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| Rev | Author | Line No. | Line |
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| 2 | pj | 1 | /* @(#)e_jn.c 5.1 93/09/24 */ |
| 2 | /* |
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| 3 | * ==================================================== |
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| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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| 5 | * |
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| 6 | * Developed at SunPro, a Sun Microsystems, Inc. business. |
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| 7 | * Permission to use, copy, modify, and distribute this |
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| 8 | * software is freely granted, provided that this notice |
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| 9 | * is preserved. |
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| 10 | * ==================================================== |
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| 11 | */ |
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| 12 | |||
| 13 | #ifndef lint |
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| 14 | static char rcsid[] = "$\Id: e_jn.c,v 1.3 1995/05/30 05:48:24 rgrimes Exp $"; |
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| 15 | #endif |
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| 16 | |||
| 17 | /* |
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| 18 | * __ieee754_jn(n, x), __ieee754_yn(n, x) |
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| 19 | * floating point Bessel's function of the 1st and 2nd kind |
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| 20 | * of order n |
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| 21 | * |
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| 22 | * Special cases: |
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| 23 | * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; |
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| 24 | * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. |
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| 25 | * Note 2. About jn(n,x), yn(n,x) |
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| 26 | * For n=0, j0(x) is called, |
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| 27 | * for n=1, j1(x) is called, |
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| 28 | * for n<x, forward recursion us used starting |
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| 29 | * from values of j0(x) and j1(x). |
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| 30 | * for n>x, a continued fraction approximation to |
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| 31 | * j(n,x)/j(n-1,x) is evaluated and then backward |
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| 32 | * recursion is used starting from a supposed value |
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| 33 | * for j(n,x). The resulting value of j(0,x) is |
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| 34 | * compared with the actual value to correct the |
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| 35 | * supposed value of j(n,x). |
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| 36 | * |
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| 37 | * yn(n,x) is similar in all respects, except |
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| 38 | * that forward recursion is used for all |
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| 39 | * values of n>1. |
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| 40 | * |
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| 41 | */ |
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| 42 | |||
| 43 | #include "math.h" |
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| 44 | #include "math_private.h" |
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| 45 | |||
| 46 | #ifdef __STDC__ |
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| 47 | static const double |
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| 48 | #else |
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| 49 | static double |
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| 50 | #endif |
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| 51 | invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ |
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| 52 | two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ |
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| 53 | one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ |
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| 54 | |||
| 55 | #ifdef __STDC__ |
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| 56 | static const double zero = 0.00000000000000000000e+00; |
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| 57 | #else |
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| 58 | static double zero = 0.00000000000000000000e+00; |
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| 59 | #endif |
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| 60 | |||
| 61 | #ifdef __STDC__ |
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| 62 | double __ieee754_jn(int n, double x) |
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| 63 | #else |
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| 64 | double __ieee754_jn(n,x) |
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| 65 | int n; double x; |
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| 66 | #endif |
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| 67 | { |
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| 68 | int32_t i,hx,ix,lx, sgn; |
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| 69 | double a, b, temp, di; |
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| 70 | double z, w; |
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| 71 | |||
| 72 | /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) |
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| 73 | * Thus, J(-n,x) = J(n,-x) |
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| 74 | */ |
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| 75 | EXTRACT_WORDS(hx,lx,x); |
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| 76 | ix = 0x7fffffff&hx; |
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| 77 | /* if J(n,NaN) is NaN */ |
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| 78 | if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; |
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| 79 | if(n<0){ |
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| 80 | n = -n; |
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| 81 | x = -x; |
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| 82 | hx ^= 0x80000000; |
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| 83 | } |
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| 84 | if(n==0) return(__ieee754_j0(x)); |
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| 85 | if(n==1) return(__ieee754_j1(x)); |
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| 86 | sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ |
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| 87 | x = fabs(x); |
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| 88 | if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */ |
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| 89 | b = zero; |
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| 90 | else if((double)n<=x) { |
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| 91 | /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ |
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| 92 | if(ix>=0x52D00000) { /* x > 2**302 */ |
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| 93 | /* (x >> n**2) |
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| 94 | * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) |
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| 95 | * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) |
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| 96 | * Let s=sin(x), c=cos(x), |
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| 97 | * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then |
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| 98 | * |
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| 99 | * n sin(xn)*sqt2 cos(xn)*sqt2 |
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| 100 | * ---------------------------------- |
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| 101 | * 0 s-c c+s |
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| 102 | * 1 -s-c -c+s |
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| 103 | * 2 -s+c -c-s |
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| 104 | * 3 s+c c-s |
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| 105 | */ |
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| 106 | switch(n&3) { |
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| 107 | case 0: temp = cos(x)+sin(x); break; |
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| 108 | case 1: temp = -cos(x)+sin(x); break; |
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| 109 | case 2: temp = -cos(x)-sin(x); break; |
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| 110 | case 3: temp = cos(x)-sin(x); break; |
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| 111 | } |
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| 112 | b = invsqrtpi*temp/sqrt(x); |
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| 113 | } else { |
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| 114 | a = __ieee754_j0(x); |
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| 115 | b = __ieee754_j1(x); |
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| 116 | for(i=1;i<n;i++){ |
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| 117 | temp = b; |
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| 118 | b = b*((double)(i+i)/x) - a; /* avoid underflow */ |
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| 119 | a = temp; |
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| 120 | } |
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| 121 | } |
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| 122 | } else { |
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| 123 | if(ix<0x3e100000) { /* x < 2**-29 */ |
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| 124 | /* x is tiny, return the first Taylor expansion of J(n,x) |
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| 125 | * J(n,x) = 1/n!*(x/2)^n - ... |
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| 126 | */ |
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| 127 | if(n>33) /* underflow */ |
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| 128 | b = zero; |
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| 129 | else { |
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| 130 | temp = x*0.5; b = temp; |
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| 131 | for (a=one,i=2;i<=n;i++) { |
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| 132 | a *= (double)i; /* a = n! */ |
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| 133 | b *= temp; /* b = (x/2)^n */ |
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| 134 | } |
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| 135 | b = b/a; |
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| 136 | } |
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| 137 | } else { |
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| 138 | /* use backward recurrence */ |
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| 139 | /* x x^2 x^2 |
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| 140 | * J(n,x)/J(n-1,x) = ---- ------ ------ ..... |
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| 141 | * 2n - 2(n+1) - 2(n+2) |
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| 142 | * |
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| 143 | * 1 1 1 |
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| 144 | * (for large x) = ---- ------ ------ ..... |
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| 145 | * 2n 2(n+1) 2(n+2) |
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| 146 | * -- - ------ - ------ - |
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| 147 | * x x x |
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| 148 | * |
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| 149 | * Let w = 2n/x and h=2/x, then the above quotient |
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| 150 | * is equal to the continued fraction: |
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| 151 | * 1 |
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| 152 | * = ----------------------- |
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| 153 | * 1 |
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| 154 | * w - ----------------- |
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| 155 | * 1 |
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| 156 | * w+h - --------- |
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| 157 | * w+2h - ... |
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| 158 | * |
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| 159 | * To determine how many terms needed, let |
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| 160 | * Q(0) = w, Q(1) = w(w+h) - 1, |
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| 161 | * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), |
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| 162 | * When Q(k) > 1e4 good for single |
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| 163 | * When Q(k) > 1e9 good for double |
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| 164 | * When Q(k) > 1e17 good for quadruple |
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| 165 | */ |
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| 166 | /* determine k */ |
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| 167 | double t,v; |
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| 168 | double q0,q1,h,tmp; int32_t k,m; |
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| 169 | w = (n+n)/(double)x; h = 2.0/(double)x; |
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| 170 | q0 = w; z = w+h; q1 = w*z - 1.0; k=1; |
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| 171 | while(q1<1.0e9) { |
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| 172 | k += 1; z += h; |
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| 173 | tmp = z*q1 - q0; |
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| 174 | q0 = q1; |
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| 175 | q1 = tmp; |
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| 176 | } |
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| 177 | m = n+n; |
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| 178 | for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); |
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| 179 | a = t; |
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| 180 | b = one; |
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| 181 | /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) |
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| 182 | * Hence, if n*(log(2n/x)) > ... |
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| 183 | * single 8.8722839355e+01 |
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| 184 | * double 7.09782712893383973096e+02 |
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| 185 | * long double 1.1356523406294143949491931077970765006170e+04 |
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| 186 | * then recurrent value may overflow and the result is |
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| 187 | * likely underflow to zero |
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| 188 | */ |
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| 189 | tmp = n; |
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| 190 | v = two/x; |
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| 191 | tmp = tmp*__ieee754_log(fabs(v*tmp)); |
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| 192 | if(tmp<7.09782712893383973096e+02) { |
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| 193 | for(i=n-1,di=(double)(i+i);i>0;i--){ |
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| 194 | temp = b; |
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| 195 | b *= di; |
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| 196 | b = b/x - a; |
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| 197 | a = temp; |
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| 198 | di -= two; |
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| 199 | } |
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| 200 | } else { |
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| 201 | for(i=n-1,di=(double)(i+i);i>0;i--){ |
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| 202 | temp = b; |
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| 203 | b *= di; |
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| 204 | b = b/x - a; |
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| 205 | a = temp; |
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| 206 | di -= two; |
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| 207 | /* scale b to avoid spurious overflow */ |
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| 208 | if(b>1e100) { |
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| 209 | a /= b; |
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| 210 | t /= b; |
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| 211 | b = one; |
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| 212 | } |
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| 213 | } |
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| 214 | } |
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| 215 | b = (t*__ieee754_j0(x)/b); |
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| 216 | } |
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| 217 | } |
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| 218 | if(sgn==1) return -b; else return b; |
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| 219 | } |
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| 220 | |||
| 221 | #ifdef __STDC__ |
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| 222 | double __ieee754_yn(int n, double x) |
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| 223 | #else |
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| 224 | double __ieee754_yn(n,x) |
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| 225 | int n; double x; |
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| 226 | #endif |
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| 227 | { |
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| 228 | int32_t i,hx,ix,lx; |
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| 229 | int32_t sign; |
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| 230 | double a, b, temp; |
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| 231 | |||
| 232 | EXTRACT_WORDS(hx,lx,x); |
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| 233 | ix = 0x7fffffff&hx; |
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| 234 | /* if Y(n,NaN) is NaN */ |
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| 235 | if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x; |
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| 236 | if((ix|lx)==0) return -one/zero; |
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| 237 | if(hx<0) return zero/zero; |
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| 238 | sign = 1; |
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| 239 | if(n<0){ |
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| 240 | n = -n; |
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| 241 | sign = 1 - ((n&1)<<1); |
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| 242 | } |
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| 243 | if(n==0) return(__ieee754_y0(x)); |
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| 244 | if(n==1) return(sign*__ieee754_y1(x)); |
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| 245 | if(ix==0x7ff00000) return zero; |
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| 246 | if(ix>=0x52D00000) { /* x > 2**302 */ |
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| 247 | /* (x >> n**2) |
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| 248 | * Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi) |
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| 249 | * Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi) |
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| 250 | * Let s=sin(x), c=cos(x), |
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| 251 | * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then |
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| 252 | * |
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| 253 | * n sin(xn)*sqt2 cos(xn)*sqt2 |
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| 254 | * ---------------------------------- |
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| 255 | * 0 s-c c+s |
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| 256 | * 1 -s-c -c+s |
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| 257 | * 2 -s+c -c-s |
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| 258 | * 3 s+c c-s |
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| 259 | */ |
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| 260 | switch(n&3) { |
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| 261 | case 0: temp = sin(x)-cos(x); break; |
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| 262 | case 1: temp = -sin(x)-cos(x); break; |
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| 263 | case 2: temp = -sin(x)+cos(x); break; |
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| 264 | case 3: temp = sin(x)+cos(x); break; |
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| 265 | } |
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| 266 | b = invsqrtpi*temp/sqrt(x); |
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| 267 | } else { |
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| 268 | u_int32_t high; |
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| 269 | a = __ieee754_y0(x); |
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| 270 | b = __ieee754_y1(x); |
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| 271 | /* quit if b is -inf */ |
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| 272 | GET_HIGH_WORD(high,b); |
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| 273 | for(i=1;i<n&&high!=0xfff00000;i++){ |
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| 274 | temp = b; |
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| 275 | b = ((double)(i+i)/x)*b - a; |
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| 276 | GET_HIGH_WORD(high,b); |
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| 277 | a = temp; |
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| 278 | } |
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| 279 | } |
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| 280 | if(sign>0) return b; else return -b; |
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| 281 | } |