237 lines
5.6 KiB
C
237 lines
5.6 KiB
C
/* e_jnf.c -- float version of e_jn.c.
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*/
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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#include <errno.h>
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#include <float.h>
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#include <math.h>
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#include <math-narrow-eval.h>
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#include <math_private.h>
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#include <fenv_private.h>
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#include <math-underflow.h>
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#include <libm-alias-finite.h>
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static const float
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two = 2.0000000000e+00, /* 0x40000000 */
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one = 1.0000000000e+00; /* 0x3F800000 */
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static const float zero = 0.0000000000e+00;
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float
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__ieee754_jnf(int n, float x)
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{
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float ret;
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{
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int32_t i,hx,ix, sgn;
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float a, b, temp, di;
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float z, w;
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/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
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* Thus, J(-n,x) = J(n,-x)
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*/
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GET_FLOAT_WORD(hx,x);
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ix = 0x7fffffff&hx;
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/* if J(n,NaN) is NaN */
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if(__builtin_expect(ix>0x7f800000, 0)) return x+x;
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if(n<0){
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n = -n;
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x = -x;
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hx ^= 0x80000000;
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}
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if(n==0) return(__ieee754_j0f(x));
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if(n==1) return(__ieee754_j1f(x));
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sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
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x = fabsf(x);
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SET_RESTORE_ROUNDF (FE_TONEAREST);
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if(__builtin_expect(ix==0||ix>=0x7f800000, 0)) /* if x is 0 or inf */
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return sgn == 1 ? -zero : zero;
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else if((float)n<=x) {
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/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
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a = __ieee754_j0f(x);
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b = __ieee754_j1f(x);
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for(i=1;i<n;i++){
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temp = b;
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b = b*((double)(i+i)/x) - a; /* avoid underflow */
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a = temp;
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}
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} else {
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if(ix<0x30800000) { /* x < 2**-29 */
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/* x is tiny, return the first Taylor expansion of J(n,x)
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* J(n,x) = 1/n!*(x/2)^n - ...
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*/
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if(n>33) /* underflow */
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b = zero;
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else {
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temp = x*(float)0.5; b = temp;
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for (a=one,i=2;i<=n;i++) {
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a *= (float)i; /* a = n! */
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b *= temp; /* b = (x/2)^n */
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}
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b = b/a;
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}
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} else {
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/* use backward recurrence */
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/* x x^2 x^2
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* J(n,x)/J(n-1,x) = ---- ------ ------ .....
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* 2n - 2(n+1) - 2(n+2)
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*
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* 1 1 1
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* (for large x) = ---- ------ ------ .....
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* 2n 2(n+1) 2(n+2)
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* -- - ------ - ------ -
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* x x x
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*
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* Let w = 2n/x and h=2/x, then the above quotient
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* is equal to the continued fraction:
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* 1
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* = -----------------------
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* 1
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* w - -----------------
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* 1
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* w+h - ---------
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* w+2h - ...
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*
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* To determine how many terms needed, let
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* Q(0) = w, Q(1) = w(w+h) - 1,
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* Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
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* When Q(k) > 1e4 good for single
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* When Q(k) > 1e9 good for double
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* When Q(k) > 1e17 good for quadruple
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*/
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/* determine k */
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float t,v;
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float q0,q1,h,tmp; int32_t k,m;
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w = (n+n)/(float)x; h = (float)2.0/(float)x;
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q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1;
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while(q1<(float)1.0e9) {
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k += 1; z += h;
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tmp = z*q1 - q0;
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q0 = q1;
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q1 = tmp;
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}
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m = n+n;
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for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
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a = t;
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b = one;
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/* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
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* Hence, if n*(log(2n/x)) > ...
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* single 8.8722839355e+01
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* double 7.09782712893383973096e+02
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* long double 1.1356523406294143949491931077970765006170e+04
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* then recurrent value may overflow and the result is
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* likely underflow to zero
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*/
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tmp = n;
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v = two/x;
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tmp = tmp*__ieee754_logf(fabsf(v*tmp));
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if(tmp<8.8721679688e+01f) {
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for(i=n-1,di=(float)(i+i);i>0;i--){
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temp = b;
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b *= di;
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b = b/x - a;
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a = temp;
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di -= two;
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}
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} else {
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for(i=n-1,di=(float)(i+i);i>0;i--){
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temp = b;
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b *= di;
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b = b/x - a;
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a = temp;
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di -= two;
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/* scale b to avoid spurious overflow */
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if(b>(float)1e10) {
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a /= b;
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t /= b;
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b = one;
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}
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}
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}
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/* j0() and j1() suffer enormous loss of precision at and
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* near zero; however, we know that their zero points never
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* coincide, so just choose the one further away from zero.
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*/
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z = __ieee754_j0f (x);
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w = __ieee754_j1f (x);
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if (fabsf (z) >= fabsf (w))
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b = (t * z / b);
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else
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b = (t * w / a);
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}
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}
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if(sgn==1) ret = -b; else ret = b;
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ret = math_narrow_eval (ret);
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}
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if (ret == 0)
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{
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ret = math_narrow_eval (copysignf (FLT_MIN, ret) * FLT_MIN);
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__set_errno (ERANGE);
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}
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else
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math_check_force_underflow (ret);
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return ret;
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}
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libm_alias_finite (__ieee754_jnf, __jnf)
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float
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__ieee754_ynf(int n, float x)
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{
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float ret;
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{
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int32_t i,hx,ix;
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uint32_t ib;
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int32_t sign;
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float a, b, temp;
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GET_FLOAT_WORD(hx,x);
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ix = 0x7fffffff&hx;
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/* if Y(n,NaN) is NaN */
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if(__builtin_expect(ix>0x7f800000, 0)) return x+x;
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sign = 1;
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if(n<0){
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n = -n;
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sign = 1 - ((n&1)<<1);
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}
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if(n==0) return(__ieee754_y0f(x));
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if(__builtin_expect(ix==0, 0))
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return -sign/zero;
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if(__builtin_expect(hx<0, 0)) return zero/(zero*x);
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SET_RESTORE_ROUNDF (FE_TONEAREST);
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if(n==1) {
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ret = sign*__ieee754_y1f(x);
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goto out;
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}
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if(__builtin_expect(ix==0x7f800000, 0)) return zero;
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a = __ieee754_y0f(x);
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b = __ieee754_y1f(x);
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/* quit if b is -inf */
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GET_FLOAT_WORD(ib,b);
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for(i=1;i<n&&ib!=0xff800000;i++){
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temp = b;
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b = ((double)(i+i)/x)*b - a;
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GET_FLOAT_WORD(ib,b);
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a = temp;
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}
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/* If B is +-Inf, set up errno accordingly. */
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if (! isfinite (b))
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__set_errno (ERANGE);
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if(sign>0) ret = b; else ret = -b;
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}
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out:
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if (isinf (ret))
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ret = copysignf (FLT_MAX, ret) * FLT_MAX;
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return ret;
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}
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libm_alias_finite (__ieee754_ynf, __ynf)
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