1239313Sdim/* origin: FreeBSD /usr/src/lib/msun/src/e_jnf.c */ 2239313Sdim/* 3239313Sdim * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. 4239313Sdim */ 5239313Sdim/* 6239313Sdim * ==================================================== 7239313Sdim * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 8239313Sdim * 9239313Sdim * Developed at SunPro, a Sun Microsystems, Inc. business. 10239313Sdim * Permission to use, copy, modify, and distribute this 11252723Sdim * software is freely granted, provided that this notice 12252723Sdim * is preserved. 13239313Sdim * ==================================================== 14239313Sdim */ 15239313Sdim 16239313Sdim#define _GNU_SOURCE 17245431Sdim#include "libm.h" 18252723Sdim 19239313Sdimfloat jnf(int n, float x) 20239313Sdim{ 21239313Sdim uint32_t ix; 22239313Sdim int nm1, sign, i; 23239313Sdim float a, b, temp; 24245431Sdim 25245431Sdim GET_FLOAT_WORD(ix, x); 26245431Sdim sign = ix>>31; 27245431Sdim ix &= 0x7fffffff; 28239313Sdim if (ix > 0x7f800000) /* nan */ 29245431Sdim return x; 30245431Sdim 31239313Sdim /* J(-n,x) = J(n,-x), use |n|-1 to avoid overflow in -n */ 32263509Sdim if (n == 0) 33239313Sdim return j0f(x); 34239313Sdim if (n < 0) { 35239313Sdim nm1 = -(n+1); 36239313Sdim x = -x; 37239313Sdim sign ^= 1; 38239313Sdim } else 39239313Sdim nm1 = n-1; 40245431Sdim if (nm1 == 0) 41239313Sdim return j1f(x); 42239313Sdim 43239313Sdim sign &= n; /* even n: 0, odd n: signbit(x) */ 44239313Sdim x = fabsf(x); 45239313Sdim if (ix == 0 || ix == 0x7f800000) /* if x is 0 or inf */ 46239313Sdim b = 0.0f; 47239313Sdim else if (nm1 < x) { 48239313Sdim /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 49252723Sdim a = j0f(x); 50252723Sdim b = j1f(x); 51252723Sdim for (i=0; i<nm1; ){ 52252723Sdim i++; 53252723Sdim temp = b; 54252723Sdim b = b*(2.0f*i/x) - a; 55252723Sdim a = temp; 56252723Sdim } 57252723Sdim } else { 58252723Sdim if (ix < 0x35800000) { /* x < 2**-20 */ 59239313Sdim /* x is tiny, return the first Taylor expansion of J(n,x) 60239313Sdim * J(n,x) = 1/n!*(x/2)^n - ... 61239313Sdim */ 62239313Sdim if (nm1 > 8) /* underflow */ 63239313Sdim nm1 = 8; 64239313Sdim temp = 0.5f * x; 65239313Sdim b = temp; 66239313Sdim a = 1.0f; 67239313Sdim for (i=2; i<=nm1+1; i++) { 68239313Sdim a *= (float)i; /* a = n! */ 69239313Sdim b *= temp; /* b = (x/2)^n */ 70239313Sdim } 71239313Sdim b = b/a; 72245431Sdim } else { 73239313Sdim /* use backward recurrence */ 74239313Sdim /* x x^2 x^2 75252723Sdim * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 76252723Sdim * 2n - 2(n+1) - 2(n+2) 77252723Sdim * 78252723Sdim * 1 1 1 79252723Sdim * (for large x) = ---- ------ ------ ..... 80239313Sdim * 2n 2(n+1) 2(n+2) 81252723Sdim * -- - ------ - ------ - 82252723Sdim * x x x 83239313Sdim * 84239313Sdim * Let w = 2n/x and h=2/x, then the above quotient 85239313Sdim * is equal to the continued fraction: 86239313Sdim * 1 87252723Sdim * = ----------------------- 88245431Sdim * 1 89239313Sdim * w - ----------------- 90239313Sdim * 1 91239313Sdim * w+h - --------- 92239313Sdim * w+2h - ... 93252723Sdim * 94252723Sdim * To determine how many terms needed, let 95252723Sdim * Q(0) = w, Q(1) = w(w+h) - 1, 96252723Sdim * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 97252723Sdim * When Q(k) > 1e4 good for single 98252723Sdim * When Q(k) > 1e9 good for double 99252723Sdim * When Q(k) > 1e17 good for quadruple 100252723Sdim */ 101263509Sdim /* determine k */ 102252723Sdim float t,q0,q1,w,h,z,tmp,nf; 103252723Sdim int k; 104263509Sdim 105252723Sdim nf = nm1+1.0f; 106252723Sdim w = 2*nf/x; 107252723Sdim h = 2/x; 108252723Sdim z = w+h; 109252723Sdim q0 = w; 110252723Sdim q1 = w*z - 1.0f; 111252723Sdim k = 1; 112252723Sdim while (q1 < 1.0e4f) { 113252723Sdim k += 1; 114252723Sdim z += h; 115252723Sdim tmp = z*q1 - q0; 116252723Sdim q0 = q1; 117252723Sdim q1 = tmp; 118252723Sdim } 119252723Sdim for (t=0.0f, i=k; i>=0; i--) 120252723Sdim t = 1.0f/(2*(i+nf)/x-t); 121252723Sdim a = t; 122252723Sdim b = 1.0f; 123252723Sdim /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 124252723Sdim * Hence, if n*(log(2n/x)) > ... 125252723Sdim * single 8.8722839355e+01 126252723Sdim * double 7.09782712893383973096e+02 127252723Sdim * long double 1.1356523406294143949491931077970765006170e+04 128252723Sdim * then recurrent value may overflow and the result is 129252723Sdim * likely underflow to zero 130252723Sdim */ 131252723Sdim tmp = nf*logf(fabsf(w)); 132252723Sdim if (tmp < 88.721679688f) { 133263509Sdim for (i=nm1; i>0; i--) { 134263509Sdim temp = b; 135263509Sdim b = 2.0f*i*b/x - a; 136263509Sdim a = temp; 137263509Sdim } 138263509Sdim } else { 139252723Sdim for (i=nm1; i>0; i--){ 140252723Sdim temp = b; 141252723Sdim b = 2.0f*i*b/x - a; 142252723Sdim a = temp; 143252723Sdim /* scale b to avoid spurious overflow */ 144252723Sdim if (b > 0x1p60f) { 145252723Sdim a /= b; 146252723Sdim t /= b; 147252723Sdim b = 1.0f; 148252723Sdim } 149252723Sdim } 150252723Sdim } 151252723Sdim z = j0f(x); 152252723Sdim w = j1f(x); 153252723Sdim if (fabsf(z) >= fabsf(w)) 154252723Sdim b = t*z/b; 155252723Sdim else 156252723Sdim b = t*w/a; 157252723Sdim } 158252723Sdim } 159252723Sdim return sign ? -b : b; 160252723Sdim} 161252723Sdim 162252723Sdimfloat ynf(int n, float x) 163252723Sdim{ 164252723Sdim uint32_t ix, ib; 165252723Sdim int nm1, sign, i; 166252723Sdim float a, b, temp; 167252723Sdim 168252723Sdim GET_FLOAT_WORD(ix, x); 169252723Sdim sign = ix>>31; 170252723Sdim ix &= 0x7fffffff; 171252723Sdim if (ix > 0x7f800000) /* nan */ 172252723Sdim return x; 173252723Sdim if (sign && ix != 0) /* x < 0 */ 174252723Sdim return 0/0.0f; 175252723Sdim if (ix == 0x7f800000) 176252723Sdim return 0.0f; 177252723Sdim 178252723Sdim if (n == 0) 179252723Sdim return y0f(x); 180252723Sdim if (n < 0) { 181252723Sdim nm1 = -(n+1); 182252723Sdim sign = n&1; 183252723Sdim } else { 184252723Sdim nm1 = n-1; 185252723Sdim sign = 0; 186252723Sdim } 187252723Sdim if (nm1 == 0) 188252723Sdim return sign ? -y1f(x) : y1f(x); 189252723Sdim 190252723Sdim a = y0f(x); 191252723Sdim b = y1f(x); 192252723Sdim /* quit if b is -inf */ 193252723Sdim GET_FLOAT_WORD(ib,b); 194252723Sdim for (i = 0; i < nm1 && ib != 0xff800000; ) { 195252723Sdim i++; 196252723Sdim temp = b; 197252723Sdim b = (2.0f*i/x)*b - a; 198252723Sdim GET_FLOAT_WORD(ib, b); 199252723Sdim a = temp; 200252723Sdim } 201252723Sdim return sign ? -b : b; 202252723Sdim} 203252723Sdim