Fix whitespace issues in bessel function routines.
PR: 229423 Submitted by: Steve Kargl <sgk@troutmask.apl.washington.edu> MFC after: 3 days
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@ -1,4 +1,3 @@
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/* @(#)e_j0.c 1.3 95/01/18 */
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/*
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* ====================================================
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@ -6,7 +5,7 @@
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*
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* Developed at SunSoft, 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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* 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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@ -33,20 +32,20 @@ __FBSDID("$FreeBSD$");
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* (To avoid cancellation, use
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* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
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* to compute the worse one.)
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*
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*
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* 3 Special cases
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* j0(nan)= nan
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* j0(0) = 1
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* j0(inf) = 0
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*
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*
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* Method -- y0(x):
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* 1. For x<2.
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* Since
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* Since
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* y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
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* therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
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* We use the following function to approximate y0,
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* y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
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* where
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* where
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* U(z) = u00 + u01*z + ... + u06*z^6
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* V(z) = 1 + v01*z + ... + v04*z^4
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* with absolute approximation error bounded by 2**-72.
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@ -71,7 +70,7 @@ huge = 1e300,
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one = 1.0,
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invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
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tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
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/* R0/S0 on [0, 2.00] */
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/* R0/S0 on [0, 2.00] */
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R02 = 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
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R03 = -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
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R04 = 1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
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@ -157,7 +156,7 @@ __ieee754_y0(double x)
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* y0(Inf) = 0.
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* y0(-Inf) = NaN and raise invalid exception.
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*/
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if(ix>=0x7ff00000) return vone/(x+x*x);
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if(ix>=0x7ff00000) return vone/(x+x*x);
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/* y0(+-0) = -inf and raise divide-by-zero exception. */
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if((ix|lx)==0) return -one/vzero;
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/* y0(x<0) = NaN and raise invalid exception. */
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@ -293,7 +292,7 @@ pzero(double x)
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s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
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return one+ r/s;
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}
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/* For x >= 8, the asymptotic expansions of qzero is
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* -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
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@ -1,4 +1,3 @@
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/* @(#)e_j1.c 1.3 95/01/18 */
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/*
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* ====================================================
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@ -6,7 +5,7 @@
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*
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* Developed at SunSoft, 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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* 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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@ -34,16 +33,16 @@ __FBSDID("$FreeBSD$");
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* (To avoid cancellation, use
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* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
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* to compute the worse one.)
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*
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*
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* 3 Special cases
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* j1(nan)= nan
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* j1(0) = 0
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* j1(inf) = 0
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*
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*
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* Method -- y1(x):
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* 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
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* 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
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* 2. For x<2.
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* Since
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* Since
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* y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
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* therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
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* We use the following function to approximate y1,
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@ -154,7 +153,7 @@ __ieee754_y1(double x)
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* y1(Inf) = 0.
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* y1(-Inf) = NaN and raise invalid exception.
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*/
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if(ix>=0x7ff00000) return vone/(x+x*x);
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if(ix>=0x7ff00000) return vone/(x+x*x);
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/* y1(+-0) = -inf and raise divide-by-zero exception. */
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if((ix|lx)==0) return -one/vzero;
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/* y1(x<0) = NaN and raise invalid exception. */
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@ -186,10 +185,10 @@ __ieee754_y1(double x)
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z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
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}
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return z;
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}
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}
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if(ix<=0x3c900000) { /* x < 2**-54 */
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return(-tpi/x);
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}
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}
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z = x*x;
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u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
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v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
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@ -287,7 +286,7 @@ pone(double x)
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s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
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return one+ r/s;
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}
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/* For x >= 8, the asymptotic expansions of qone is
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* 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
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@ -32,7 +32,7 @@ huge = 1e30,
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one = 1.0,
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invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
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tpi = 6.3661974669e-01, /* 0x3f22f983 */
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/* R0/S0 on [0,2] */
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/* R0/S0 on [0,2] */
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r00 = -6.2500000000e-02, /* 0xbd800000 */
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r01 = 1.4070566976e-03, /* 0x3ab86cfd */
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r02 = -1.5995563444e-05, /* 0xb7862e36 */
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@ -1,4 +1,3 @@
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/* @(#)e_jn.c 1.4 95/01/18 */
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/*
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* ====================================================
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@ -6,7 +5,7 @@
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*
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* Developed at SunSoft, 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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* 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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@ -18,7 +17,7 @@ __FBSDID("$FreeBSD$");
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* __ieee754_jn(n, x), __ieee754_yn(n, x)
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* floating point Bessel's function of the 1st and 2nd kind
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* of order n
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*
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*
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* Special cases:
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* y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
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* y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
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@ -37,7 +36,6 @@ __FBSDID("$FreeBSD$");
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* yn(n,x) is similar in all respects, except
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* that forward recursion is used for all
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* values of n>1.
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*
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*/
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#include "math.h"
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@ -66,7 +64,7 @@ __ieee754_jn(int n, double x)
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ix = 0x7fffffff&hx;
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/* if J(n,NaN) is NaN */
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if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
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if(n<0){
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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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@ -77,14 +75,14 @@ __ieee754_jn(int n, double x)
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x = fabs(x);
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if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
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b = zero;
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else if((double)n<=x) {
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else if((double)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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if(ix>=0x52D00000) { /* x > 2**302 */
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/* (x >> n**2)
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/* (x >> n**2)
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then
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*
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* n sin(xn)*sqt2 cos(xn)*sqt2
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* ----------------------------------
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@ -100,7 +98,7 @@ __ieee754_jn(int n, double x)
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case 3: temp = cos(x)-sin(x); break;
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}
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b = invsqrtpi*temp/sqrt(x);
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} else {
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} else {
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a = __ieee754_j0(x);
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b = __ieee754_j1(x);
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for(i=1;i<n;i++){
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@ -111,7 +109,7 @@ __ieee754_jn(int n, double x)
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}
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} else {
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if(ix<0x3e100000) { /* x < 2**-29 */
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/* x is tiny, return the first Taylor expansion of J(n,x)
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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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@ -126,14 +124,14 @@ __ieee754_jn(int n, double x)
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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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/* 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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* 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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* -- - ------ - ------ -
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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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@ -149,9 +147,9 @@ __ieee754_jn(int n, double x)
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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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* 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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double t,v;
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@ -237,11 +235,11 @@ __ieee754_yn(int n, double x)
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if(n==1) return(sign*__ieee754_y1(x));
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if(ix==0x7ff00000) return zero;
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if(ix>=0x52D00000) { /* x > 2**302 */
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/* (x >> n**2)
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/* (x >> n**2)
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2), then
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*
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* n sin(xn)*sqt2 cos(xn)*sqt2
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* ----------------------------------
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