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/* e_j1f.c -- float version of e_j1.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 float ponef(float), qonef(float); |
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static const float |
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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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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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r03 = 4.9672799207e-08, /* 0x335557d2 */ |
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s01 = 1.9153760746e-02, /* 0x3c9ce859 */ |
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s02 = 1.8594678841e-04, /* 0x3942fab6 */ |
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s03 = 1.1771846857e-06, /* 0x359dffc2 */ |
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s04 = 5.0463624390e-09, /* 0x31ad6446 */ |
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s05 = 1.2354227016e-11; /* 0x2d59567e */ |
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static const float zero = 0.0; |
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float |
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j1f(float x) |
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{ |
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float z, s,c,ss,cc,r,u,v,y; |
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int32_t hx,ix; |
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GET_FLOAT_WORD(hx,x); |
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ix = hx&0x7fffffff; |
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if(ix>=0x7f800000) return one/x; |
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y = fabsf(x); |
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if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
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s = sinf(y); |
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c = cosf(y); |
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ss = -s-c; |
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cc = s-c; |
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if(ix<0x7f000000) { /* make sure y+y not overflow */ |
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z = cosf(y+y); |
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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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* j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x) |
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* y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x) |
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*/ |
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if(ix>0x80000000U) z = (invsqrtpi*cc)/sqrtf(y); |
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else { |
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u = ponef(y); v = qonef(y); |
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z = invsqrtpi*(u*cc-v*ss)/sqrtf(y); |
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} |
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if(hx<0) return -z; |
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else return z; |
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} |
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if(ix<0x32000000) { /* |x|<2**-27 */ |
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if(huge+x>one) return (float)0.5*x;/* inexact if x!=0 necessary */ |
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} |
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z = x*x; |
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r = z*(r00+z*(r01+z*(r02+z*r03))); |
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s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05)))); |
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r *= x; |
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return(x*(float)0.5+r/s); |
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} |
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DEF_NONSTD(j1f); |
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static const float U0[5] = { |
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-1.9605709612e-01, /* 0xbe48c331 */ |
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5.0443872809e-02, /* 0x3d4e9e3c */ |
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-1.9125689287e-03, /* 0xbafaaf2a */ |
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2.3525259166e-05, /* 0x37c5581c */ |
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-9.1909917899e-08, /* 0xb3c56003 */ |
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}; |
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static const float V0[5] = { |
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1.9916731864e-02, /* 0x3ca3286a */ |
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2.0255257550e-04, /* 0x3954644b */ |
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1.3560879779e-06, /* 0x35b602d4 */ |
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6.2274145840e-09, /* 0x31d5f8eb */ |
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1.6655924903e-11, /* 0x2d9281cf */ |
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}; |
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float |
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y1f(float x) |
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{ |
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float z, s,c,ss,cc,u,v; |
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int32_t hx,ix; |
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GET_FLOAT_WORD(hx,x); |
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ix = 0x7fffffff&hx; |
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/* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */ |
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if(ix>=0x7f800000) return one/(x+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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if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
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s = sinf(x); |
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c = cosf(x); |
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ss = -s-c; |
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cc = s-c; |
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if(ix<0x7f000000) { /* make sure x+x not overflow */ |
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z = cosf(x+x); |
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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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/* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0)) |
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* where x0 = x-3pi/4 |
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* Better formula: |
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* cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/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) * (cos(x) + sin(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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if(ix>0x48000000) z = (invsqrtpi*ss)/sqrtf(x); |
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else { |
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u = ponef(x); v = qonef(x); |
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z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); |
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} |
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return z; |
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} |
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if(ix<=0x24800000) { /* x < 2**-54 */ |
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return(-tpi/x); |
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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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return(x*(u/v) + tpi*(j1f(x)*logf(x)-one/x)); |
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} |
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DEF_NONSTD(y1f); |
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/* For x >= 8, the asymptotic expansions of pone is |
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* 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x. |
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* We approximate pone by |
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* pone(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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* | pone(x)-1-R/S | <= 2 ** ( -60.06) |
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*/ |
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static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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0.0000000000e+00, /* 0x00000000 */ |
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1.1718750000e-01, /* 0x3df00000 */ |
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1.3239480972e+01, /* 0x4153d4ea */ |
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4.1205184937e+02, /* 0x43ce06a3 */ |
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3.8747453613e+03, /* 0x45722bed */ |
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7.9144794922e+03, /* 0x45f753d6 */ |
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}; |
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static const float ps8[5] = { |
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1.1420736694e+02, /* 0x42e46a2c */ |
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3.6509309082e+03, /* 0x45642ee5 */ |
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3.6956207031e+04, /* 0x47105c35 */ |
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9.7602796875e+04, /* 0x47bea166 */ |
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3.0804271484e+04, /* 0x46f0a88b */ |
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}; |
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static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
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1.3199052094e-11, /* 0x2d68333f */ |
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1.1718749255e-01, /* 0x3defffff */ |
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6.8027510643e+00, /* 0x40d9b023 */ |
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1.0830818176e+02, /* 0x42d89dca */ |
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5.1763616943e+02, /* 0x440168b7 */ |
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5.2871520996e+02, /* 0x44042dc6 */ |
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}; |
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static const float ps5[5] = { |
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5.9280597687e+01, /* 0x426d1f55 */ |
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9.9140142822e+02, /* 0x4477d9b1 */ |
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5.3532670898e+03, /* 0x45a74a23 */ |
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7.8446904297e+03, /* 0x45f52586 */ |
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1.5040468750e+03, /* 0x44bc0180 */ |
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}; |
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static const float pr3[6] = { |
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3.0250391081e-09, /* 0x314fe10d */ |
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1.1718686670e-01, /* 0x3defffab */ |
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3.9329774380e+00, /* 0x407bb5e7 */ |
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3.5119403839e+01, /* 0x420c7a45 */ |
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9.1055007935e+01, /* 0x42b61c2a */ |
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4.8559066772e+01, /* 0x42423c7c */ |
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}; |
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static const float ps3[5] = { |
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3.4791309357e+01, /* 0x420b2a4d */ |
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3.3676245117e+02, /* 0x43a86198 */ |
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1.0468714600e+03, /* 0x4482dbe3 */ |
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8.9081134033e+02, /* 0x445eb3ed */ |
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1.0378793335e+02, /* 0x42cf936c */ |
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}; |
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static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
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1.0771083225e-07, /* 0x33e74ea8 */ |
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1.1717621982e-01, /* 0x3deffa16 */ |
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2.3685150146e+00, /* 0x401795c0 */ |
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1.2242610931e+01, /* 0x4143e1bc */ |
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1.7693971634e+01, /* 0x418d8d41 */ |
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5.0735230446e+00, /* 0x40a25a4d */ |
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}; |
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static const float ps2[5] = { |
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2.1436485291e+01, /* 0x41ab7dec */ |
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1.2529022980e+02, /* 0x42fa9499 */ |
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2.3227647400e+02, /* 0x436846c7 */ |
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1.1767937469e+02, /* 0x42eb5bd7 */ |
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8.3646392822e+00, /* 0x4105d590 */ |
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}; |
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static float |
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ponef(float x) |
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{ |
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const float *p,*q; |
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float z,r,s; |
226 |
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int32_t ix; |
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GET_FLOAT_WORD(ix,x); |
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ix &= 0x7fffffff; |
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if(ix>=0x41000000) {p = pr8; q= ps8;} |
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else if(ix>=0x40f71c58){p = pr5; q= ps5;} |
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else if(ix>=0x4036db68){p = pr3; q= ps3;} |
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else if(ix>=0x40000000){p = pr2; q= ps2;} |
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z = one/(x*x); |
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r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
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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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239 |
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240 |
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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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* We approximate pone by |
243 |
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* qone(x) = s*(0.375 + (R/S)) |
244 |
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* where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10 |
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* S = 1 + qs1*s^2 + ... + qs6*s^12 |
246 |
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* and |
247 |
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* | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13) |
248 |
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*/ |
249 |
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250 |
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static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
251 |
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0.0000000000e+00, /* 0x00000000 */ |
252 |
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-1.0253906250e-01, /* 0xbdd20000 */ |
253 |
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-1.6271753311e+01, /* 0xc1822c8d */ |
254 |
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-7.5960174561e+02, /* 0xc43de683 */ |
255 |
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-1.1849806641e+04, /* 0xc639273a */ |
256 |
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-4.8438511719e+04, /* 0xc73d3683 */ |
257 |
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}; |
258 |
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static const float qs8[6] = { |
259 |
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1.6139537048e+02, /* 0x43216537 */ |
260 |
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7.8253862305e+03, /* 0x45f48b17 */ |
261 |
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1.3387534375e+05, /* 0x4802bcd6 */ |
262 |
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7.1965775000e+05, /* 0x492fb29c */ |
263 |
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6.6660125000e+05, /* 0x4922be94 */ |
264 |
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-2.9449025000e+05, /* 0xc88fcb48 */ |
265 |
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}; |
266 |
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267 |
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static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
268 |
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-2.0897993405e-11, /* 0xadb7d219 */ |
269 |
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-1.0253904760e-01, /* 0xbdd1fffe */ |
270 |
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-8.0564479828e+00, /* 0xc100e736 */ |
271 |
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-1.8366960144e+02, /* 0xc337ab6b */ |
272 |
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-1.3731937256e+03, /* 0xc4aba633 */ |
273 |
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-2.6124443359e+03, /* 0xc523471c */ |
274 |
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}; |
275 |
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static const float qs5[6] = { |
276 |
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8.1276550293e+01, /* 0x42a28d98 */ |
277 |
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1.9917987061e+03, /* 0x44f8f98f */ |
278 |
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1.7468484375e+04, /* 0x468878f8 */ |
279 |
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4.9851425781e+04, /* 0x4742bb6d */ |
280 |
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2.7948074219e+04, /* 0x46da5826 */ |
281 |
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-4.7191835938e+03, /* 0xc5937978 */ |
282 |
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}; |
283 |
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284 |
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static const float qr3[6] = { |
285 |
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-5.0783124372e-09, /* 0xb1ae7d4f */ |
286 |
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-1.0253783315e-01, /* 0xbdd1ff5b */ |
287 |
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-4.6101160049e+00, /* 0xc0938612 */ |
288 |
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-5.7847221375e+01, /* 0xc267638e */ |
289 |
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-2.2824453735e+02, /* 0xc3643e9a */ |
290 |
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-2.1921012878e+02, /* 0xc35b35cb */ |
291 |
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}; |
292 |
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static const float qs3[6] = { |
293 |
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4.7665153503e+01, /* 0x423ea91e */ |
294 |
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6.7386511230e+02, /* 0x4428775e */ |
295 |
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3.3801528320e+03, /* 0x45534272 */ |
296 |
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5.5477290039e+03, /* 0x45ad5dd5 */ |
297 |
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1.9031191406e+03, /* 0x44ede3d0 */ |
298 |
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-1.3520118713e+02, /* 0xc3073381 */ |
299 |
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}; |
300 |
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301 |
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static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
302 |
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-1.7838172539e-07, /* 0xb43f8932 */ |
303 |
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-1.0251704603e-01, /* 0xbdd1f475 */ |
304 |
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-2.7522056103e+00, /* 0xc0302423 */ |
305 |
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-1.9663616180e+01, /* 0xc19d4f16 */ |
306 |
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-4.2325313568e+01, /* 0xc2294d1f */ |
307 |
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-2.1371921539e+01, /* 0xc1aaf9b2 */ |
308 |
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}; |
309 |
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static const float qs2[6] = { |
310 |
|
|
2.9533363342e+01, /* 0x41ec4454 */ |
311 |
|
|
2.5298155212e+02, /* 0x437cfb47 */ |
312 |
|
|
7.5750280762e+02, /* 0x443d602e */ |
313 |
|
|
7.3939318848e+02, /* 0x4438d92a */ |
314 |
|
|
1.5594900513e+02, /* 0x431bf2f2 */ |
315 |
|
|
-4.9594988823e+00, /* 0xc09eb437 */ |
316 |
|
|
}; |
317 |
|
|
|
318 |
|
|
static float |
319 |
|
|
qonef(float x) |
320 |
|
|
{ |
321 |
|
|
const float *p,*q; |
322 |
|
|
float s,r,z; |
323 |
|
|
int32_t ix; |
324 |
|
|
GET_FLOAT_WORD(ix,x); |
325 |
|
|
ix &= 0x7fffffff; |
326 |
|
|
if(ix>=0x40200000) {p = qr8; q= qs8;} |
327 |
|
|
else if(ix>=0x40f71c58){p = qr5; q= qs5;} |
328 |
|
|
else if(ix>=0x4036db68){p = qr3; q= qs3;} |
329 |
|
|
else if(ix>=0x40000000){p = qr2; q= qs2;} |
330 |
|
|
z = one/(x*x); |
331 |
|
|
r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
332 |
|
|
s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); |
333 |
|
|
return ((float).375 + r/s)/x; |
334 |
|
|
} |