mirror of
https://sourceware.org/git/glibc.git
synced 2024-12-27 13:10:29 +00:00
4918e5f4cd
The Bessel functions of the second type (Yn) should raise the "divide by zero" exception when input is zero (both positive and negative). Current code gives the right output, but fails to set the exception. This error is exposed for float, double, and long double when linking with -lieee. Without this flag, the error is not exposed, because the wrappers for these functions, which use __kernel_standard functionality, set the exception as expected. Tested for powerpc64le. [BZ #21134] * sysdeps/ieee754/dbl-64/e_j0.c (__ieee754_y0): Raise the "divide by zero" exception when the input is zero. * sysdeps/ieee754/dbl-64/e_j1.c (__ieee754_y1): Likewise. * sysdeps/ieee754/flt-32/e_j0f.c (__ieee754_y0f): Likewise. * sysdeps/ieee754/flt-32/e_j1f.c (__ieee754_y1f): Likewise. * sysdeps/ieee754/ldbl-128/e_j0l.c (__ieee754_y0l): Likewise. * sysdeps/ieee754/ldbl-128/e_j1l.c (__ieee754_y1l): Likewise.
459 lines
15 KiB
C
459 lines
15 KiB
C
/* @(#)e_j0.c 5.1 93/09/24 */
|
|
/*
|
|
* ====================================================
|
|
* 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.
|
|
* ====================================================
|
|
*/
|
|
/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/26,
|
|
for performance improvement on pipelined processors.
|
|
*/
|
|
|
|
/* __ieee754_j0(x), __ieee754_y0(x)
|
|
* Bessel function of the first and second kinds of order zero.
|
|
* Method -- j0(x):
|
|
* 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
|
|
* 2. Reduce x to |x| since j0(x)=j0(-x), and
|
|
* for x in (0,2)
|
|
* j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x;
|
|
* (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
|
|
* for x in (2,inf)
|
|
* j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
|
|
* where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
|
|
* as follow:
|
|
* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
|
|
* = 1/sqrt(2) * (cos(x) + sin(x))
|
|
* sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/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.)
|
|
*
|
|
* 3 Special cases
|
|
* j0(nan)= nan
|
|
* j0(0) = 1
|
|
* j0(inf) = 0
|
|
*
|
|
* Method -- y0(x):
|
|
* 1. For x<2.
|
|
* Since
|
|
* y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
|
|
* therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
|
|
* We use the following function to approximate y0,
|
|
* y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
|
|
* where
|
|
* U(z) = u00 + u01*z + ... + u06*z^6
|
|
* V(z) = 1 + v01*z + ... + v04*z^4
|
|
* with absolute approximation error bounded by 2**-72.
|
|
* Note: For tiny x, U/V = u0 and j0(x)~1, hence
|
|
* y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
|
|
* 2. For x>=2.
|
|
* y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
|
|
* where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
|
|
* by the method mentioned above.
|
|
* 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
|
|
*/
|
|
|
|
#include <math.h>
|
|
#include <math_private.h>
|
|
|
|
static double pzero (double), qzero (double);
|
|
|
|
static const double
|
|
huge = 1e300,
|
|
one = 1.0,
|
|
invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
|
|
tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
|
|
/* R0/S0 on [0, 2.00] */
|
|
R[] = { 0.0, 0.0, 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
|
|
-1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
|
|
1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
|
|
-4.61832688532103189199e-09 }, /* 0xBE33D5E7, 0x73D63FCE */
|
|
S[] = { 0.0, 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
|
|
1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
|
|
5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
|
|
1.16614003333790000205e-09 }; /* 0x3E1408BC, 0xF4745D8F */
|
|
|
|
static const double zero = 0.0;
|
|
|
|
double
|
|
__ieee754_j0 (double x)
|
|
{
|
|
double z, s, c, ss, cc, r, u, v, r1, r2, s1, s2, z2, z4;
|
|
int32_t hx, ix;
|
|
|
|
GET_HIGH_WORD (hx, x);
|
|
ix = hx & 0x7fffffff;
|
|
if (ix >= 0x7ff00000)
|
|
return one / (x * x);
|
|
x = fabs (x);
|
|
if (ix >= 0x40000000) /* |x| >= 2.0 */
|
|
{
|
|
__sincos (x, &s, &c);
|
|
ss = s - c;
|
|
cc = s + c;
|
|
if (ix < 0x7fe00000) /* make sure x+x not overflow */
|
|
{
|
|
z = -__cos (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_sqrt (x);
|
|
else
|
|
{
|
|
u = pzero (x); v = qzero (x);
|
|
z = invsqrtpi * (u * cc - v * ss) / __ieee754_sqrt (x);
|
|
}
|
|
return z;
|
|
}
|
|
if (ix < 0x3f200000) /* |x| < 2**-13 */
|
|
{
|
|
math_force_eval (huge + x); /* raise inexact if x != 0 */
|
|
if (ix < 0x3e400000)
|
|
return one; /* |x|<2**-27 */
|
|
else
|
|
return one - 0.25 * x * x;
|
|
}
|
|
z = x * x;
|
|
r1 = z * R[2]; z2 = z * z;
|
|
r2 = R[3] + z * R[4]; z4 = z2 * z2;
|
|
r = r1 + z2 * r2 + z4 * R[5];
|
|
s1 = one + z * S[1];
|
|
s2 = S[2] + z * S[3];
|
|
s = s1 + z2 * s2 + z4 * S[4];
|
|
if (ix < 0x3FF00000) /* |x| < 1.00 */
|
|
{
|
|
return one + z * (-0.25 + (r / s));
|
|
}
|
|
else
|
|
{
|
|
u = 0.5 * x;
|
|
return ((one + u) * (one - u) + z * (r / s));
|
|
}
|
|
}
|
|
strong_alias (__ieee754_j0, __j0_finite)
|
|
|
|
static const double
|
|
U[] = { -7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
|
|
1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
|
|
-1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
|
|
3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
|
|
-3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
|
|
1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
|
|
-3.98205194132103398453e-11 }, /* 0xBDC5E43D, 0x693FB3C8 */
|
|
V[] = { 1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
|
|
7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
|
|
2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
|
|
4.41110311332675467403e-10 }; /* 0x3DFE5018, 0x3BD6D9EF */
|
|
|
|
double
|
|
__ieee754_y0 (double x)
|
|
{
|
|
double z, s, c, ss, cc, u, v, z2, z4, z6, u1, u2, u3, v1, v2;
|
|
int32_t hx, ix, lx;
|
|
|
|
EXTRACT_WORDS (hx, lx, x);
|
|
ix = 0x7fffffff & hx;
|
|
/* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf. */
|
|
if (ix >= 0x7ff00000)
|
|
return one / (x + x * x);
|
|
if ((ix | lx) == 0)
|
|
return -1 / zero; /* -inf and divide by zero 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.
|
|
*/
|
|
__sincos (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 < 0x7fe00000) /* make sure x+x not overflow */
|
|
{
|
|
z = -__cos (x + x);
|
|
if ((s * c) < zero)
|
|
cc = z / ss;
|
|
else
|
|
ss = z / cc;
|
|
}
|
|
if (ix > 0x48000000)
|
|
z = (invsqrtpi * ss) / __ieee754_sqrt (x);
|
|
else
|
|
{
|
|
u = pzero (x); v = qzero (x);
|
|
z = invsqrtpi * (u * ss + v * cc) / __ieee754_sqrt (x);
|
|
}
|
|
return z;
|
|
}
|
|
if (ix <= 0x3e400000) /* x < 2**-27 */
|
|
{
|
|
return (U[0] + tpi * __ieee754_log (x));
|
|
}
|
|
z = x * x;
|
|
u1 = U[0] + z * U[1]; z2 = z * z;
|
|
u2 = U[2] + z * U[3]; z4 = z2 * z2;
|
|
u3 = U[4] + z * U[5]; z6 = z4 * z2;
|
|
u = u1 + z2 * u2 + z4 * u3 + z6 * U[6];
|
|
v1 = one + z * V[0];
|
|
v2 = V[1] + z * V[2];
|
|
v = v1 + z2 * v2 + z4 * V[3];
|
|
return (u / v + tpi * (__ieee754_j0 (x) * __ieee754_log (x)));
|
|
}
|
|
strong_alias (__ieee754_y0, __y0_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 double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
|
|
0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
|
|
-7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
|
|
-8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
|
|
-2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
|
|
-2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
|
|
-5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
|
|
};
|
|
static const double pS8[5] = {
|
|
1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
|
|
3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
|
|
4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
|
|
1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
|
|
4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
|
|
};
|
|
|
|
static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
|
|
-1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
|
|
-7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
|
|
-4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
|
|
-6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
|
|
-3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
|
|
-3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
|
|
};
|
|
static const double pS5[5] = {
|
|
6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
|
|
1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
|
|
5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
|
|
9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
|
|
2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
|
|
};
|
|
|
|
static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
|
|
-2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
|
|
-7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
|
|
-2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
|
|
-2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
|
|
-5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
|
|
-3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
|
|
};
|
|
static const double pS3[5] = {
|
|
3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
|
|
3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
|
|
1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
|
|
1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
|
|
1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
|
|
};
|
|
|
|
static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
|
|
-8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
|
|
-7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
|
|
-1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
|
|
-7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
|
|
-1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
|
|
-3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
|
|
};
|
|
static const double pS2[5] = {
|
|
2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
|
|
1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
|
|
2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
|
|
1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
|
|
1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
|
|
};
|
|
|
|
static double
|
|
pzero (double x)
|
|
{
|
|
const double *p, *q;
|
|
double z, r, s, z2, z4, r1, r2, r3, s1, s2, s3;
|
|
int32_t ix;
|
|
GET_HIGH_WORD (ix, x);
|
|
ix &= 0x7fffffff;
|
|
/* ix >= 0x40000000 for all calls to this function. */
|
|
if (ix >= 0x41b00000)
|
|
{
|
|
return one;
|
|
}
|
|
else if (ix >= 0x40200000)
|
|
{
|
|
p = pR8; q = pS8;
|
|
}
|
|
else if (ix >= 0x40122E8B)
|
|
{
|
|
p = pR5; q = pS5;
|
|
}
|
|
else if (ix >= 0x4006DB6D)
|
|
{
|
|
p = pR3; q = pS3;
|
|
}
|
|
else
|
|
{
|
|
p = pR2; q = pS2;
|
|
}
|
|
z = one / (x * x);
|
|
r1 = p[0] + z * p[1]; z2 = z * z;
|
|
r2 = p[2] + z * p[3]; z4 = z2 * z2;
|
|
r3 = p[4] + z * p[5];
|
|
r = r1 + z2 * r2 + z4 * r3;
|
|
s1 = one + z * q[0];
|
|
s2 = q[1] + z * q[2];
|
|
s3 = q[3] + z * q[4];
|
|
s = s1 + z2 * s2 + z4 * s3;
|
|
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 double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
|
|
0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
|
|
7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
|
|
1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
|
|
5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
|
|
8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
|
|
3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
|
|
};
|
|
static const double qS8[6] = {
|
|
1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
|
|
8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
|
|
1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
|
|
8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
|
|
8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
|
|
-3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
|
|
};
|
|
|
|
static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
|
|
1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
|
|
7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
|
|
5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
|
|
1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
|
|
1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
|
|
1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
|
|
};
|
|
static const double qS5[6] = {
|
|
8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
|
|
2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
|
|
1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
|
|
5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
|
|
3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
|
|
-5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
|
|
};
|
|
|
|
static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
|
|
4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
|
|
7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
|
|
3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
|
|
4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
|
|
1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
|
|
1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
|
|
};
|
|
static const double qS3[6] = {
|
|
4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
|
|
7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
|
|
3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
|
|
6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
|
|
2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
|
|
-1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
|
|
};
|
|
|
|
static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
|
|
1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
|
|
7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
|
|
1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
|
|
1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
|
|
3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
|
|
1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
|
|
};
|
|
static const double qS2[6] = {
|
|
3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
|
|
2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
|
|
8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
|
|
8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
|
|
2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
|
|
-5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
|
|
};
|
|
|
|
static double
|
|
qzero (double x)
|
|
{
|
|
const double *p, *q;
|
|
double s, r, z, z2, z4, z6, r1, r2, r3, s1, s2, s3;
|
|
int32_t ix;
|
|
GET_HIGH_WORD (ix, x);
|
|
ix &= 0x7fffffff;
|
|
/* ix >= 0x40000000 for all calls to this function. */
|
|
if (ix >= 0x41b00000)
|
|
{
|
|
return -.125 / x;
|
|
}
|
|
else if (ix >= 0x40200000)
|
|
{
|
|
p = qR8; q = qS8;
|
|
}
|
|
else if (ix >= 0x40122E8B)
|
|
{
|
|
p = qR5; q = qS5;
|
|
}
|
|
else if (ix >= 0x4006DB6D)
|
|
{
|
|
p = qR3; q = qS3;
|
|
}
|
|
else
|
|
{
|
|
p = qR2; q = qS2;
|
|
}
|
|
z = one / (x * x);
|
|
r1 = p[0] + z * p[1]; z2 = z * z;
|
|
r2 = p[2] + z * p[3]; z4 = z2 * z2;
|
|
r3 = p[4] + z * p[5]; z6 = z4 * z2;
|
|
r = r1 + z2 * r2 + z4 * r3;
|
|
s1 = one + z * q[0];
|
|
s2 = q[1] + z * q[2];
|
|
s3 = q[3] + z * q[4];
|
|
s = s1 + z2 * s2 + z4 * s3 + z6 * q[5];
|
|
return (-.125 + r / s) / x;
|
|
}
|