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/* e_jnf.c -- float version of e_jn.c. |
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* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. |
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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 "math.h" |
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#include "math_private.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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jnf(int n, float x) |
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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(ix>0x7f800000) 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(j0f(x)); |
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if(n==1) return(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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if(ix==0||ix>=0x7f800000) /* if x is 0 or inf */ |
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b = 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 = j0f(x); |
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b = j1f(x); |
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for(i=1;i<n;i++){ |
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temp = b; |
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b = b*((float)(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*logf(fabsf(v*tmp)); |
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if(tmp<(float)8.8721679688e+01) { |
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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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b = (t*j0f(x)/b); |
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} |
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} |
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if(sgn==1) return -b; else return b; |
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} |
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float |
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ynf(int n, float x) |
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{ |
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int32_t i,hx,ix,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(ix>0x7f800000) return x+x; |
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if(ix==0) return -one/zero; |
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if(hx<0) return zero/zero; |
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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(y0f(x)); |
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if(n==1) return(sign*y1f(x)); |
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if(ix==0x7f800000) return zero; |
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a = y0f(x); |
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b = 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 = ((float)(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(sign>0) return b; else return -b; |
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} |