336 lines
10 KiB
C
336 lines
10 KiB
C
/* e_j0f.c -- float version of e_j0.c.
|
|
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
|
*/
|
|
|
|
/*
|
|
* ====================================================
|
|
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
|
*
|
|
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
|
* Permission to use, copy, modify, and distribute this
|
|
* software is freely granted, provided that this notice
|
|
* is preserved.
|
|
* ====================================================
|
|
*/
|
|
|
|
#include <math.h>
|
|
#include <math_private.h>
|
|
|
|
static float pzerof(float), qzerof(float);
|
|
|
|
static const float
|
|
huge = 1e30,
|
|
one = 1.0,
|
|
invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
|
|
tpi = 6.3661974669e-01, /* 0x3f22f983 */
|
|
/* R0/S0 on [0, 2.00] */
|
|
R02 = 1.5625000000e-02, /* 0x3c800000 */
|
|
R03 = -1.8997929874e-04, /* 0xb947352e */
|
|
R04 = 1.8295404516e-06, /* 0x35f58e88 */
|
|
R05 = -4.6183270541e-09, /* 0xb19eaf3c */
|
|
S01 = 1.5619102865e-02, /* 0x3c7fe744 */
|
|
S02 = 1.1692678527e-04, /* 0x38f53697 */
|
|
S03 = 5.1354652442e-07, /* 0x3509daa6 */
|
|
S04 = 1.1661400734e-09; /* 0x30a045e8 */
|
|
|
|
static const float zero = 0.0;
|
|
|
|
float
|
|
__ieee754_j0f(float x)
|
|
{
|
|
float z, s,c,ss,cc,r,u,v;
|
|
int32_t hx,ix;
|
|
|
|
GET_FLOAT_WORD(hx,x);
|
|
ix = hx&0x7fffffff;
|
|
if(ix>=0x7f800000) return one/(x*x);
|
|
x = fabsf(x);
|
|
if(ix >= 0x40000000) { /* |x| >= 2.0 */
|
|
__sincosf (x, &s, &c);
|
|
ss = s-c;
|
|
cc = s+c;
|
|
if(ix<0x7f000000) { /* make sure x+x not overflow */
|
|
z = -__cosf(x+x);
|
|
if ((s*c)<zero) cc = z/ss;
|
|
else ss = z/cc;
|
|
}
|
|
/*
|
|
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
|
|
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
|
|
*/
|
|
if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(x);
|
|
else {
|
|
u = pzerof(x); v = qzerof(x);
|
|
z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrtf(x);
|
|
}
|
|
return z;
|
|
}
|
|
if(ix<0x39000000) { /* |x| < 2**-13 */
|
|
math_force_eval(huge+x); /* raise inexact if x != 0 */
|
|
if(ix<0x32000000) return one; /* |x|<2**-27 */
|
|
else return one - (float)0.25*x*x;
|
|
}
|
|
z = x*x;
|
|
r = z*(R02+z*(R03+z*(R04+z*R05)));
|
|
s = one+z*(S01+z*(S02+z*(S03+z*S04)));
|
|
if(ix < 0x3F800000) { /* |x| < 1.00 */
|
|
return one + z*((float)-0.25+(r/s));
|
|
} else {
|
|
u = (float)0.5*x;
|
|
return((one+u)*(one-u)+z*(r/s));
|
|
}
|
|
}
|
|
strong_alias (__ieee754_j0f, __j0f_finite)
|
|
|
|
static const float
|
|
u00 = -7.3804296553e-02, /* 0xbd9726b5 */
|
|
u01 = 1.7666645348e-01, /* 0x3e34e80d */
|
|
u02 = -1.3818567619e-02, /* 0xbc626746 */
|
|
u03 = 3.4745343146e-04, /* 0x39b62a69 */
|
|
u04 = -3.8140706238e-06, /* 0xb67ff53c */
|
|
u05 = 1.9559013964e-08, /* 0x32a802ba */
|
|
u06 = -3.9820518410e-11, /* 0xae2f21eb */
|
|
v01 = 1.2730483897e-02, /* 0x3c509385 */
|
|
v02 = 7.6006865129e-05, /* 0x389f65e0 */
|
|
v03 = 2.5915085189e-07, /* 0x348b216c */
|
|
v04 = 4.4111031494e-10; /* 0x2ff280c2 */
|
|
|
|
float
|
|
__ieee754_y0f(float x)
|
|
{
|
|
float z, s,c,ss,cc,u,v;
|
|
int32_t hx,ix;
|
|
|
|
GET_FLOAT_WORD(hx,x);
|
|
ix = 0x7fffffff&hx;
|
|
/* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf. */
|
|
if(ix>=0x7f800000) return one/(x+x*x);
|
|
if(ix==0) return -HUGE_VALF+x; /* -inf and overflow exception. */
|
|
if(hx<0) return zero/(zero*x);
|
|
if(ix >= 0x40000000) { /* |x| >= 2.0 */
|
|
/* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
|
|
* where x0 = x-pi/4
|
|
* Better formula:
|
|
* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
|
|
* = 1/sqrt(2) * (sin(x) + cos(x))
|
|
* sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
|
|
* = 1/sqrt(2) * (sin(x) - cos(x))
|
|
* To avoid cancellation, use
|
|
* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
|
|
* to compute the worse one.
|
|
*/
|
|
__sincosf (x, &s, &c);
|
|
ss = s-c;
|
|
cc = s+c;
|
|
/*
|
|
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
|
|
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
|
|
*/
|
|
if(ix<0x7f000000) { /* make sure x+x not overflow */
|
|
z = -__cosf(x+x);
|
|
if ((s*c)<zero) cc = z/ss;
|
|
else ss = z/cc;
|
|
}
|
|
if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrtf(x);
|
|
else {
|
|
u = pzerof(x); v = qzerof(x);
|
|
z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrtf(x);
|
|
}
|
|
return z;
|
|
}
|
|
if(ix<=0x32000000) { /* x < 2**-27 */
|
|
return(u00 + tpi*__ieee754_logf(x));
|
|
}
|
|
z = x*x;
|
|
u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
|
|
v = one+z*(v01+z*(v02+z*(v03+z*v04)));
|
|
return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
|
|
}
|
|
strong_alias (__ieee754_y0f, __y0f_finite)
|
|
|
|
/* The asymptotic expansions of pzero is
|
|
* 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
|
|
* For x >= 2, We approximate pzero by
|
|
* pzero(x) = 1 + (R/S)
|
|
* where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
|
|
* S = 1 + pS0*s^2 + ... + pS4*s^10
|
|
* and
|
|
* | pzero(x)-1-R/S | <= 2 ** ( -60.26)
|
|
*/
|
|
static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
|
|
0.0000000000e+00, /* 0x00000000 */
|
|
-7.0312500000e-02, /* 0xbd900000 */
|
|
-8.0816707611e+00, /* 0xc1014e86 */
|
|
-2.5706311035e+02, /* 0xc3808814 */
|
|
-2.4852163086e+03, /* 0xc51b5376 */
|
|
-5.2530439453e+03, /* 0xc5a4285a */
|
|
};
|
|
static const float pS8[5] = {
|
|
1.1653436279e+02, /* 0x42e91198 */
|
|
3.8337448730e+03, /* 0x456f9beb */
|
|
4.0597855469e+04, /* 0x471e95db */
|
|
1.1675296875e+05, /* 0x47e4087c */
|
|
4.7627726562e+04, /* 0x473a0bba */
|
|
};
|
|
static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
|
|
-1.1412546255e-11, /* 0xad48c58a */
|
|
-7.0312492549e-02, /* 0xbd8fffff */
|
|
-4.1596107483e+00, /* 0xc0851b88 */
|
|
-6.7674766541e+01, /* 0xc287597b */
|
|
-3.3123129272e+02, /* 0xc3a59d9b */
|
|
-3.4643338013e+02, /* 0xc3ad3779 */
|
|
};
|
|
static const float pS5[5] = {
|
|
6.0753936768e+01, /* 0x42730408 */
|
|
1.0512523193e+03, /* 0x44836813 */
|
|
5.9789707031e+03, /* 0x45bad7c4 */
|
|
9.6254453125e+03, /* 0x461665c8 */
|
|
2.4060581055e+03, /* 0x451660ee */
|
|
};
|
|
|
|
static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
|
|
-2.5470459075e-09, /* 0xb12f081b */
|
|
-7.0311963558e-02, /* 0xbd8fffb8 */
|
|
-2.4090321064e+00, /* 0xc01a2d95 */
|
|
-2.1965976715e+01, /* 0xc1afba52 */
|
|
-5.8079170227e+01, /* 0xc2685112 */
|
|
-3.1447946548e+01, /* 0xc1fb9565 */
|
|
};
|
|
static const float pS3[5] = {
|
|
3.5856033325e+01, /* 0x420f6c94 */
|
|
3.6151397705e+02, /* 0x43b4c1ca */
|
|
1.1936077881e+03, /* 0x44953373 */
|
|
1.1279968262e+03, /* 0x448cffe6 */
|
|
1.7358093262e+02, /* 0x432d94b8 */
|
|
};
|
|
|
|
static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
|
|
-8.8753431271e-08, /* 0xb3be98b7 */
|
|
-7.0303097367e-02, /* 0xbd8ffb12 */
|
|
-1.4507384300e+00, /* 0xbfb9b1cc */
|
|
-7.6356959343e+00, /* 0xc0f4579f */
|
|
-1.1193166733e+01, /* 0xc1331736 */
|
|
-3.2336456776e+00, /* 0xc04ef40d */
|
|
};
|
|
static const float pS2[5] = {
|
|
2.2220300674e+01, /* 0x41b1c32d */
|
|
1.3620678711e+02, /* 0x430834f0 */
|
|
2.7047027588e+02, /* 0x43873c32 */
|
|
1.5387539673e+02, /* 0x4319e01a */
|
|
1.4657617569e+01, /* 0x416a859a */
|
|
};
|
|
|
|
static float
|
|
pzerof(float x)
|
|
{
|
|
const float *p,*q;
|
|
float z,r,s;
|
|
int32_t ix;
|
|
GET_FLOAT_WORD(ix,x);
|
|
ix &= 0x7fffffff;
|
|
if(ix>=0x41000000) {p = pR8; q= pS8;}
|
|
else if(ix>=0x40f71c58){p = pR5; q= pS5;}
|
|
else if(ix>=0x4036db68){p = pR3; q= pS3;}
|
|
else if(ix>=0x40000000){p = pR2; q= pS2;}
|
|
z = one/(x*x);
|
|
r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
|
|
s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
|
|
return one+ r/s;
|
|
}
|
|
|
|
|
|
/* For x >= 8, the asymptotic expansions of qzero is
|
|
* -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
|
|
* We approximate pzero by
|
|
* qzero(x) = s*(-1.25 + (R/S))
|
|
* where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
|
|
* S = 1 + qS0*s^2 + ... + qS5*s^12
|
|
* and
|
|
* | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
|
|
*/
|
|
static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
|
|
0.0000000000e+00, /* 0x00000000 */
|
|
7.3242187500e-02, /* 0x3d960000 */
|
|
1.1768206596e+01, /* 0x413c4a93 */
|
|
5.5767340088e+02, /* 0x440b6b19 */
|
|
8.8591972656e+03, /* 0x460a6cca */
|
|
3.7014625000e+04, /* 0x471096a0 */
|
|
};
|
|
static const float qS8[6] = {
|
|
1.6377603149e+02, /* 0x4323c6aa */
|
|
8.0983447266e+03, /* 0x45fd12c2 */
|
|
1.4253829688e+05, /* 0x480b3293 */
|
|
8.0330925000e+05, /* 0x49441ed4 */
|
|
8.4050156250e+05, /* 0x494d3359 */
|
|
-3.4389928125e+05, /* 0xc8a7eb69 */
|
|
};
|
|
|
|
static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
|
|
1.8408595828e-11, /* 0x2da1ec79 */
|
|
7.3242180049e-02, /* 0x3d95ffff */
|
|
5.8356351852e+00, /* 0x40babd86 */
|
|
1.3511157227e+02, /* 0x43071c90 */
|
|
1.0272437744e+03, /* 0x448067cd */
|
|
1.9899779053e+03, /* 0x44f8bf4b */
|
|
};
|
|
static const float qS5[6] = {
|
|
8.2776611328e+01, /* 0x42a58da0 */
|
|
2.0778142090e+03, /* 0x4501dd07 */
|
|
1.8847289062e+04, /* 0x46933e94 */
|
|
5.6751113281e+04, /* 0x475daf1d */
|
|
3.5976753906e+04, /* 0x470c88c1 */
|
|
-5.3543427734e+03, /* 0xc5a752be */
|
|
};
|
|
|
|
static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
|
|
4.3774099900e-09, /* 0x3196681b */
|
|
7.3241114616e-02, /* 0x3d95ff70 */
|
|
3.3442313671e+00, /* 0x405607e3 */
|
|
4.2621845245e+01, /* 0x422a7cc5 */
|
|
1.7080809021e+02, /* 0x432acedf */
|
|
1.6673394775e+02, /* 0x4326bbe4 */
|
|
};
|
|
static const float qS3[6] = {
|
|
4.8758872986e+01, /* 0x42430916 */
|
|
7.0968920898e+02, /* 0x44316c1c */
|
|
3.7041481934e+03, /* 0x4567825f */
|
|
6.4604252930e+03, /* 0x45c9e367 */
|
|
2.5163337402e+03, /* 0x451d4557 */
|
|
-1.4924745178e+02, /* 0xc3153f59 */
|
|
};
|
|
|
|
static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
|
|
1.5044444979e-07, /* 0x342189db */
|
|
7.3223426938e-02, /* 0x3d95f62a */
|
|
1.9981917143e+00, /* 0x3fffc4bf */
|
|
1.4495602608e+01, /* 0x4167edfd */
|
|
3.1666231155e+01, /* 0x41fd5471 */
|
|
1.6252708435e+01, /* 0x4182058c */
|
|
};
|
|
static const float qS2[6] = {
|
|
3.0365585327e+01, /* 0x41f2ecb8 */
|
|
2.6934811401e+02, /* 0x4386ac8f */
|
|
8.4478375244e+02, /* 0x44533229 */
|
|
8.8293585205e+02, /* 0x445cbbe5 */
|
|
2.1266638184e+02, /* 0x4354aa98 */
|
|
-5.3109550476e+00, /* 0xc0a9f358 */
|
|
};
|
|
|
|
static float
|
|
qzerof(float x)
|
|
{
|
|
const float *p,*q;
|
|
float s,r,z;
|
|
int32_t ix;
|
|
GET_FLOAT_WORD(ix,x);
|
|
ix &= 0x7fffffff;
|
|
if(ix>=0x41000000) {p = qR8; q= qS8;}
|
|
else if(ix>=0x40f71c58){p = qR5; q= qS5;}
|
|
else if(ix>=0x4036db68){p = qR3; q= qS3;}
|
|
else if(ix>=0x40000000){p = qR2; q= qS2;}
|
|
z = one/(x*x);
|
|
r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
|
|
s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
|
|
return (-(float).125 + r/s)/x;
|
|
}
|