if(ix>0x7f800000) return x+x; |
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| 56 | if(n<0){ |
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| 57 | n = -n; |
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| 58 | x = -x; |
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| 59 | hx ^= 0x80000000; |
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| 60 | } |
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| 61 | if(n==0) return(__ieee754_j0f(x)); |
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| 62 | if(n==1) return(__ieee754_j1f(x)); |
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| 63 | sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ |
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| 64 | x = fabsf(x); |
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| 65 | if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */ |
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| 66 | b = zero; |
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| 67 | else if((float)n<=x) { |
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| 68 | /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ |
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| 69 | a = __ieee754_j0f(x); |
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| 70 | b = __ieee754_j1f(x); |
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| 71 | for(i=1;i<n;i++){ |
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| 72 | temp = b; |
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| 73 | b = b*((float)(i+i)/x) - a; /* avoid underflow */ |
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| 74 | a = temp; |
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| 75 | } |
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| 76 | } else { |
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| 77 | if(ix<0x30800000) { /* x < 2**-29 */ |
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| 78 | /* x is tiny, return the first Taylor expansion of J(n,x) |
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| 79 | * J(n,x) = 1/n!*(x/2)^n - ... |
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| 80 | */ |
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| 81 | if(n>33) /* underflow */ |
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| 82 | b = zero; |
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| 83 | else { |
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| 84 | temp = x*(float)0.5; b = temp; |
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| 85 | for (a=one,i=2;i<=n;i++) { |
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| 86 | a *= (float)i; /* a = n! */ |
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| 87 | b *= temp; /* b = (x/2)^n */ |
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| 88 | } |
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| 89 | b = b/a; |
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| 90 | } |
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| 91 | } else { |
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| 92 | /* use backward recurrence */ |
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| 93 | /* x x^2 x^2 |
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| 94 | * J(n,x)/J(n-1,x) = ---- ------ ------ ..... |
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| 95 | * 2n - 2(n+1) - 2(n+2) |
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| 96 | * |
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| 97 | * 1 1 1 |
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| 98 | * (for large x) = ---- ------ ------ ..... |
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| 99 | * 2n 2(n+1) 2(n+2) |
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| 100 | * -- - ------ - ------ - |
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| 101 | * x x x |
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| 102 | * |
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| 103 | * Let w = 2n/x and h=2/x, then the above quotient |
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| 104 | * is equal to the continued fraction: |
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| 105 | * 1 |
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| 106 | * = ----------------------- |
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| 107 | * 1 |
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| 108 | * w - ----------------- |
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| 109 | * 1 |
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| 110 | * w+h - --------- |
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| 111 | * w+2h - ... |
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| 112 | * |
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| 113 | * To determine how many terms needed, let |
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| 114 | * Q(0) = w, Q(1) = w(w+h) - 1, |
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| 115 | * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), |
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| 116 | * When Q(k) > 1e4 good for single |
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| 117 | * When Q(k) > 1e9 good for double |
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| 118 | * When Q(k) > 1e17 good for quadruple |
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| 119 | */ |
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| 120 | /* determine k */ |
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| 121 | float t,v; |
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| 122 | float q0,q1,h,tmp; int32_t k,m; |
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| 123 | w = (n+n)/(float)x; h = (float)2.0/(float)x; |
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| 124 | q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1; |
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| 125 | while(q1<(float)1.0e9) { |
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| 126 | k += 1; z += h; |
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| 127 | tmp = z*q1 - q0; |
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| 128 | q0 = q1; |
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| 129 | q1 = tmp; |
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| 130 | } |
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| 131 | m = n+n; |
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| 132 | for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); |
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| 133 | a = t; |
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| 134 | b = one; |
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| 135 | /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) |
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| 136 | * Hence, if n*(log(2n/x)) > ... |
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| 137 | * single 8.8722839355e+01 |
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| 138 | * double 7.09782712893383973096e+02 |
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| 139 | * long double 1.1356523406294143949491931077970765006170e+04 |
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| 140 | * then recurrent value may overflow and the result is |
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| 141 | * likely underflow to zero |
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| 142 | */ |
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| 143 | tmp = n; |
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| 144 | v = two/x; |
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| 145 | tmp = tmp*__ieee754_logf(fabsf(v*tmp)); |
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| 146 | if(tmp<(float)8.8721679688e+01) { |
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| 147 | for(i=n-1,di=(float)(i+i);i>0;i--){ |
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| 148 | temp = b; |
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| 149 | b *= di; |
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| 150 | b = b/x - a; |
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| 151 | a = temp; |
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| 152 | di -= two; |
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| 153 | } |
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| 154 | } else { |
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| 155 | for(i=n-1,di=(float)(i+i);i>0;i--){ |
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| 156 | temp = b; |
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| 157 | b *= di; |
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| 158 | b = b/x - a; |
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| 159 | a = temp; |
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| 160 | di -= two; |
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| 161 | /* scale b to avoid spurious overflow */ |
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| 162 | if(b>(float)1e10) { |
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| 163 | a /= b; |
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| 164 | t /= b; |
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| 165 | b = one; |
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| 166 | } |
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| 167 | } |
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| 168 | } |
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| 169 | b = (t*__ieee754_j0f(x)/b); |
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| 170 | } |
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| 171 | } |
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| 172 | if(sgn==1) return -b; else return b; |
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| 173 | } |
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| 174 | |||
| 175 | #ifdef __STDC__ |
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| 176 | float __ieee754_ynf(int n, float x) |
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| 177 | #else |
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| 178 | float __ieee754_ynf(n,x) |
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| 179 | int n; float x; |
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| 180 | #endif |
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| 181 | { |
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| 182 | int32_t i,hx,ix,ib; |
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| 183 | int32_t sign; |
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| 184 | float a, b, temp; |
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| 185 | |||
| 186 | GET_FLOAT_WORD(hx,x); |
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| 187 | ix = 0x7fffffff&hx; |
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| 188 | /* if Y(n,NaN) is NaN */ |
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| 189 | if(ix>0x7f800000) return x+x; |
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| 190 | if(ix==0) return -one/zero; |
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| 191 | if(hx<0) return zero/zero; |
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| 192 | sign = 1; |
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| 193 | if(n<0){ |
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| 194 | n = -n; |
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| 195 | sign = 1 - ((n&1)<<1); |
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| 196 | } |
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| 197 | if(n==0) return(__ieee754_y0f(x)); |
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| 198 | if(n==1) return(sign*__ieee754_y1f(x)); |
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| 199 | if(ix==0x7f800000) return zero; |
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| 200 | |||
| 201 | a = __ieee754_y0f(x); |
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| 202 | b = __ieee754_y1f(x); |
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| 203 | /* quit if b is -inf */ |
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| 204 | GET_FLOAT_WORD(ib,b); |
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| 205 | for(i=1;i<n&&ib!=0xff800000;i++){ |
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| 206 | temp = b; |
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| 207 | b = ((float)(i+i)/x)*b - a; |
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| 208 | GET_FLOAT_WORD(ib,b); |
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| 209 | a = temp; |
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| 210 | } |
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| 211 | if(sign>0) return b; else return -b; |
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| 212 | } |