| 1 | /* e_jnf.c -- float version of e_jn.c.
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| 2 | * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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| 3 | */
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| 4 |
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| 5 | /*
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| 6 | * ====================================================
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| 7 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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| 8 | *
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| 9 | * Developed at SunPro, a Sun Microsystems, Inc. business.
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| 10 | * Permission to use, copy, modify, and distribute this
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| 11 | * software is freely granted, provided that this notice
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| 12 | * is preserved.
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| 13 | * ====================================================
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| 14 | */
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| 15 |
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| 16 | #ifndef lint
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| 17 | static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_jnf.c,v 1.8 2002/05/28 18:15:04 alfred Exp $";
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| 18 | #endif
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| 19 |
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| 20 | #include "math.h"
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| 21 | #include "math_private.h"
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| 22 |
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| 23 | static const float
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| 24 | invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
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| 25 | two = 2.0000000000e+00, /* 0x40000000 */
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| 26 | one = 1.0000000000e+00; /* 0x3F800000 */
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| 27 |
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| 28 | static const float zero = 0.0000000000e+00;
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| 29 |
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| 30 | float
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| 31 | __ieee754_jnf(int n, float x)
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| 32 | {
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| 33 | int32_t i,hx,ix, sgn;
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| 34 | float a, b, temp, di;
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| 35 | float z, w;
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| 36 |
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| 37 | /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
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| 38 | * Thus, J(-n,x) = J(n,-x)
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| 39 | */
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| 40 | GET_FLOAT_WORD(hx,x);
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| 41 | ix = 0x7fffffff&hx;
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| 42 | /* if J(n,NaN) is NaN */
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| 43 | if(ix>0x7f800000) return x+x;
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| 44 | if(n<0){
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| 45 | n = -n;
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| 46 | x = -x;
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| 47 | hx ^= 0x80000000;
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| 48 | }
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| 49 | if(n==0) return(__ieee754_j0f(x));
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| 50 | if(n==1) return(__ieee754_j1f(x));
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| 51 | sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
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| 52 | x = fabsf(x);
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| 53 | if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */
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| 54 | b = zero;
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| 55 | else if((float)n<=x) {
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| 56 | /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
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| 57 | a = __ieee754_j0f(x);
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| 58 | b = __ieee754_j1f(x);
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| 59 | for(i=1;i<n;i++){
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| 60 | temp = b;
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| 61 | b = b*((float)(i+i)/x) - a; /* avoid underflow */
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| 62 | a = temp;
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| 63 | }
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| 64 | } else {
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| 65 | if(ix<0x30800000) { /* x < 2**-29 */
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| 66 | /* x is tiny, return the first Taylor expansion of J(n,x)
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| 67 | * J(n,x) = 1/n!*(x/2)^n - ...
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| 68 | */
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| 69 | if(n>33) /* underflow */
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| 70 | b = zero;
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| 71 | else {
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| 72 | temp = x*(float)0.5; b = temp;
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| 73 | for (a=one,i=2;i<=n;i++) {
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| 74 | a *= (float)i; /* a = n! */
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| 75 | b *= temp; /* b = (x/2)^n */
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| 76 | }
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| 77 | b = b/a;
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| 78 | }
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| 79 | } else {
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| 80 | /* use backward recurrence */
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| 81 | /* x x^2 x^2
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| 82 | * J(n,x)/J(n-1,x) = ---- ------ ------ .....
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| 83 | * 2n - 2(n+1) - 2(n+2)
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| 84 | *
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| 85 | * 1 1 1
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| 86 | * (for large x) = ---- ------ ------ .....
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| 87 | * 2n 2(n+1) 2(n+2)
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| 88 | * -- - ------ - ------ -
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| 89 | * x x x
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| 90 | *
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| 91 | * Let w = 2n/x and h=2/x, then the above quotient
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| 92 | * is equal to the continued fraction:
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| 93 | * 1
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| 94 | * = -----------------------
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| 95 | * 1
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| 96 | * w - -----------------
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| 97 | * 1
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| 98 | * w+h - ---------
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| 99 | * w+2h - ...
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| 100 | *
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| 101 | * To determine how many terms needed, let
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| 102 | * Q(0) = w, Q(1) = w(w+h) - 1,
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| 103 | * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
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| 104 | * When Q(k) > 1e4 good for single
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| 105 | * When Q(k) > 1e9 good for double
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| 106 | * When Q(k) > 1e17 good for quadruple
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| 107 | */
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| 108 | /* determine k */
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| 109 | float t,v;
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| 110 | float q0,q1,h,tmp; int32_t k,m;
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| 111 | w = (n+n)/(float)x; h = (float)2.0/(float)x;
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| 112 | q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1;
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| 113 | while(q1<(float)1.0e9) {
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| 114 | k += 1; z += h;
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| 115 | tmp = z*q1 - q0;
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| 116 | q0 = q1;
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| 117 | q1 = tmp;
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| 118 | }
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| 119 | m = n+n;
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| 120 | for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
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| 121 | a = t;
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| 122 | b = one;
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| 123 | /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
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| 124 | * Hence, if n*(log(2n/x)) > ...
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| 125 | * single 8.8722839355e+01
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| 126 | * double 7.09782712893383973096e+02
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| 127 | * long double 1.1356523406294143949491931077970765006170e+04
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| 128 | * then recurrent value may overflow and the result is
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| 129 | * likely underflow to zero
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| 130 | */
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| 131 | tmp = n;
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| 132 | v = two/x;
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| 133 | tmp = tmp*__ieee754_logf(fabsf(v*tmp));
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| 134 | if(tmp<(float)8.8721679688e+01) {
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| 135 | for(i=n-1,di=(float)(i+i);i>0;i--){
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| 136 | temp = b;
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| 137 | b *= di;
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| 138 | b = b/x - a;
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| 139 | a = temp;
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| 140 | di -= two;
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| 141 | }
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| 142 | } else {
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| 143 | for(i=n-1,di=(float)(i+i);i>0;i--){
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| 144 | temp = b;
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| 145 | b *= di;
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| 146 | b = b/x - a;
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| 147 | a = temp;
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| 148 | di -= two;
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| 149 | /* scale b to avoid spurious overflow */
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| 150 | if(b>(float)1e10) {
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| 151 | a /= b;
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| 152 | t /= b;
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| 153 | b = one;
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| 154 | }
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| 155 | }
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| 156 | }
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| 157 | b = (t*__ieee754_j0f(x)/b);
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| 158 | }
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| 159 | }
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| 160 | if(sgn==1) return -b; else return b;
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| 161 | }
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| 162 |
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| 163 | float
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| 164 | __ieee754_ynf(int n, float x)
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| 165 | {
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| 166 | int32_t i,hx,ix,ib;
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| 167 | int32_t sign;
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| 168 | float a, b, temp;
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| 169 |
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| 170 | GET_FLOAT_WORD(hx,x);
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| 171 | ix = 0x7fffffff&hx;
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| 172 | /* if Y(n,NaN) is NaN */
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| 173 | if(ix>0x7f800000) return x+x;
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| 174 | if(ix==0) return -one/zero;
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| 175 | if(hx<0) return zero/zero;
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| 176 | sign = 1;
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| 177 | if(n<0){
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| 178 | n = -n;
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| 179 | sign = 1 - ((n&1)<<1);
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| 180 | }
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| 181 | if(n==0) return(__ieee754_y0f(x));
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| 182 | if(n==1) return(sign*__ieee754_y1f(x));
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| 183 | if(ix==0x7f800000) return zero;
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| 184 |
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| 185 | a = __ieee754_y0f(x);
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| 186 | b = __ieee754_y1f(x);
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| 187 | /* quit if b is -inf */
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| 188 | GET_FLOAT_WORD(ib,b);
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| 189 | for(i=1;i<n&&ib!=0xff800000;i++){
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| 190 | temp = b;
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| 191 | b = ((float)(i+i)/x)*b - a;
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| 192 | GET_FLOAT_WORD(ib,b);
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| 193 | a = temp;
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| 194 | }
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| 195 | if(sign>0) return b; else return -b;
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| 196 | }
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