/* ef_jn.c -- float version of e_jn.c. * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #include "fdlibm.h" #ifdef __STDC__ static const float #else static float #endif invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ two = 2.0000000000e+00, /* 0x40000000 */ one = 1.0000000000e+00; /* 0x3F800000 */ #ifdef __STDC__ static const float zero = 0.0000000000e+00; #else static float zero = 0.0000000000e+00; #endif #ifdef __STDC__ float __ieee754_jnf(int n, float x) #else float __ieee754_jnf(n,x) int n; float x; #endif { __int32_t i,hx,ix, sgn; float a, b, temp, di; float z, w; /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) * Thus, J(-n,x) = J(n,-x) */ GET_FLOAT_WORD(hx,x); ix = 0x7fffffff&hx; /* if J(n,NaN) is NaN */ if(FLT_UWORD_IS_NAN(ix)) return x+x; if(n<0){ n = -n; x = -x; hx ^= 0x80000000; } if(n==0) return(__ieee754_j0f(x)); if(n==1) return(__ieee754_j1f(x)); sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */ x = fabsf(x); if(FLT_UWORD_IS_ZERO(ix)||FLT_UWORD_IS_INFINITE(ix)) b = zero; else if((float)n<=x) { /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ a = __ieee754_j0f(x); b = __ieee754_j1f(x); for(i=1;i<n;i++){ temp = b; b = b*((float)(i+i)/x) - a; /* avoid underflow */ a = temp; } } else { if(ix<0x30800000) { /* x < 2**-29 */ /* x is tiny, return the first Taylor expansion of J(n,x) * J(n,x) = 1/n!*(x/2)^n - ... */ if(n>33) /* underflow */ b = zero; else { temp = x*(float)0.5; b = temp; for (a=one,i=2;i<=n;i++) { a *= (float)i; /* a = n! */ b *= temp; /* b = (x/2)^n */ } b = b/a; } } else { /* use backward recurrence */ /* x x^2 x^2 * J(n,x)/J(n-1,x) = ---- ------ ------ ..... * 2n - 2(n+1) - 2(n+2) * * 1 1 1 * (for large x) = ---- ------ ------ ..... * 2n 2(n+1) 2(n+2) * -- - ------ - ------ - * x x x * * Let w = 2n/x and h=2/x, then the above quotient * is equal to the continued fraction: * 1 * = ----------------------- * 1 * w - ----------------- * 1 * w+h - --------- * w+2h - ... * * To determine how many terms needed, let * Q(0) = w, Q(1) = w(w+h) - 1, * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), * When Q(k) > 1e4 good for single * When Q(k) > 1e9 good for double * When Q(k) > 1e17 good for quadruple */ /* determine k */ float t,v; float q0,q1,h,tmp; __int32_t k,m; w = (n+n)/(float)x; h = (float)2.0/(float)x; q0 = w; z = w+h; q1 = w*z - (float)1.0; k=1; while(q1<(float)1.0e9) { k += 1; z += h; tmp = z*q1 - q0; q0 = q1; q1 = tmp; } m = n+n; for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t); a = t; b = one; /* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n) * Hence, if n*(log(2n/x)) > ... * single 8.8722839355e+01 * double 7.09782712893383973096e+02 * long double 1.1356523406294143949491931077970765006170e+04 * then recurrent value may overflow and the result is * likely underflow to zero */ tmp = n; v = two/x; tmp = tmp*__ieee754_logf(fabsf(v*tmp)); if(tmp<(float)8.8721679688e+01) { for(i=n-1,di=(float)(i+i);i>0;i--){ temp = b; b *= di; b = b/x - a; a = temp; di -= two; } } else { for(i=n-1,di=(float)(i+i);i>0;i--){ temp = b; b *= di; b = b/x - a; a = temp; di -= two; /* scale b to avoid spurious overflow */ if(b>(float)1e10) { a /= b; t /= b; b = one; } } } b = (t*__ieee754_j0f(x)/b); } } if(sgn==1) return -b; else return b; } #ifdef __STDC__ float __ieee754_ynf(int n, float x) #else float __ieee754_ynf(n,x) int n; float x; #endif { __int32_t i,hx,ix,ib; __int32_t sign; float a, b, temp; GET_FLOAT_WORD(hx,x); ix = 0x7fffffff&hx; /* if Y(n,NaN) is NaN */ if(FLT_UWORD_IS_NAN(ix)) return x+x; if(FLT_UWORD_IS_ZERO(ix)) return -one/zero; if(hx<0) return zero/zero; sign = 1; if(n<0){ n = -n; sign = 1 - ((n&1)<<1); } if(n==0) return(__ieee754_y0f(x)); if(n==1) return(sign*__ieee754_y1f(x)); if(FLT_UWORD_IS_INFINITE(ix)) return zero; a = __ieee754_y0f(x); b = __ieee754_y1f(x); /* quit if b is -inf */ GET_FLOAT_WORD(ib,b); for(i=1;i<n&&ib!=0xff800000;i++){ temp = b; b = ((float)(i+i)/x)*b - a; GET_FLOAT_WORD(ib,b); a = temp; } if(sign>0) return b; else return -b; }