[388] | 1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_jnf.c */
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| 2 | /*
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| 3 | * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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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 | #define _GNU_SOURCE
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| 17 | #include "libm.h"
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| 18 |
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| 19 | float jnf(int n, float x)
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| 20 | {
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| 21 | uint32_t ix;
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| 22 | int nm1, sign, i;
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| 23 | float a, b, temp;
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| 24 |
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| 25 | GET_FLOAT_WORD(ix, x);
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| 26 | sign = ix>>31;
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| 27 | ix &= 0x7fffffff;
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| 28 | if (ix > 0x7f800000) /* nan */
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| 29 | return x;
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| 30 |
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| 31 | /* J(-n,x) = J(n,-x), use |n|-1 to avoid overflow in -n */
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| 32 | if (n == 0)
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| 33 | return j0f(x);
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| 34 | if (n < 0) {
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| 35 | nm1 = -(n+1);
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| 36 | x = -x;
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| 37 | sign ^= 1;
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| 38 | } else
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| 39 | nm1 = n-1;
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| 40 | if (nm1 == 0)
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| 41 | return j1f(x);
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| 42 |
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| 43 | sign &= n; /* even n: 0, odd n: signbit(x) */
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| 44 | x = fabsf(x);
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| 45 | if (ix == 0 || ix == 0x7f800000) /* if x is 0 or inf */
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| 46 | b = 0.0f;
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| 47 | else if (nm1 < x) {
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| 48 | /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
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| 49 | a = j0f(x);
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| 50 | b = j1f(x);
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| 51 | for (i=0; i<nm1; ){
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| 52 | i++;
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| 53 | temp = b;
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| 54 | b = b*(2.0f*i/x) - a;
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| 55 | a = temp;
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| 56 | }
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| 57 | } else {
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| 58 | if (ix < 0x35800000) { /* x < 2**-20 */
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| 59 | /* x is tiny, return the first Taylor expansion of J(n,x)
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| 60 | * J(n,x) = 1/n!*(x/2)^n - ...
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| 61 | */
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| 62 | if (nm1 > 8) /* underflow */
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| 63 | nm1 = 8;
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| 64 | temp = 0.5f * x;
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| 65 | b = temp;
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| 66 | a = 1.0f;
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| 67 | for (i=2; i<=nm1+1; i++) {
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| 68 | a *= (float)i; /* a = n! */
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| 69 | b *= temp; /* b = (x/2)^n */
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| 70 | }
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| 71 | b = b/a;
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| 72 | } else {
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| 73 | /* use backward recurrence */
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| 74 | /* x x^2 x^2
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| 75 | * J(n,x)/J(n-1,x) = ---- ------ ------ .....
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| 76 | * 2n - 2(n+1) - 2(n+2)
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| 77 | *
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| 78 | * 1 1 1
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| 79 | * (for large x) = ---- ------ ------ .....
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| 80 | * 2n 2(n+1) 2(n+2)
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| 81 | * -- - ------ - ------ -
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| 82 | * x x x
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| 83 | *
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| 84 | * Let w = 2n/x and h=2/x, then the above quotient
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| 85 | * is equal to the continued fraction:
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| 86 | * 1
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| 87 | * = -----------------------
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| 88 | * 1
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| 89 | * w - -----------------
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| 90 | * 1
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| 91 | * w+h - ---------
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| 92 | * w+2h - ...
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| 93 | *
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| 94 | * To determine how many terms needed, let
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| 95 | * Q(0) = w, Q(1) = w(w+h) - 1,
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| 96 | * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
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| 97 | * When Q(k) > 1e4 good for single
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| 98 | * When Q(k) > 1e9 good for double
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| 99 | * When Q(k) > 1e17 good for quadruple
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| 100 | */
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| 101 | /* determine k */
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| 102 | float t,q0,q1,w,h,z,tmp,nf;
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| 103 | int k;
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| 104 |
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| 105 | nf = nm1+1.0f;
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| 106 | w = 2*nf/x;
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| 107 | h = 2/x;
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| 108 | z = w+h;
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| 109 | q0 = w;
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| 110 | q1 = w*z - 1.0f;
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| 111 | k = 1;
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| 112 | while (q1 < 1.0e4f) {
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| 113 | k += 1;
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| 114 | 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 | for (t=0.0f, i=k; i>=0; i--)
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| 120 | t = 1.0f/(2*(i+nf)/x-t);
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| 121 | a = t;
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| 122 | b = 1.0f;
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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 = nf*logf(fabsf(w));
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| 132 | if (tmp < 88.721679688f) {
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| 133 | for (i=nm1; i>0; i--) {
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| 134 | temp = b;
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| 135 | b = 2.0f*i*b/x - a;
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| 136 | a = temp;
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| 137 | }
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| 138 | } else {
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| 139 | for (i=nm1; i>0; i--){
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| 140 | temp = b;
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| 141 | b = 2.0f*i*b/x - a;
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| 142 | a = temp;
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| 143 | /* scale b to avoid spurious overflow */
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| 144 | if (b > 0x1p60f) {
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| 145 | a /= b;
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| 146 | t /= b;
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| 147 | b = 1.0f;
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| 148 | }
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| 149 | }
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| 150 | }
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| 151 | z = j0f(x);
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| 152 | w = j1f(x);
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| 153 | if (fabsf(z) >= fabsf(w))
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| 154 | b = t*z/b;
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| 155 | else
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| 156 | b = t*w/a;
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| 157 | }
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| 158 | }
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| 159 | return sign ? -b : b;
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| 160 | }
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| 161 |
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| 162 | float ynf(int n, float x)
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| 163 | {
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| 164 | uint32_t ix, ib;
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| 165 | int nm1, sign, i;
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| 166 | float a, b, temp;
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| 167 |
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| 168 | GET_FLOAT_WORD(ix, x);
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| 169 | sign = ix>>31;
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| 170 | ix &= 0x7fffffff;
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| 171 | if (ix > 0x7f800000) /* nan */
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| 172 | return x;
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| 173 | if (sign && ix != 0) /* x < 0 */
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| 174 | return 0/0.0f;
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| 175 | if (ix == 0x7f800000)
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| 176 | return 0.0f;
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| 177 |
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| 178 | if (n == 0)
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| 179 | return y0f(x);
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| 180 | if (n < 0) {
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| 181 | nm1 = -(n+1);
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| 182 | sign = n&1;
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| 183 | } else {
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| 184 | nm1 = n-1;
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| 185 | sign = 0;
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| 186 | }
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| 187 | if (nm1 == 0)
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| 188 | return sign ? -y1f(x) : y1f(x);
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| 189 |
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| 190 | a = y0f(x);
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| 191 | b = y1f(x);
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| 192 | /* quit if b is -inf */
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| 193 | GET_FLOAT_WORD(ib,b);
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| 194 | for (i = 0; i < nm1 && ib != 0xff800000; ) {
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| 195 | i++;
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| 196 | temp = b;
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| 197 | b = (2.0f*i/x)*b - a;
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| 198 | GET_FLOAT_WORD(ib, b);
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| 199 | a = temp;
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| 200 | }
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| 201 | return sign ? -b : b;
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| 202 | }
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