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 16 #define _GNU_SOURCE 17 #include "libm.h" 18 19 float jnf(int n, float x) 20 { 21 uint32_t ix; 22 int nm1, sign, i; 23 float a, b, temp; 24 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); 34 if (n < 0) { 35 nm1 = -(n+1); 36 x = -x; 37 sign ^= 1; 38 } else 39 nm1 = n-1; 40 if (nm1 == 0) 41 return j1f(x); 42 43 sign &= n; /* even n: 0, odd n: signbit(x) */ 44 x = fabsf(x); 45 if (ix == 0 || ix == 0x7f800000) /* if x is 0 or inf */ 46 b = 0.0f; 47 else if (nm1 < x) { 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); 51 for (i=0; i<nm1; ){ 52 i++; 53 temp = b; 54 b = b*(2.0f*i/x) - a; 55 a = temp; 56 } 57 } else { 58 if (ix < 0x35800000) { /* x < 2**-20 */ 59 /* x is tiny, return the first Taylor expansion of J(n,x) 60 * J(n,x) = 1/n!*(x/2)^n - ... 61 */ 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 */ 70 } 71 b = b/a; 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 */ 102 float t,q0,q1,w,h,z,tmp,nf; 103 int k; 104 105 nf = nm1+1.0f; 106 w = 2*nf/x; 107 h = 2/x; 108 z = w+h; 109 q0 = w; 110 q1 = w*z - 1.0f; 111 k = 1; 112 while (q1 < 1.0e4f) { 113 k += 1; 114 z += h; 115 tmp = z*q1 - q0; 116 q0 = q1; 117 q1 = tmp; 118 } 119 for (t=0.0f, i=k; i>=0; i--) 120 t = 1.0f/(2*(i+nf)/x-t); 121 a = t; 122 b = 1.0f; 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 */ 131 tmp = nf*logf(fabsf(w)); 132 if (tmp < 88.721679688f) { 133 for (i=nm1; i>0; i--) { 134 temp = b; 135 b = 2.0f*i*b/x - a; 136 a = temp; 137 } 138 } else { 139 for (i=nm1; i>0; i--){ 140 temp = b; 141 b = 2.0f*i*b/x - a; 142 a = temp; 143 /* scale b to avoid spurious overflow */ 144 if (b > 0x1p60f) { 145 a /= b; 146 t /= b; 147 b = 1.0f; 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 } 159 return sign ? -b : b; 160 } 161 162 float ynf(int n, float x) 163 { 164 uint32_t ix, ib; 165 int nm1, sign, i; 166 float a, b, temp; 167 168 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; 175 if (ix == 0x7f800000) 176 return 0.0f; 177 178 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); 189 190 a = y0f(x); 191 b = y1f(x); 192 /* quit if b is -inf */ 193 GET_FLOAT_WORD(ib,b); 194 for (i = 0; i < nm1 && ib != 0xff800000; ) { 195 i++; 196 temp = b; 197 b = (2.0f*i/x)*b - a; 198 GET_FLOAT_WORD(ib, b); 199 a = temp; 200 } 201 return sign ? -b : b; 202 } 203