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/* @(#)e_j0.c 5.1 93/09/24 */ |
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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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/* j0(x), y0(x) |
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* Bessel function of the first and second kinds of order zero. |
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* Method -- j0(x): |
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* 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ... |
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* 2. Reduce x to |x| since j0(x)=j0(-x), and |
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* for x in (0,2) |
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* j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x; |
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* (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 ) |
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* for x in (2,inf) |
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* j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) |
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* where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) |
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* as follow: |
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* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) |
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* = 1/sqrt(2) * (cos(x) + sin(x)) |
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* sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4) |
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* = 1/sqrt(2) * (sin(x) - cos(x)) |
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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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* 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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* Method -- y0(x): |
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* 1. For x<2. |
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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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* 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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* Note: For tiny x, U/V = u0 and j0(x)~1, hence |
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* y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27) |
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* 2. For x>=2. |
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* y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0)) |
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* where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) |
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* by the method mentioned above. |
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* 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0. |
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*/ |
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#include "math.h" |
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#include "math_private.h" |
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static double pzero(double), qzero(double); |
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static const double |
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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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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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R05 = -4.61832688532103189199e-09, /* 0xBE33D5E7, 0x73D63FCE */ |
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S01 = 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */ |
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S02 = 1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */ |
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S03 = 5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */ |
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S04 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */ |
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static const double zero = 0.0; |
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double |
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j0(double x) |
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{ |
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double z, s,c,ss,cc,r,u,v; |
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int32_t hx,ix; |
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GET_HIGH_WORD(hx,x); |
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ix = hx&0x7fffffff; |
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✗✓ |
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if(ix>=0x7ff00000) return one/(x*x); |
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x = fabs(x); |
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✓✓ |
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if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
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s = sin(x); |
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c = cos(x); |
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ss = s-c; |
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cc = s+c; |
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✓✗ |
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if(ix<0x7fe00000) { /* make sure x+x not overflow */ |
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z = -cos(x+x); |
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✓✓ |
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if ((s*c)<zero) cc = z/ss; |
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else ss = z/cc; |
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} |
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/* |
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* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) |
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* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) |
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*/ |
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✗✓ |
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if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(x); |
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else { |
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u = pzero(x); v = qzero(x); |
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z = invsqrtpi*(u*cc-v*ss)/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<0x3f200000) { /* |x| < 2**-13 */ |
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✓✗ |
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if(huge+x>one) { /* raise inexact if x != 0 */ |
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✓✗ |
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if(ix<0x3e400000) return one; /* |x|<2**-27 */ |
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else return one - 0.25*x*x; |
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} |
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} |
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z = x*x; |
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r = z*(R02+z*(R03+z*(R04+z*R05))); |
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s = one+z*(S01+z*(S02+z*(S03+z*S04))); |
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✓✓ |
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if(ix < 0x3FF00000) { /* |x| < 1.00 */ |
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return one + z*(-0.25+(r/s)); |
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} else { |
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u = 0.5*x; |
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return((one+u)*(one-u)+z*(r/s)); |
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} |
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} |
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DEF_NONSTD(j0); |
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static const double |
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u00 = -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */ |
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u01 = 1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */ |
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u02 = -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */ |
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u03 = 3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */ |
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u04 = -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */ |
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u05 = 1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */ |
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u06 = -3.98205194132103398453e-11, /* 0xBDC5E43D, 0x693FB3C8 */ |
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v01 = 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */ |
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v02 = 7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */ |
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v03 = 2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */ |
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v04 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */ |
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double |
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y0(double x) |
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{ |
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double z, s,c,ss,cc,u,v; |
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int32_t hx,ix,lx; |
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EXTRACT_WORDS(hx,lx,x); |
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ix = 0x7fffffff&hx; |
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/* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */ |
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✗✓ |
5856 |
if(ix>=0x7ff00000) return one/(x+x*x); |
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✗✓ |
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if((ix|lx)==0) return -one/zero; |
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✗✓ |
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if(hx<0) return zero/zero; |
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✓✓ |
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if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
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/* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) |
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* where x0 = x-pi/4 |
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* Better formula: |
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* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) |
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* = 1/sqrt(2) * (sin(x) + cos(x)) |
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* sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) |
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* = 1/sqrt(2) * (sin(x) - cos(x)) |
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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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s = sin(x); |
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c = cos(x); |
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ss = s-c; |
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cc = s+c; |
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/* |
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* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) |
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* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) |
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*/ |
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✓✗ |
3822 |
if(ix<0x7fe00000) { /* make sure x+x not overflow */ |
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z = -cos(x+x); |
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✓✓ |
5148 |
if ((s*c)<zero) cc = z/ss; |
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else ss = z/cc; |
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} |
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✗✓ |
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if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x); |
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else { |
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u = pzero(x); v = qzero(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<=0x3e400000) { /* x < 2**-27 */ |
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return(u00 + tpi*log(x)); |
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} |
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z = x*x; |
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u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); |
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v = one+z*(v01+z*(v02+z*(v03+z*v04))); |
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return(u/v + tpi*(j0(x)*log(x))); |
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} |
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DEF_NONSTD(y0); |
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/* The asymptotic expansions of pzero is |
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* 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. |
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* For x >= 2, We approximate pzero by |
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* pzero(x) = 1 + (R/S) |
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* where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 |
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* S = 1 + pS0*s^2 + ... + pS4*s^10 |
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* and |
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* | pzero(x)-1-R/S | <= 2 ** ( -60.26) |
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*/ |
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static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ |
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-7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */ |
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-8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */ |
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-2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */ |
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-2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */ |
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-5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */ |
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}; |
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static const double pS8[5] = { |
214 |
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1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */ |
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3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */ |
216 |
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4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */ |
217 |
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1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */ |
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4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */ |
219 |
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}; |
220 |
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221 |
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static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
222 |
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-1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */ |
223 |
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-7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */ |
224 |
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-4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */ |
225 |
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-6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */ |
226 |
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-3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */ |
227 |
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-3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */ |
228 |
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}; |
229 |
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static const double pS5[5] = { |
230 |
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6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */ |
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1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */ |
232 |
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5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */ |
233 |
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9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */ |
234 |
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2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */ |
235 |
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}; |
236 |
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237 |
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static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
238 |
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-2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */ |
239 |
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-7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */ |
240 |
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-2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */ |
241 |
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-2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */ |
242 |
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-5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */ |
243 |
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-3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */ |
244 |
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}; |
245 |
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static const double pS3[5] = { |
246 |
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3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */ |
247 |
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3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */ |
248 |
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1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */ |
249 |
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1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */ |
250 |
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1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */ |
251 |
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}; |
252 |
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253 |
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static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
254 |
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-8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */ |
255 |
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-7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */ |
256 |
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-1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */ |
257 |
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-7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */ |
258 |
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-1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */ |
259 |
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-3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */ |
260 |
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}; |
261 |
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static const double pS2[5] = { |
262 |
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2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */ |
263 |
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1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */ |
264 |
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2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */ |
265 |
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1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */ |
266 |
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1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */ |
267 |
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}; |
268 |
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269 |
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static double |
270 |
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pzero(double x) |
271 |
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{ |
272 |
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const double *p,*q; |
273 |
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double z,r,s; |
274 |
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int32_t ix; |
275 |
|
16428 |
GET_HIGH_WORD(ix,x); |
276 |
|
8214 |
ix &= 0x7fffffff; |
277 |
✓✓ |
13914 |
if(ix>=0x40200000) {p = pR8; q= pS8;} |
278 |
✓✓ |
3708 |
else if(ix>=0x40122E8B){p = pR5; q= pS5;} |
279 |
✓✓ |
2064 |
else if(ix>=0x4006DB6D){p = pR3; q= pS3;} |
280 |
✓✗ |
1152 |
else if(ix>=0x40000000){p = pR2; q= pS2;} |
281 |
|
8214 |
z = one/(x*x); |
282 |
|
8214 |
r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
283 |
|
8214 |
s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); |
284 |
|
8214 |
return one+ r/s; |
285 |
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} |
286 |
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287 |
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288 |
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/* For x >= 8, the asymptotic expansions of qzero is |
289 |
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* -1/8 s + 75/1024 s^3 - ..., where s = 1/x. |
290 |
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* We approximate pzero by |
291 |
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* qzero(x) = s*(-1.25 + (R/S)) |
292 |
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* where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 |
293 |
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* S = 1 + qS0*s^2 + ... + qS5*s^12 |
294 |
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* and |
295 |
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* | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) |
296 |
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*/ |
297 |
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static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
298 |
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0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ |
299 |
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7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */ |
300 |
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1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */ |
301 |
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5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */ |
302 |
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8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */ |
303 |
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3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */ |
304 |
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}; |
305 |
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static const double qS8[6] = { |
306 |
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1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */ |
307 |
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8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */ |
308 |
|
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1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */ |
309 |
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8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */ |
310 |
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8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */ |
311 |
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-3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */ |
312 |
|
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}; |
313 |
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|
314 |
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static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
315 |
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1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */ |
316 |
|
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7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */ |
317 |
|
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5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */ |
318 |
|
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1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */ |
319 |
|
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1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */ |
320 |
|
|
1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */ |
321 |
|
|
}; |
322 |
|
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static const double qS5[6] = { |
323 |
|
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8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */ |
324 |
|
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2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */ |
325 |
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1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */ |
326 |
|
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5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */ |
327 |
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3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */ |
328 |
|
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-5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */ |
329 |
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|
}; |
330 |
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|
331 |
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static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
332 |
|
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4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */ |
333 |
|
|
7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */ |
334 |
|
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3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */ |
335 |
|
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4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */ |
336 |
|
|
1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */ |
337 |
|
|
1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */ |
338 |
|
|
}; |
339 |
|
|
static const double qS3[6] = { |
340 |
|
|
4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */ |
341 |
|
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7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */ |
342 |
|
|
3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */ |
343 |
|
|
6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */ |
344 |
|
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2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */ |
345 |
|
|
-1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */ |
346 |
|
|
}; |
347 |
|
|
|
348 |
|
|
static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
349 |
|
|
1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */ |
350 |
|
|
7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */ |
351 |
|
|
1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */ |
352 |
|
|
1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */ |
353 |
|
|
3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */ |
354 |
|
|
1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */ |
355 |
|
|
}; |
356 |
|
|
static const double qS2[6] = { |
357 |
|
|
3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */ |
358 |
|
|
2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */ |
359 |
|
|
8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */ |
360 |
|
|
8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */ |
361 |
|
|
2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */ |
362 |
|
|
-5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */ |
363 |
|
|
}; |
364 |
|
|
|
365 |
|
|
static double |
366 |
|
|
qzero(double x) |
367 |
|
|
{ |
368 |
|
|
const double *p,*q; |
369 |
|
|
double s,r,z; |
370 |
|
|
int32_t ix; |
371 |
|
16428 |
GET_HIGH_WORD(ix,x); |
372 |
|
8214 |
ix &= 0x7fffffff; |
373 |
✓✓ |
13914 |
if(ix>=0x40200000) {p = qR8; q= qS8;} |
374 |
✓✓ |
3708 |
else if(ix>=0x40122E8B){p = qR5; q= qS5;} |
375 |
✓✓ |
2064 |
else if(ix>=0x4006DB6D){p = qR3; q= qS3;} |
376 |
✓✗ |
1152 |
else if(ix>=0x40000000){p = qR2; q= qS2;} |
377 |
|
8214 |
z = one/(x*x); |
378 |
|
8214 |
r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
379 |
|
8214 |
s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); |
380 |
|
8214 |
return (-.125 + r/s)/x; |
381 |
|
|
} |