12116Sjkh/* e_jnf.c -- float version of e_jn.c. 22116Sjkh * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. 32116Sjkh */ 42116Sjkh 52116Sjkh/* 62116Sjkh * ==================================================== 72116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 82116Sjkh * 92116Sjkh * Developed at SunPro, a Sun Microsystems, Inc. business. 102116Sjkh * Permission to use, copy, modify, and distribute this 118870Srgrimes * software is freely granted, provided that this notice 122116Sjkh * is preserved. 132116Sjkh * ==================================================== 142116Sjkh */ 152116Sjkh 16176451Sdas#include <sys/cdefs.h> 17176451Sdas__FBSDID("$FreeBSD$"); 182116Sjkh 192116Sjkh#include "math.h" 202116Sjkh#include "math_private.h" 212116Sjkh 222116Sjkhstatic const float 232116Sjkhtwo = 2.0000000000e+00, /* 0x40000000 */ 242116Sjkhone = 1.0000000000e+00; /* 0x3F800000 */ 252116Sjkh 262116Sjkhstatic const float zero = 0.0000000000e+00; 272116Sjkh 2897413Salfredfloat 2997413Salfred__ieee754_jnf(int n, float x) 302116Sjkh{ 312116Sjkh int32_t i,hx,ix, sgn; 322116Sjkh float a, b, temp, di; 332116Sjkh float z, w; 342116Sjkh 352116Sjkh /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) 362116Sjkh * Thus, J(-n,x) = J(n,-x) 372116Sjkh */ 382116Sjkh GET_FLOAT_WORD(hx,x); 392116Sjkh ix = 0x7fffffff&hx; 402116Sjkh /* if J(n,NaN) is NaN */ 412116Sjkh if(ix>0x7f800000) return x+x; 428870Srgrimes if(n<0){ 432116Sjkh n = -n; 442116Sjkh x = -x; 452116Sjkh hx ^= 0x80000000; 462116Sjkh } 472116Sjkh if(n==0) return(__ieee754_j0f(x)); 482116Sjkh if(n==1) return(__ieee754_j1f(x)); 492116Sjkh sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ 502116Sjkh x = fabsf(x); 512116Sjkh if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */ 522116Sjkh b = zero; 538870Srgrimes else if((float)n<=x) { 542116Sjkh /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ 552116Sjkh a = __ieee754_j0f(x); 562116Sjkh b = __ieee754_j1f(x); 572116Sjkh for(i=1;i<n;i++){ 582116Sjkh temp = b; 592116Sjkh b = b*((float)(i+i)/x) - a; /* avoid underflow */ 602116Sjkh a = temp; 612116Sjkh } 622116Sjkh } else { 632116Sjkh if(ix<0x30800000) { /* x < 2**-29 */ 648870Srgrimes /* x is tiny, return the first Taylor expansion of J(n,x) 652116Sjkh * J(n,x) = 1/n!*(x/2)^n - ... 662116Sjkh */ 672116Sjkh if(n>33) /* underflow */ 682116Sjkh b = zero; 692116Sjkh else { 702116Sjkh temp = x*(float)0.5; b = temp; 712116Sjkh for (a=one,i=2;i<=n;i++) { 722116Sjkh a *= (float)i; /* a = n! */ 732116Sjkh b *= temp; /* b = (x/2)^n */ 742116Sjkh } 752116Sjkh b = b/a; 762116Sjkh } 772116Sjkh } else { 782116Sjkh /* use backward recurrence */ 798870Srgrimes /* x x^2 x^2 802116Sjkh * J(n,x)/J(n-1,x) = ---- ------ ------ ..... 812116Sjkh * 2n - 2(n+1) - 2(n+2) 822116Sjkh * 838870Srgrimes * 1 1 1 842116Sjkh * (for large x) = ---- ------ ------ ..... 852116Sjkh * 2n 2(n+1) 2(n+2) 868870Srgrimes * -- - ------ - ------ - 872116Sjkh * x x x 882116Sjkh * 892116Sjkh * Let w = 2n/x and h=2/x, then the above quotient 902116Sjkh * is equal to the continued fraction: 912116Sjkh * 1 922116Sjkh * = ----------------------- 932116Sjkh * 1 942116Sjkh * w - ----------------- 952116Sjkh * 1 962116Sjkh * w+h - --------- 972116Sjkh * w+2h - ... 982116Sjkh * 992116Sjkh * To determine how many terms needed, let 1002116Sjkh * Q(0) = w, Q(1) = w(w+h) - 1, 1012116Sjkh * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), 1028870Srgrimes * When Q(k) > 1e4 good for single 1038870Srgrimes * When Q(k) > 1e9 good for double 1048870Srgrimes * When Q(k) > 1e17 good for quadruple 1052116Sjkh */ 1062116Sjkh /* determine k */ 1072116Sjkh float t,v; 1082116Sjkh float q0,q1,h,tmp; int32_t k,m; 1092116Sjkh w = (n+n)/(float)x; h = (float)2.0/(float)x; 1102116Sjkh q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1; 1112116Sjkh while(q1<(float)1.0e9) { 1122116Sjkh k += 1; z += h; 1132116Sjkh tmp = z*q1 - q0; 1142116Sjkh q0 = q1; 1152116Sjkh q1 = tmp; 1162116Sjkh } 1172116Sjkh m = n+n; 1182116Sjkh for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); 1192116Sjkh a = t; 1202116Sjkh b = one; 1212116Sjkh /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) 1222116Sjkh * Hence, if n*(log(2n/x)) > ... 1232116Sjkh * single 8.8722839355e+01 1242116Sjkh * double 7.09782712893383973096e+02 1252116Sjkh * long double 1.1356523406294143949491931077970765006170e+04 1268870Srgrimes * then recurrent value may overflow and the result is 1272116Sjkh * likely underflow to zero 1282116Sjkh */ 1292116Sjkh tmp = n; 1302116Sjkh v = two/x; 1312116Sjkh tmp = tmp*__ieee754_logf(fabsf(v*tmp)); 1322116Sjkh if(tmp<(float)8.8721679688e+01) { 1332116Sjkh for(i=n-1,di=(float)(i+i);i>0;i--){ 1342116Sjkh temp = b; 1352116Sjkh b *= di; 1362116Sjkh b = b/x - a; 1372116Sjkh a = temp; 1382116Sjkh di -= two; 1392116Sjkh } 1402116Sjkh } else { 1412116Sjkh for(i=n-1,di=(float)(i+i);i>0;i--){ 1422116Sjkh temp = b; 1432116Sjkh b *= di; 1442116Sjkh b = b/x - a; 1452116Sjkh a = temp; 1462116Sjkh di -= two; 1472116Sjkh /* scale b to avoid spurious overflow */ 1482116Sjkh if(b>(float)1e10) { 1492116Sjkh a /= b; 1502116Sjkh t /= b; 1512116Sjkh b = one; 1522116Sjkh } 1532116Sjkh } 1542116Sjkh } 155215237Suqs z = __ieee754_j0f(x); 156215237Suqs w = __ieee754_j1f(x); 157215237Suqs if (fabsf(z) >= fabsf(w)) 158215237Suqs b = (t*z/b); 159215237Suqs else 160215237Suqs b = (t*w/a); 1612116Sjkh } 1622116Sjkh } 1632116Sjkh if(sgn==1) return -b; else return b; 1642116Sjkh} 1652116Sjkh 16697413Salfredfloat 16797413Salfred__ieee754_ynf(int n, float x) 1682116Sjkh{ 1692116Sjkh int32_t i,hx,ix,ib; 1702116Sjkh int32_t sign; 1712116Sjkh float a, b, temp; 1722116Sjkh 1732116Sjkh GET_FLOAT_WORD(hx,x); 1742116Sjkh ix = 0x7fffffff&hx; 1752116Sjkh /* if Y(n,NaN) is NaN */ 1762116Sjkh if(ix>0x7f800000) return x+x; 1772116Sjkh if(ix==0) return -one/zero; 1782116Sjkh if(hx<0) return zero/zero; 1792116Sjkh sign = 1; 1802116Sjkh if(n<0){ 1812116Sjkh n = -n; 1827658Sbde sign = 1 - ((n&1)<<1); 1832116Sjkh } 1842116Sjkh if(n==0) return(__ieee754_y0f(x)); 1852116Sjkh if(n==1) return(sign*__ieee754_y1f(x)); 1862116Sjkh if(ix==0x7f800000) return zero; 1872116Sjkh 1882116Sjkh a = __ieee754_y0f(x); 1892116Sjkh b = __ieee754_y1f(x); 1902116Sjkh /* quit if b is -inf */ 1912116Sjkh GET_FLOAT_WORD(ib,b); 1928870Srgrimes for(i=1;i<n&&ib!=0xff800000;i++){ 1932116Sjkh temp = b; 1942116Sjkh b = ((float)(i+i)/x)*b - a; 1952116Sjkh GET_FLOAT_WORD(ib,b); 1962116Sjkh a = temp; 1972116Sjkh } 1982116Sjkh if(sign>0) return b; else return -b; 1992116Sjkh} 200