Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_jnf.c */ |
| 2 | /* |
| 3 | * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. |
| 4 | */ |
| 5 | /* |
| 6 | * ==================================================== |
| 7 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| 8 | * |
| 9 | * Developed at SunPro, a Sun Microsystems, Inc. business. |
| 10 | * Permission to use, copy, modify, and distribute this |
| 11 | * software is freely granted, provided that this notice |
| 12 | * is preserved. |
| 13 | * ==================================================== |
| 14 | */ |
| 15 | |
Rich Felker | de7db6e | 2012-03-16 21:16:32 -0400 | [diff] [blame] | 16 | #define _GNU_SOURCE |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 17 | #include "libm.h" |
| 18 | |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 19 | float jnf(int n, float x) |
| 20 | { |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 21 | uint32_t ix; |
| 22 | int nm1, sign, i; |
| 23 | float a, b, temp; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 24 | |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 25 | GET_FLOAT_WORD(ix, x); |
| 26 | sign = ix>>31; |
| 27 | ix &= 0x7fffffff; |
| 28 | if (ix > 0x7f800000) /* nan */ |
| 29 | return x; |
| 30 | |
| 31 | /* J(-n,x) = J(n,-x), use |n|-1 to avoid overflow in -n */ |
| 32 | if (n == 0) |
| 33 | return j0f(x); |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 34 | if (n < 0) { |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 35 | nm1 = -(n+1); |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 36 | x = -x; |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 37 | sign ^= 1; |
| 38 | } else |
| 39 | nm1 = n-1; |
| 40 | if (nm1 == 0) |
| 41 | return j1f(x); |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 42 | |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 43 | sign &= n; /* even n: 0, odd n: signbit(x) */ |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 44 | x = fabsf(x); |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 45 | if (ix == 0 || ix == 0x7f800000) /* if x is 0 or inf */ |
nsz | 0cbb654 | 2012-03-19 23:41:19 +0100 | [diff] [blame] | 46 | b = 0.0f; |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 47 | else if (nm1 < x) { |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 48 | /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ |
| 49 | a = j0f(x); |
| 50 | b = j1f(x); |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 51 | for (i=0; i<nm1; ){ |
| 52 | i++; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 53 | temp = b; |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 54 | b = b*(2.0f*i/x) - a; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 55 | a = temp; |
| 56 | } |
| 57 | } else { |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 58 | if (ix < 0x35800000) { /* x < 2**-20 */ |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 59 | /* x is tiny, return the first Taylor expansion of J(n,x) |
| 60 | * J(n,x) = 1/n!*(x/2)^n - ... |
| 61 | */ |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 62 | if (nm1 > 8) /* underflow */ |
| 63 | nm1 = 8; |
| 64 | temp = 0.5f * x; |
| 65 | b = temp; |
| 66 | a = 1.0f; |
| 67 | for (i=2; i<=nm1+1; i++) { |
| 68 | a *= (float)i; /* a = n! */ |
| 69 | b *= temp; /* b = (x/2)^n */ |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 70 | } |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 71 | b = b/a; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 72 | } else { |
| 73 | /* use backward recurrence */ |
| 74 | /* x x^2 x^2 |
| 75 | * J(n,x)/J(n-1,x) = ---- ------ ------ ..... |
| 76 | * 2n - 2(n+1) - 2(n+2) |
| 77 | * |
| 78 | * 1 1 1 |
| 79 | * (for large x) = ---- ------ ------ ..... |
| 80 | * 2n 2(n+1) 2(n+2) |
| 81 | * -- - ------ - ------ - |
| 82 | * x x x |
| 83 | * |
| 84 | * Let w = 2n/x and h=2/x, then the above quotient |
| 85 | * is equal to the continued fraction: |
| 86 | * 1 |
| 87 | * = ----------------------- |
| 88 | * 1 |
| 89 | * w - ----------------- |
| 90 | * 1 |
| 91 | * w+h - --------- |
| 92 | * w+2h - ... |
| 93 | * |
| 94 | * To determine how many terms needed, let |
| 95 | * Q(0) = w, Q(1) = w(w+h) - 1, |
| 96 | * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), |
| 97 | * When Q(k) > 1e4 good for single |
| 98 | * When Q(k) > 1e9 good for double |
| 99 | * When Q(k) > 1e17 good for quadruple |
| 100 | */ |
| 101 | /* determine k */ |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 102 | float t,q0,q1,w,h,z,tmp,nf; |
| 103 | int k; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 104 | |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 105 | nf = nm1+1.0f; |
| 106 | w = 2*nf/x; |
| 107 | h = 2/x; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 108 | z = w+h; |
| 109 | q0 = w; |
nsz | 8d0a6f7 | 2012-03-13 20:24:23 +0100 | [diff] [blame] | 110 | q1 = w*z - 1.0f; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 111 | k = 1; |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 112 | while (q1 < 1.0e4f) { |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 113 | k += 1; |
| 114 | z += h; |
| 115 | tmp = z*q1 - q0; |
| 116 | q0 = q1; |
| 117 | q1 = tmp; |
| 118 | } |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 119 | for (t=0.0f, i=k; i>=0; i--) |
| 120 | t = 1.0f/(2*(i+nf)/x-t); |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 121 | a = t; |
nsz | 0cbb654 | 2012-03-19 23:41:19 +0100 | [diff] [blame] | 122 | b = 1.0f; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 123 | /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) |
| 124 | * Hence, if n*(log(2n/x)) > ... |
| 125 | * single 8.8722839355e+01 |
| 126 | * double 7.09782712893383973096e+02 |
| 127 | * long double 1.1356523406294143949491931077970765006170e+04 |
| 128 | * then recurrent value may overflow and the result is |
| 129 | * likely underflow to zero |
| 130 | */ |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 131 | tmp = nf*logf(fabsf(w)); |
nsz | 8d0a6f7 | 2012-03-13 20:24:23 +0100 | [diff] [blame] | 132 | if (tmp < 88.721679688f) { |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 133 | for (i=nm1; i>0; i--) { |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 134 | temp = b; |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 135 | b = 2.0f*i*b/x - a; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 136 | a = temp; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 137 | } |
| 138 | } else { |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 139 | for (i=nm1; i>0; i--){ |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 140 | temp = b; |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 141 | b = 2.0f*i*b/x - a; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 142 | a = temp; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 143 | /* scale b to avoid spurious overflow */ |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 144 | if (b > 0x1p60f) { |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 145 | a /= b; |
| 146 | t /= b; |
nsz | 0cbb654 | 2012-03-19 23:41:19 +0100 | [diff] [blame] | 147 | b = 1.0f; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 148 | } |
| 149 | } |
| 150 | } |
| 151 | z = j0f(x); |
| 152 | w = j1f(x); |
| 153 | if (fabsf(z) >= fabsf(w)) |
| 154 | b = t*z/b; |
| 155 | else |
| 156 | b = t*w/a; |
| 157 | } |
| 158 | } |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 159 | return sign ? -b : b; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 160 | } |
| 161 | |
| 162 | float ynf(int n, float x) |
| 163 | { |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 164 | uint32_t ix, ib; |
| 165 | int nm1, sign, i; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 166 | float a, b, temp; |
| 167 | |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 168 | GET_FLOAT_WORD(ix, x); |
| 169 | sign = ix>>31; |
| 170 | ix &= 0x7fffffff; |
| 171 | if (ix > 0x7f800000) /* nan */ |
| 172 | return x; |
| 173 | if (sign && ix != 0) /* x < 0 */ |
| 174 | return 0/0.0f; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 175 | if (ix == 0x7f800000) |
nsz | 0cbb654 | 2012-03-19 23:41:19 +0100 | [diff] [blame] | 176 | return 0.0f; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 177 | |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 178 | if (n == 0) |
| 179 | return y0f(x); |
| 180 | if (n < 0) { |
| 181 | nm1 = -(n+1); |
| 182 | sign = n&1; |
| 183 | } else { |
| 184 | nm1 = n-1; |
| 185 | sign = 0; |
| 186 | } |
| 187 | if (nm1 == 0) |
| 188 | return sign ? -y1f(x) : y1f(x); |
| 189 | |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 190 | a = y0f(x); |
| 191 | b = y1f(x); |
| 192 | /* quit if b is -inf */ |
| 193 | GET_FLOAT_WORD(ib,b); |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 194 | for (i = 0; i < nm1 && ib != 0xff800000; ) { |
| 195 | i++; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 196 | temp = b; |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 197 | b = (2.0f*i/x)*b - a; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 198 | GET_FLOAT_WORD(ib, b); |
| 199 | a = temp; |
| 200 | } |
Szabolcs Nagy | 5652d70 | 2013-01-01 22:20:45 +0100 | [diff] [blame] | 201 | return sign ? -b : b; |
Rich Felker | b69f695 | 2012-03-13 01:17:53 -0400 | [diff] [blame] | 202 | } |