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/* e_j0f.c -- float version of e_j0.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 pzerof(float), qzerof(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, 2.00] */ |
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R02 = 1.5625000000e-02, /* 0x3c800000 */ |
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R03 = -1.8997929874e-04, /* 0xb947352e */ |
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R04 = 1.8295404516e-06, /* 0x35f58e88 */ |
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R05 = -4.6183270541e-09, /* 0xb19eaf3c */ |
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S01 = 1.5619102865e-02, /* 0x3c7fe744 */ |
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S02 = 1.1692678527e-04, /* 0x38f53697 */ |
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S03 = 5.1354652442e-07, /* 0x3509daa6 */ |
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S04 = 1.1661400734e-09; /* 0x30a045e8 */ |
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static const float zero = 0.0; |
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float |
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j0f(float x) |
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{ |
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float z, s,c,ss,cc,r,u,v; |
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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*x); |
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x = fabsf(x); |
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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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/* |
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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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if(ix>0x80000000U) z = (invsqrtpi*cc)/sqrtf(x); |
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else { |
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u = pzerof(x); v = qzerof(x); |
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z = invsqrtpi*(u*cc-v*ss)/sqrtf(x); |
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} |
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return z; |
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} |
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if(ix<0x39000000) { /* |x| < 2**-13 */ |
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if(huge+x>one) { /* raise inexact if x != 0 */ |
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if(ix<0x32000000) return one; /* |x|<2**-27 */ |
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else return one - (float)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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if(ix < 0x3F800000) { /* |x| < 1.00 */ |
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return one + z*((float)-0.25+(r/s)); |
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} else { |
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u = (float)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(j0f); |
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static const float |
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u00 = -7.3804296553e-02, /* 0xbd9726b5 */ |
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u01 = 1.7666645348e-01, /* 0x3e34e80d */ |
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u02 = -1.3818567619e-02, /* 0xbc626746 */ |
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u03 = 3.4745343146e-04, /* 0x39b62a69 */ |
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u04 = -3.8140706238e-06, /* 0xb67ff53c */ |
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u05 = 1.9559013964e-08, /* 0x32a802ba */ |
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u06 = -3.9820518410e-11, /* 0xae2f21eb */ |
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v01 = 1.2730483897e-02, /* 0x3c509385 */ |
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v02 = 7.6006865129e-05, /* 0x389f65e0 */ |
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v03 = 2.5915085189e-07, /* 0x348b216c */ |
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v04 = 4.4111031494e-10; /* 0x2ff280c2 */ |
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float |
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y0f(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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/* Y0(NaN) is NaN, y0(-inf) is Nan, y0(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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/* 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 = 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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/* |
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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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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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if(ix>0x80000000U) z = (invsqrtpi*ss)/sqrtf(x); |
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else { |
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u = pzerof(x); v = qzerof(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<=0x32000000) { /* x < 2**-27 */ |
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return(u00 + tpi*logf(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*(j0f(x)*logf(x))); |
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} |
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DEF_NONSTD(y0f); |
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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 float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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0.0000000000e+00, /* 0x00000000 */ |
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-7.0312500000e-02, /* 0xbd900000 */ |
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-8.0816707611e+00, /* 0xc1014e86 */ |
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-2.5706311035e+02, /* 0xc3808814 */ |
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-2.4852163086e+03, /* 0xc51b5376 */ |
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-5.2530439453e+03, /* 0xc5a4285a */ |
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}; |
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static const float pS8[5] = { |
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1.1653436279e+02, /* 0x42e91198 */ |
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3.8337448730e+03, /* 0x456f9beb */ |
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4.0597855469e+04, /* 0x471e95db */ |
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1.1675296875e+05, /* 0x47e4087c */ |
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4.7627726562e+04, /* 0x473a0bba */ |
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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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-1.1412546255e-11, /* 0xad48c58a */ |
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-7.0312492549e-02, /* 0xbd8fffff */ |
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-4.1596107483e+00, /* 0xc0851b88 */ |
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-6.7674766541e+01, /* 0xc287597b */ |
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-3.3123129272e+02, /* 0xc3a59d9b */ |
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-3.4643338013e+02, /* 0xc3ad3779 */ |
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}; |
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static const float pS5[5] = { |
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6.0753936768e+01, /* 0x42730408 */ |
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1.0512523193e+03, /* 0x44836813 */ |
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5.9789707031e+03, /* 0x45bad7c4 */ |
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9.6254453125e+03, /* 0x461665c8 */ |
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2.4060581055e+03, /* 0x451660ee */ |
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}; |
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static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
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-2.5470459075e-09, /* 0xb12f081b */ |
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-7.0311963558e-02, /* 0xbd8fffb8 */ |
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-2.4090321064e+00, /* 0xc01a2d95 */ |
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-2.1965976715e+01, /* 0xc1afba52 */ |
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-5.8079170227e+01, /* 0xc2685112 */ |
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-3.1447946548e+01, /* 0xc1fb9565 */ |
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}; |
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static const float pS3[5] = { |
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3.5856033325e+01, /* 0x420f6c94 */ |
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3.6151397705e+02, /* 0x43b4c1ca */ |
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1.1936077881e+03, /* 0x44953373 */ |
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1.1279968262e+03, /* 0x448cffe6 */ |
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1.7358093262e+02, /* 0x432d94b8 */ |
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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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-8.8753431271e-08, /* 0xb3be98b7 */ |
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-7.0303097367e-02, /* 0xbd8ffb12 */ |
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-1.4507384300e+00, /* 0xbfb9b1cc */ |
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-7.6356959343e+00, /* 0xc0f4579f */ |
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-1.1193166733e+01, /* 0xc1331736 */ |
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-3.2336456776e+00, /* 0xc04ef40d */ |
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}; |
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static const float pS2[5] = { |
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2.2220300674e+01, /* 0x41b1c32d */ |
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1.3620678711e+02, /* 0x430834f0 */ |
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2.7047027588e+02, /* 0x43873c32 */ |
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1.5387539673e+02, /* 0x4319e01a */ |
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1.4657617569e+01, /* 0x416a859a */ |
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}; |
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226 |
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static float |
227 |
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pzerof(float x) |
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{ |
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const float *p,*q; |
230 |
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float z,r,s; |
231 |
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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;} |
237 |
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else if(ix>=0x40000000){p = pR2; q= pS2;} |
238 |
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z = one/(x*x); |
239 |
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r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
240 |
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s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); |
241 |
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return one+ r/s; |
242 |
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} |
243 |
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244 |
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245 |
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/* For x >= 8, the asymptotic expansions of qzero is |
246 |
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* -1/8 s + 75/1024 s^3 - ..., where s = 1/x. |
247 |
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* We approximate pzero by |
248 |
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* qzero(x) = s*(-1.25 + (R/S)) |
249 |
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* where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 |
250 |
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* S = 1 + qS0*s^2 + ... + qS5*s^12 |
251 |
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* and |
252 |
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* | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) |
253 |
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*/ |
254 |
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static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
255 |
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0.0000000000e+00, /* 0x00000000 */ |
256 |
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7.3242187500e-02, /* 0x3d960000 */ |
257 |
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1.1768206596e+01, /* 0x413c4a93 */ |
258 |
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5.5767340088e+02, /* 0x440b6b19 */ |
259 |
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8.8591972656e+03, /* 0x460a6cca */ |
260 |
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3.7014625000e+04, /* 0x471096a0 */ |
261 |
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}; |
262 |
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static const float qS8[6] = { |
263 |
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1.6377603149e+02, /* 0x4323c6aa */ |
264 |
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8.0983447266e+03, /* 0x45fd12c2 */ |
265 |
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1.4253829688e+05, /* 0x480b3293 */ |
266 |
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8.0330925000e+05, /* 0x49441ed4 */ |
267 |
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8.4050156250e+05, /* 0x494d3359 */ |
268 |
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-3.4389928125e+05, /* 0xc8a7eb69 */ |
269 |
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}; |
270 |
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271 |
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static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
272 |
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1.8408595828e-11, /* 0x2da1ec79 */ |
273 |
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7.3242180049e-02, /* 0x3d95ffff */ |
274 |
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5.8356351852e+00, /* 0x40babd86 */ |
275 |
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1.3511157227e+02, /* 0x43071c90 */ |
276 |
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1.0272437744e+03, /* 0x448067cd */ |
277 |
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1.9899779053e+03, /* 0x44f8bf4b */ |
278 |
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}; |
279 |
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static const float qS5[6] = { |
280 |
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8.2776611328e+01, /* 0x42a58da0 */ |
281 |
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2.0778142090e+03, /* 0x4501dd07 */ |
282 |
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1.8847289062e+04, /* 0x46933e94 */ |
283 |
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5.6751113281e+04, /* 0x475daf1d */ |
284 |
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3.5976753906e+04, /* 0x470c88c1 */ |
285 |
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-5.3543427734e+03, /* 0xc5a752be */ |
286 |
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}; |
287 |
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288 |
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static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
289 |
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4.3774099900e-09, /* 0x3196681b */ |
290 |
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7.3241114616e-02, /* 0x3d95ff70 */ |
291 |
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3.3442313671e+00, /* 0x405607e3 */ |
292 |
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4.2621845245e+01, /* 0x422a7cc5 */ |
293 |
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1.7080809021e+02, /* 0x432acedf */ |
294 |
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1.6673394775e+02, /* 0x4326bbe4 */ |
295 |
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}; |
296 |
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static const float qS3[6] = { |
297 |
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4.8758872986e+01, /* 0x42430916 */ |
298 |
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7.0968920898e+02, /* 0x44316c1c */ |
299 |
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3.7041481934e+03, /* 0x4567825f */ |
300 |
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6.4604252930e+03, /* 0x45c9e367 */ |
301 |
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2.5163337402e+03, /* 0x451d4557 */ |
302 |
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-1.4924745178e+02, /* 0xc3153f59 */ |
303 |
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}; |
304 |
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305 |
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static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
306 |
|
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1.5044444979e-07, /* 0x342189db */ |
307 |
|
|
7.3223426938e-02, /* 0x3d95f62a */ |
308 |
|
|
1.9981917143e+00, /* 0x3fffc4bf */ |
309 |
|
|
1.4495602608e+01, /* 0x4167edfd */ |
310 |
|
|
3.1666231155e+01, /* 0x41fd5471 */ |
311 |
|
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1.6252708435e+01, /* 0x4182058c */ |
312 |
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}; |
313 |
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static const float qS2[6] = { |
314 |
|
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3.0365585327e+01, /* 0x41f2ecb8 */ |
315 |
|
|
2.6934811401e+02, /* 0x4386ac8f */ |
316 |
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|
8.4478375244e+02, /* 0x44533229 */ |
317 |
|
|
8.8293585205e+02, /* 0x445cbbe5 */ |
318 |
|
|
2.1266638184e+02, /* 0x4354aa98 */ |
319 |
|
|
-5.3109550476e+00, /* 0xc0a9f358 */ |
320 |
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}; |
321 |
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|
|
322 |
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static float |
323 |
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qzerof(float x) |
324 |
|
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{ |
325 |
|
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const float *p,*q; |
326 |
|
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float s,r,z; |
327 |
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int32_t ix; |
328 |
|
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GET_FLOAT_WORD(ix,x); |
329 |
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ix &= 0x7fffffff; |
330 |
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if(ix>=0x41000000) {p = qR8; q= qS8;} |
331 |
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else if(ix>=0x40f71c58){p = qR5; q= qS5;} |
332 |
|
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else if(ix>=0x4036db68){p = qR3; q= qS3;} |
333 |
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else if(ix>=0x40000000){p = qR2; q= qS2;} |
334 |
|
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z = one/(x*x); |
335 |
|
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r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
336 |
|
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s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); |
337 |
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return (-(float).125 + r/s)/x; |
338 |
|
|
} |