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/* e_j1f.c -- float version of e_j1.c.
*/
/*
* ====================================================
* 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 <errno.h>
#include <float.h>
#include <math.h>
#include <math-narrow-eval.h>
#include <math_private.h>
#include <fenv_private.h>
#include <math-underflow.h>
#include <libm-alias-finite.h>
#include <reduce_aux.h>
static float ponef(float), qonef(float);
static const float
huge = 1e30,
one = 1.0,
invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
tpi = 6.3661974669e-01, /* 0x3f22f983 */
/* R0/S0 on [0,2] */
r00 = -6.2500000000e-02, /* 0xbd800000 */
r01 = 1.4070566976e-03, /* 0x3ab86cfd */
r02 = -1.5995563444e-05, /* 0xb7862e36 */
r03 = 4.9672799207e-08, /* 0x335557d2 */
s01 = 1.9153760746e-02, /* 0x3c9ce859 */
s02 = 1.8594678841e-04, /* 0x3942fab6 */
s03 = 1.1771846857e-06, /* 0x359dffc2 */
s04 = 5.0463624390e-09, /* 0x31ad6446 */
s05 = 1.2354227016e-11; /* 0x2d59567e */
static const float zero = 0.0;
/* This is the nearest approximation of the first positive zero of j1. */
#define FIRST_ZERO_J1 0x3.d4eabp+0f
#define SMALL_SIZE 64
/* The following table contains successive zeros of j1 and degree-3
polynomial approximations of j1 around these zeros: Pj[0] for the first
positive zero (3.831705), Pj[1] for the second one (7.015586), and so on.
Each line contains:
{x0, xmid, x1, p0, p1, p2, p3}
where [x0,x1] is the interval around the zero, xmid is the binary32 number
closest to the zero, and p0+p1*x+p2*x^2+p3*x^3 is the approximation
polynomial. Each polynomial was generated using Sollya on the interval
[x0,x1] around the corresponding zero where the error exceeds 9 ulps
for the alternate code. Degree 3 is enough to get an error at most
9 ulps, except around the first zero.
*/
static const float Pj[SMALL_SIZE][7] = {
/* For index 0, we use a degree-4 polynomial generated by Sollya, with the
coefficient of degree 4 hard-coded in j1f_near_root(). */
{ 0x1.e09e5ep+1, 0x1.ea7558p+1, 0x1.ef7352p+1, -0x8.4f069p-28,
-0x6.71b3d8p-4, 0xd.744a2p-8, 0xd.acd9p-8/*, -0x1.3e51aap-8*/ }, /* 0 */
{ 0x1.bdb4c2p+2, 0x1.c0ff6p+2, 0x1.c27a8cp+2, 0xe.c2858p-28,
0x4.cd464p-4, -0x5.79b71p-8, -0xc.11124p-8 }, /* 1 */
{ 0x1.43b214p+3, 0x1.458d0ep+3, 0x1.460ccep+3, -0x1.e7acecp-24,
-0x3.feca9p-4, 0x3.2470f8p-8, 0xa.625b7p-8 }, /* 2 */
{ 0x1.a9c98p+3, 0x1.aa5bbp+3, 0x1.aaa4d8p+3, 0x1.698158p-24,
0x3.7e666cp-4, -0x2.1900ap-8, -0x9.2755p-8 }, /* 3 */
{ 0x1.073be4p+4, 0x1.0787b4p+4, 0x1.07aed8p+4, -0x1.f5f658p-24,
-0x3.24b8ep-4, 0x1.86e35cp-8, 0x8.4e4bbp-8 }, /* 4 */
{ 0x1.39ae2ap+4, 0x1.39da8ep+4, 0x1.39f3dap+4, -0x1.4e744p-24,
0x2.e18a24p-4, -0x1.2ccd16p-8, -0x7.a27ep-8 }, /* 5 */
{ 0x1.6bfa46p+4, 0x1.6c294ep+4, 0x1.6c412p+4, 0xa.3fb7fp-28,
-0x2.acc9c4p-4, 0xf.0b783p-12, 0x7.1c0d3p-8 }, /* 6 */
{ 0x1.9e42bep+4, 0x1.9e757p+4, 0x1.9e876ep+4, -0x2.29f6f4p-24,
0x2.81f21p-4, -0xc.641bp-12, -0x6.a7ea58p-8 }, /* 7 */
{ 0x1.d08a3ep+4, 0x1.d0bfdp+4, 0x1.d0cd3cp+4, -0x1.b5d196p-24,
-0x2.5e40e4p-4, 0xa.7059fp-12, 0x6.4d6bfp-8 }, /* 8 */
{ 0x1.017794p+5, 0x1.018476p+5, 0x1.018b8cp+5, -0x4.0e001p-24,
0x2.3febep-4, -0x8.f23aap-12, -0x6.0102cp-8 }, /* 9 */
{ 0x1.1a9e78p+5, 0x1.1aa89p+5, 0x1.1aaf88p+5, 0x3.b26f2p-24,
-0x2.25babp-4, 0x7.c6d948p-12, 0x5.a1d988p-8 }, /* 10 */
{ 0x1.33bddep+5, 0x1.33cc52p+5, 0x1.33d2e4p+5, -0xf.c8cdap-28,
0x2.0ed05p-4, -0x6.d97dbp-12, -0x5.8da498p-8 }, /* 11 */
{ 0x1.4ce7cp+5, 0x1.4cefdp+5, 0x1.4cf7d4p+5, -0x3.9940e4p-24,
-0x1.fa8b4p-4, 0x6.16108p-12, 0x5.4355e8p-8 }, /* 12 */
{ 0x1.6603e8p+5, 0x1.661316p+5, 0x1.66173ap+5, 0x9.da15dp-28,
0x1.e8727ep-4, -0x5.742468p-12, -0x5.117c28p-8 }, /* 13 */
{ 0x1.7f2ebcp+5, 0x1.7f3632p+5, 0x1.7f3a7ep+5, -0x3.39b218p-24,
-0x1.d8293ap-4, 0x4.ee3348p-12, 0x4.f9bep-8 }, /* 14 */
{ 0x1.9850e6p+5, 0x1.985928p+5, 0x1.985d9ep+5, -0x3.7b5108p-24,
0x1.c96702p-4, -0x4.7b0d08p-12, -0x4.c784a8p-8 }, /* 15 */
{ 0x1.b172e8p+5, 0x1.b17c04p+5, 0x1.b1805cp+5, -0x1.91e43ep-24,
-0x1.bbf246p-4, 0x4.18ad78p-12, 0x4.9bfae8p-8 }, /* 16 */
{ 0x1.ca955ap+5, 0x1.ca9ec6p+5, 0x1.caa2a4p+5, 0x1.28453cp-24,
0x1.af9cb4p-4, -0x3.c3a494p-12, -0x4.78b69p-8 }, /* 17 */
{ 0x1.e3bc94p+5, 0x1.e3c174p+5, 0x1.e3c64p+5, -0x2.e7fef4p-24,
-0x1.a4407ep-4, 0x3.79b228p-12, 0x4.874f7p-8 }, /* 18 */
{ 0x1.fcdf16p+5, 0x1.fce40ep+5, 0x1.fce71p+5, -0x3.23b2fcp-24,
0x1.99be76p-4, -0x3.39ad7cp-12, -0x4.92a55p-8 }, /* 19 */
{ 0x1.0afe34p+6, 0x1.0b034ep+6, 0x1.0b054ap+6, -0xd.19e93p-28,
-0x1.8ffc9cp-4, 0x2.fee7f8p-12, 0x4.2d33b8p-8 }, /* 20 */
{ 0x1.179344p+6, 0x1.17948ep+6, 0x1.1795bp+6, 0x1.c2ac48p-24,
0x1.86e51cp-4, -0x2.cc5abp-12, -0x4.866d08p-8 }, /* 21 */
{ 0x1.24231ep+6, 0x1.2425c8p+6, 0x1.2426e2p+6, -0xd.31027p-28,
-0x1.7e656ep-4, 0x2.9db23cp-12, 0x3.cc63c8p-8 }, /* 22 */
{ 0x1.30b5a8p+6, 0x1.30b6fep+6, 0x1.30b84ep+6, 0x5.b5e53p-24,
0x1.766dc2p-4, -0x2.754cfcp-12, -0x3.c39bb4p-8 }, /* 23 */
{ 0x1.3d46ccp+6, 0x1.3d482ep+6, 0x1.3d495ep+6, -0x1.340a8ap-24,
-0x1.6ef07ep-4, 0x2.4ff9d4p-12, 0x3.9b36e4p-8 }, /* 24 */
{ 0x1.49d688p+6, 0x1.49d95ap+6, 0x1.49dabep+6, -0x3.ba66p-24,
0x1.67e1dcp-4, -0x2.2f32b8p-12, -0x3.e2aaf4p-8 }, /* 25 */
{ 0x1.566916p+6, 0x1.566a84p+6, 0x1.566bcp+6, 0x6.47ca5p-28,
-0x1.61379ap-4, 0x2.1096acp-12, 0x4.2d0968p-8 }, /* 26 */
{ 0x1.62f8dap+6, 0x1.62fbaap+6, 0x1.62fc9cp+6, -0x2.12affp-24,
0x1.5ae8c4p-4, -0x1.f32444p-12, -0x3.9e592p-8 }, /* 27 */
{ 0x1.6f89e6p+6, 0x1.6f8ccep+6, 0x1.6f8e34p+6, -0x7.4853ap-28,
-0x1.54ed76p-4, 0x1.db004ap-12, 0x3.907034p-8 }, /* 28 */
{ 0x1.7c1c6ap+6, 0x1.7c1deep+6, 0x1.7c1f4cp+6, -0x4.f0a998p-24,
0x1.4f3ebcp-4, -0x1.c26808p-12, -0x2.da8df8p-8 }, /* 29 */
{ 0x1.88adaep+6, 0x1.88af0ep+6, 0x1.88afc4p+6, -0x1.80c246p-24,
-0x1.49d668p-4, 0x1.aebc26p-12, 0x3.af7b5cp-8 }, /* 30 */
{ 0x1.953d7p+6, 0x1.95402ap+6, 0x1.9540ep+6, -0x2.22aff8p-24,
0x1.44aefap-4, -0x1.99f25p-12, -0x3.5e9198p-8 }, /* 31 */
{ 0x1.a1d01ep+6, 0x1.a1d146p+6, 0x1.a1d20ap+6, -0x3.aad6d4p-24,
-0x1.3fc386p-4, 0x1.892858p-12, 0x3.fe0184p-8 }, /* 32 */
{ 0x1.ae60ecp+6, 0x1.ae625ep+6, 0x1.ae6326p+6, -0x2.010be4p-24,
0x1.3b0fa4p-4, -0x1.7539ap-12, -0x2.b2c9bp-8 }, /* 33 */
{ 0x1.baf234p+6, 0x1.baf376p+6, 0x1.baf442p+6, -0xd.4fd17p-32,
-0x1.368f5cp-4, 0x1.6734e4p-12, 0x3.59f514p-8 }, /* 34 */
{ 0x1.c782e6p+6, 0x1.c7848cp+6, 0x1.c78516p+6, -0xa.d662dp-28,
0x1.323f18p-4, -0x1.571c02p-12, -0x3.2be5bp-8 }, /* 35 */
{ 0x1.d4144ep+6, 0x1.d415ap+6, 0x1.d41622p+6, 0x4.9f217p-24,
-0x1.2e1b9ap-4, 0x1.4a2edap-12, 0x3.a4e96p-8 }, /* 36 */
{ 0x1.e0a5ep+6, 0x1.e0a6b4p+6, 0x1.e0a788p+6, -0x2.d167p-24,
0x1.2a21eep-4, -0x1.3c4b46p-12, -0x4.9e0978p-8 }, /* 37 */
{ 0x1.ed36eep+6, 0x1.ed37c8p+6, 0x1.ed3892p+6, -0x4.15a83p-24,
-0x1.264f66p-4, 0x1.31dea4p-12, 0x3.d125ecp-8 }, /* 38 */
{ 0x1.f9c77p+6, 0x1.f9c8d8p+6, 0x1.f9c9acp+6, -0x2.a5bbbp-24,
0x1.22a192p-4, -0x1.25e59ep-12, -0x2.ef6934p-8 }, /* 39 */
{ 0x1.032c54p+7, 0x1.032cf4p+7, 0x1.032d66p+7, 0x4.e828bp-24,
-0x1.1f1634p-4, 0x1.1c2394p-12, 0x3.6d744cp-8 }, /* 40 */
{ 0x1.09750cp+7, 0x1.09757cp+7, 0x1.0975b6p+7, -0x3.28a3bcp-24,
0x1.1bab3ep-4, -0x1.1569cep-12, -0x5.84da7p-8 }, /* 41 */
{ 0x1.0fbd9ap+7, 0x1.0fbe04p+7, 0x1.0fbe5ep+7, -0x2.2f667p-24,
-0x1.185eccp-4, 0x1.07f42cp-12, 0x2.87896cp-8 }, /* 42 */
{ 0x1.160628p+7, 0x1.16068ap+7, 0x1.1606cep+7, -0x6.9097dp-24,
0x1.152f28p-4, -0x1.0227fep-12, -0x5.da6e6p-8 }, /* 43 */
{ 0x1.1c4e9ap+7, 0x1.1c4f12p+7, 0x1.1c4f7cp+7, -0x5.1b408p-24,
-0x1.121abp-4, 0xf.6efcp-16, 0x2.c5e954p-8 }, /* 44 */
{ 0x1.2296aap+7, 0x1.229798p+7, 0x1.2297d4p+7, 0x2.70d0dp-24,
0x1.0f1ffp-4, -0xf.523f5p-16, -0x3.5c0568p-8 }, /* 45 */
{ 0x1.28dfa4p+7, 0x1.28e01ep+7, 0x1.28e054p+7, -0x2.7c176p-24,
-0x1.0c3d8ap-4, 0xe.8329ap-16, 0x3.5eb34p-8 }, /* 46 */
{ 0x1.2f282ap+7, 0x1.2f28a4p+7, 0x1.2f28dep+7, 0x4.fd6368p-24,
0x1.097236p-4, -0xe.17299p-16, -0x3.73a2e4p-8 }, /* 47 */
{ 0x1.3570bp+7, 0x1.357128p+7, 0x1.35716p+7, 0x6.b05f68p-24,
-0x1.06bccap-4, 0xd.527b8p-16, 0x2.b8bf9cp-8 }, /* 48 */
{ 0x1.3bb932p+7, 0x1.3bb9aep+7, 0x1.3bb9eap+7, 0x4.0d622p-28,
0x1.041c28p-4, -0xd.0ac11p-16, -0x1.65f2ccp-8 }, /* 49 */
{ 0x1.4201b6p+7, 0x1.420232p+7, 0x1.42027p+7, 0x7.0d98cp-24,
-0x1.018f52p-4, 0xc.c4d8ep-16, 0x2.8f250cp-8 }, /* 50 */
{ 0x1.484a78p+7, 0x1.484ab8p+7, 0x1.484af4p+7, 0x3.93d10cp-24,
0xf.f154fp-8, -0xc.7b7fep-16, -0x3.6b6e4cp-8 }, /* 51 */
{ 0x1.4e92c8p+7, 0x1.4e933cp+7, 0x1.4e9368p+7, 0xd.88185p-32,
-0xf.cad3fp-8, 0xc.1462p-16, 0x2.bd66p-8 }, /* 52 */
{ 0x1.54db84p+7, 0x1.54dbcp+7, 0x1.54dbf4p+7, -0x1.fe6b92p-24,
0xf.a564cp-8, -0xb.c4e6cp-16, -0x3.d51decp-8 }, /* 53 */
{ 0x1.5b23c4p+7, 0x1.5b2444p+7, 0x1.5b2486p+7, 0x2.6137f4p-24,
-0xf.80faep-8, 0xb.5199ep-16, 0x1.9ca85ap-8 }, /* 54 */
{ 0x1.616c62p+7, 0x1.616cc8p+7, 0x1.616d0ap+7, -0x1.55468p-24,
0xf.5d8acp-8, -0xb.21d16p-16, -0x1.b8809ap-8 }, /* 55 */
{ 0x1.67b4fp+7, 0x1.67b54cp+7, 0x1.67b588p+7, -0x1.08c6bep-24,
-0xf.3b096p-8, 0xa.e65efp-16, 0x3.642304p-8 }, /* 56 */
{ 0x1.6dfd8ep+7, 0x1.6dfddp+7, 0x1.6dfe0ap+7, 0x4.9ebfa8p-24,
0xf.196c7p-8, -0xa.ba8c8p-16, -0x5.ad6a2p-8 }, /* 57 */
{ 0x1.74461p+7, 0x1.744652p+7, 0x1.744692p+7, 0x5.a4017p-24,
-0xe.f8aa5p-8, 0xa.49748p-16, 0x2.a86498p-8 }, /* 58 */
{ 0x1.7a8e5ep+7, 0x1.7a8ed6p+7, 0x1.7a8ef8p+7, 0x3.bcb2a8p-28,
0xe.d8b9dp-8, -0x9.c43bep-16, -0x1.e7124ap-8 }, /* 59 */
{ 0x1.80d6cep+7, 0x1.80d75ap+7, 0x1.80d78ap+7, -0x7.1091fp-24,
-0xe.b9925p-8, 0x9.c43dap-16, 0x1.aba86p-8 }, /* 60 */
{ 0x1.871f58p+7, 0x1.871fdcp+7, 0x1.87201ep+7, 0x2.ca1cf4p-28,
0xe.9b2bep-8, -0x9.843b3p-16, -0x2.093e68p-8 }, /* 61 */
{ 0x1.8d67e8p+7, 0x1.8d685ep+7, 0x1.8d688ep+7, 0x5.aa8908p-24,
-0xe.7d7ecp-8, 0x9.501a8p-16, 0x2.54a754p-8 }, /* 62 */
{ 0x1.93b09cp+7, 0x1.93b0e2p+7, 0x1.93b10ep+7, 0x3.d9cd9cp-24,
0xe.6083ap-8, -0x9.45dadp-16, -0x5.112908p-8 }, /* 63 */
};
/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
j1f(x) ~ sqrt(2/(pi*x))*beta1(x)*cos(x-3pi/4-alpha1(x))
where beta1(x) = 1 + 3/(16*x^2) - 99/(512*x^4)
and alpha1(x) = -3/(8*x) + 21/(128*x^3) - 1899/(5120*x^5). */
static float
j1f_asympt (float x)
{
float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */
if (x < 0)
{
x = -x;
cst = -cst;
}
double y = 1.0 / (double) x;
double y2 = y * y;
double beta1 = 1.0f + y2 * (0x3p-4 - 0x3.18p-4 * y2);
double alpha1;
alpha1 = y * (-0x6p-4 + y2 * (0x2.ap-4 - 0x5.ef33333333334p-4 * y2));
double h;
int n;
h = reduce_aux (x, &n, alpha1);
n--; /* Subtract pi/2. */
/* Now x - 3pi/4 - alpha1 = h + n*pi/2 mod (2*pi). */
float xr = (float) h;
n = n & 3;
float t = cst / sqrtf (x) * (float) beta1;
if (n == 0)
return t * __cosf (xr);
else if (n == 2) /* cos(x+pi) = -cos(x) */
return -t * __cosf (xr);
else if (n == 1) /* cos(x+pi/2) = -sin(x) */
return -t * __sinf (xr);
else /* cos(x+3pi/2) = sin(x) */
return t * __sinf (xr);
}
/* Special code for x near a root of j1.
z is the value computed by the generic code.
For small x, we use a polynomial approximating j1 around its root.
For large x, we use an asymptotic formula (j1f_asympt). */
static float
j1f_near_root (float x, float z)
{
float index_f, sign = 1.0f;
int index;
if (x < 0)
{
x = -x;
sign = -1.0f;
}
index_f = roundf ((x - FIRST_ZERO_J1) / M_PIf);
if (index_f >= SMALL_SIZE)
return sign * j1f_asympt (x);
index = (int) index_f;
const float *p = Pj[index];
float x0 = p[0];
float x1 = p[2];
/* If not in the interval [x0,x1] around xmid, return the value z. */
if (! (x0 <= x && x <= x1))
return z;
float xmid = p[1];
float y = x - xmid;
float p6 = (index > 0) ? p[6] : p[6] + y * -0x1.3e51aap-8f;
return sign * (p[3] + y * (p[4] + y * (p[5] + y * p6)));
}
float
__ieee754_j1f(float x)
{
float z, s,c,ss,cc,r,u,v,y;
int32_t hx,ix;
GET_FLOAT_WORD(hx,x);
ix = hx&0x7fffffff;
if(__builtin_expect(ix>=0x7f800000, 0)) return one/x;
y = fabsf(x);
if(ix >= 0x40000000) { /* |x| >= 2.0 */
SET_RESTORE_ROUNDF (FE_TONEAREST);
__sincosf (y, &s, &c);
ss = -s-c;
cc = s-c;
if (ix >= 0x7f000000)
/* x >= 2^127: use asymptotic expansion. */
return j1f_asympt (x);
/* Now we are sure that x+x cannot overflow. */
z = __cosf(y+y);
if ((s*c)>zero) cc = z/ss;
else ss = z/cc;
/*
* j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
* y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
*/
if (ix <= 0x5c000000)
{
u = ponef(y); v = qonef(y);
cc = u*cc-v*ss;
}
z = (invsqrtpi * cc) / sqrtf(y);
/* Adjust sign of z. */
z = (hx < 0) ? -z : z;
/* The following threshold is optimal: for x=0x1.e09e5ep+1
and rounding upwards, cc=0x1.b79638p-4 and z is 10 ulps
far from the correctly rounded value. */
float threshold = 0x1.b79638p-4;
if (fabsf (cc) > threshold)
return z;
else
return j1f_near_root (x, z);
}
if(__builtin_expect(ix<0x32000000, 0)) { /* |x|<2**-27 */
if(huge+x>one) { /* inexact if x!=0 necessary */
float ret = math_narrow_eval ((float) 0.5 * x);
math_check_force_underflow (ret);
if (ret == 0 && x != 0)
__set_errno (ERANGE);
return ret;
}
}
z = x*x;
r = z*(r00+z*(r01+z*(r02+z*r03)));
s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
r *= x;
return(x*(float)0.5+r/s);
}
libm_alias_finite (__ieee754_j1f, __j1f)
static const float U0[5] = {
-1.9605709612e-01, /* 0xbe48c331 */
5.0443872809e-02, /* 0x3d4e9e3c */
-1.9125689287e-03, /* 0xbafaaf2a */
2.3525259166e-05, /* 0x37c5581c */
-9.1909917899e-08, /* 0xb3c56003 */
};
static const float V0[5] = {
1.9916731864e-02, /* 0x3ca3286a */
2.0255257550e-04, /* 0x3954644b */
1.3560879779e-06, /* 0x35b602d4 */
6.2274145840e-09, /* 0x31d5f8eb */
1.6655924903e-11, /* 0x2d9281cf */
};
/* This is the nearest approximation of the first zero of y1. */
#define FIRST_ZERO_Y1 0x2.3277dcp+0f
/* The following table contains successive zeros of y1 and degree-3
polynomial approximations of y1 around these zeros: Py[0] for the first
positive zero (2.197141), Py[1] for the second one (5.429681), and so on.
Each line contains:
{x0, xmid, x1, p0, p1, p2, p3}
where [x0,x1] is the interval around the zero, xmid is the binary32 number
closest to the zero, and p0+p1*x+p2*x^2+p3*x^3 is the approximation
polynomial. Each polynomial was generated using Sollya on the interval
[x0,x1] around the corresponding zero where the error exceeds 9 ulps
for the alternate code. Degree 3 is enough, except for the first roots.
*/
static const float Py[SMALL_SIZE][7] = {
/* For index 0, we use a degree-5 polynomial generated by Sollya, with the
coefficients of degree 4 and 5 hard-coded in y1f_near_root(). */
{ 0x1.f7f16ap+0, 0x1.193beep+1, 0x1.2105dcp+1, 0xb.96749p-28,
0x8.55241p-4, -0x1.e570bp-4, -0x8.68b61p-8
/*, -0x1.28043p-8, 0x2.50e83p-8*/ }, /* 0 */
/* For index 1, we use a degree-4 polynomial generated by Sollya, with the
coefficient of degree 4 hard-coded in y1f_near_root(). */
{ 0x1.55c6d2p+2, 0x1.5b7fe4p+2, 0x1.5cf8cap+2, 0x1.3c7822p-24,
-0x5.71f158p-4, 0x8.05cb4p-8, 0xd.0b15p-8/*, -0xf.ff6b8p-12*/ }, /* 1 */
{ 0x1.113c6p+3, 0x1.13127ap+3, 0x1.1387dcp+3, -0x1.f3ad8ep-24,
0x4.57e66p-4, -0x4.0afb58p-8, -0xb.29207p-8 }, /* 2 */
{ 0x1.76e7dep+3, 0x1.77f914p+3, 0x1.786a6ap+3, -0xd.5608fp-28,
-0x3.b829d4p-4, 0x2.8852cp-8, 0x9.b70e3p-8 }, /* 3 */
{ 0x1.dc2794p+3, 0x1.dcb7d8p+3, 0x1.dd032p+3, -0xe.a7c04p-28,
0x3.4e0458p-4, -0x1.c64b18p-8, -0x8.b0e7fp-8 }, /* 4 */
{ 0x1.20874p+4, 0x1.20b1c6p+4, 0x1.20c71p+4, 0x1.c2626p-24,
-0x3.00f03cp-4, 0x1.54f806p-8, 0x7.f9cf9p-8 }, /* 5 */
{ 0x1.52d848p+4, 0x1.530254p+4, 0x1.531962p+4, -0x1.9503ecp-24,
0x2.c5b29cp-4, -0x1.0bf28p-8, -0x7.562e58p-8 }, /* 6 */
{ 0x1.851e64p+4, 0x1.854fa4p+4, 0x1.85679p+4, -0x2.8d40fcp-24,
-0x2.96547p-4, 0xd.9c38bp-12, 0x6.dcbf8p-8 }, /* 7 */
{ 0x1.b7808ep+4, 0x1.b79acep+4, 0x1.b7b2a8p+4, -0x2.36df5cp-24,
0x2.6f55ap-4, -0xb.57f9fp-12, -0x6.82569p-8 }, /* 8 */
{ 0x1.e9c8fp+4, 0x1.e9e48p+4, 0x1.e9f24p+4, 0xd.e2eb7p-28,
-0x2.4e8104p-4, 0x9.a4be2p-12, 0x6.2541fp-8 }, /* 9 */
{ 0x1.0e0808p+5, 0x1.0e169p+5, 0x1.0e1d92p+5, -0x2.3070f4p-24,
0x2.325e4cp-4, -0x8.53604p-12, -0x5.ca03a8p-8 }, /* 10 */
{ 0x1.272e08p+5, 0x1.273a7cp+5, 0x1.2741fcp+5, -0x3.525508p-24,
-0x2.19e7dcp-4, 0x7.49d1dp-12, 0x5.9cb02p-8 }, /* 11 */
{ 0x1.404ec6p+5, 0x1.405e18p+5, 0x1.4065cep+5, -0xe.6e158p-28,
0x2.046174p-4, -0x6.71b3dp-12, -0x5.4c3c8p-8 }, /* 12 */
{ 0x1.5971dcp+5, 0x1.598178p+5, 0x1.598592p+5, 0x1.e72698p-24,
-0x1.f13fb2p-4, 0x5.c0f938p-12, 0x5.28ca78p-8 }, /* 13 */
{ 0x1.729c4ep+5, 0x1.72a4a8p+5, 0x1.72a8eap+5, -0x1.5bed9cp-24,
0x1.e018dcp-4, -0x5.2f11e8p-12, -0x5.16ce48p-8 }, /* 14 */
{ 0x1.8bbf4ep+5, 0x1.8bc7b2p+5, 0x1.8bcc1p+5, -0x3.6b654cp-24,
-0x1.d09b2p-4, 0x4.b1747p-12, 0x4.bd22fp-8 }, /* 15 */
{ 0x1.a4e272p+5, 0x1.a4ea9ap+5, 0x1.a4eef4p+5, 0x1.6f11bp-24,
0x1.c28612p-4, -0x4.47462p-12, -0x4.947c5p-8 }, /* 16 */
{ 0x1.be08bep+5, 0x1.be0d68p+5, 0x1.be1088p+5, -0x2.0bc074p-24,
-0x1.b5a622p-4, 0x3.ed52d4p-12, 0x4.b76fc8p-8 }, /* 17 */
{ 0x1.d7272ap+5, 0x1.d7301ep+5, 0x1.d734aep+5, -0x2.87dd4p-24,
0x1.a9d184p-4, -0x3.9cf494p-12, -0x4.6303ep-8 }, /* 18 */
{ 0x1.f0499ap+5, 0x1.f052c4p+5, 0x1.f05758p+5, -0x2.fb964p-24,
-0x1.9ee5eep-4, 0x3.5800dp-12, 0x4.4e9f9p-8 }, /* 19 */
{ 0x1.04b63ap+6, 0x1.04baacp+6, 0x1.04bc92p+6, 0x2.cf5adp-24,
0x1.94c6f4p-4, -0x3.1a83e4p-12, -0x4.2311fp-8 }, /* 20 */
{ 0x1.1146dp+6, 0x1.114beep+6, 0x1.114e12p+6, 0x3.6766fp-24,
-0x1.8b5cccp-4, 0x2.e4a4e4p-12, 0x4.20bf9p-8 }, /* 21 */
{ 0x1.1dda8cp+6, 0x1.1ddd2cp+6, 0x1.1dde7ap+6, 0x3.501424p-24,
0x1.829356p-4, -0x2.b47524p-12, -0x4.04bf18p-8 }, /* 22 */
{ 0x1.2a6bcp+6, 0x1.2a6e64p+6, 0x1.2a6faap+6, -0x5.c05808p-24,
-0x1.7a597ep-4, 0x2.8a0498p-12, 0x4.187258p-8 }, /* 23 */
{ 0x1.36fcd6p+6, 0x1.36ff96p+6, 0x1.3700f6p+6, 0x7.1e1478p-28,
0x1.72a09ap-4, -0x2.61a7fp-12, -0x3.c0b54p-8 }, /* 24 */
{ 0x1.438f46p+6, 0x1.4390c4p+6, 0x1.4392p+6, 0x3.e36e6cp-24,
-0x1.6b5c06p-4, 0x2.3f612p-12, 0x4.18f868p-8 }, /* 25 */
{ 0x1.501f4cp+6, 0x1.5021fp+6, 0x1.50235p+6, 0x1.3f9e5ap-24,
0x1.6480c4p-4, -0x2.1f28fcp-12, -0x3.bb4e3cp-8 }, /* 26 */
{ 0x1.5cb07cp+6, 0x1.5cb318p+6, 0x1.5cb464p+6, -0x2.39e41cp-24,
-0x1.5e0544p-4, 0x2.0189f4p-12, 0x3.8b55acp-8 }, /* 27 */
{ 0x1.694166p+6, 0x1.69443cp+6, 0x1.694594p+6, -0x2.912f84p-24,
0x1.57e12p-4, -0x1.e6fabep-12, -0x3.850174p-8 }, /* 28 */
{ 0x1.75d27cp+6, 0x1.75d55ep+6, 0x1.75d67ep+6, 0x3.d5b00cp-24,
-0x1.520ceep-4, 0x1.d0286ep-12, 0x3.8e7d1p-8 }, /* 29 */
{ 0x1.82653ep+6, 0x1.82667ep+6, 0x1.82674p+6, -0x3.1726ecp-24,
0x1.4c8222p-4, -0x1.b98206p-12, -0x3.f34978p-8 }, /* 30 */
{ 0x1.8ef4b4p+6, 0x1.8ef79cp+6, 0x1.8ef888p+6, 0x1.949e22p-24,
-0x1.473ae6p-4, 0x1.a47388p-12, 0x3.69eefcp-8 }, /* 31 */
{ 0x1.9b8728p+6, 0x1.9b88b8p+6, 0x1.9b896cp+6, -0x5.5553bp-28,
0x1.42320ap-4, -0x1.90f0b8p-12, -0x3.6565p-8 }, /* 32 */
{ 0x1.a8183cp+6, 0x1.a819d2p+6, 0x1.a81aecp+6, 0x3.2df7ecp-28,
-0x1.3d62e4p-4, 0x1.7dae28p-12, 0x2.9eb128p-8 }, /* 33 */
{ 0x1.b4aa1cp+6, 0x1.b4aaeap+6, 0x1.b4abb8p+6, -0x1.e13fcep-24,
0x1.38c948p-4, -0x1.6eb0ecp-12, -0x1.f9ddf8p-8 }, /* 34 */
{ 0x1.c13a7ap+6, 0x1.c13c02p+6, 0x1.c13cbp+6, -0x3.ad9974p-24,
-0x1.34616ep-4, 0x1.5e36ecp-12, 0x2.a9fc5p-8 }, /* 35 */
{ 0x1.cdcb76p+6, 0x1.cdcd16p+6, 0x1.cdcde4p+6, -0x3.6977e8p-24,
0x1.3027fp-4, -0x1.4f703p-12, -0x2.9817d4p-8 }, /* 36 */
{ 0x1.da5cdep+6, 0x1.da5e2ap+6, 0x1.da5efp+6, 0x4.654cbp-24,
-0x1.2c19b6p-4, 0x1.455982p-12, 0x3.f1c564p-8 }, /* 37 */
{ 0x1.e6edccp+6, 0x1.e6ef3ep+6, 0x1.e6f00ap+6, 0x8.825c8p-32,
0x1.2833eep-4, -0x1.39097p-12, -0x3.b2646p-8 }, /* 38 */
{ 0x1.f37f72p+6, 0x1.f3805p+6, 0x1.f3812ap+6, -0x2.0d11d8p-28,
-0x1.24740ap-4, 0x1.2c16p-12, 0x1.fc3804p-8 }, /* 39 */
{ 0x1.000842p+7, 0x1.0008bp+7, 0x1.000908p+7, -0x4.4e495p-24,
0x1.20d7b6p-4, -0x1.20816p-12, -0x2.d1ebe8p-8 }, /* 40 */
{ 0x1.06505cp+7, 0x1.065138p+7, 0x1.06518p+7, 0x4.81c1c8p-24,
-0x1.1d5ccap-4, 0x1.17ad5ap-12, 0x2.fda33p-8 }, /* 41 */
{ 0x1.0c98dap+7, 0x1.0c99cp+7, 0x1.0c9a28p+7, -0xe.99386p-28,
0x1.1a015p-4, -0x1.0bd50ap-12, -0x2.9dfb68p-8 }, /* 42 */
{ 0x1.12e212p+7, 0x1.12e248p+7, 0x1.12e29p+7, -0x6.16f1c8p-24,
-0x1.16c37ap-4, 0x1.0303dcp-12, 0x4.34316p-8 }, /* 43 */
{ 0x1.192a68p+7, 0x1.192acep+7, 0x1.192b02p+7, -0x1.129336p-24,
0x1.13a19ep-4, -0xf.bd247p-16, -0x3.851d18p-8 }, /* 44 */
{ 0x1.1f727p+7, 0x1.1f7354p+7, 0x1.1f73ap+7, 0x5.19c09p-24,
-0x1.109a32p-4, 0xf.09644p-16, 0x2.d78194p-8 }, /* 45 */
{ 0x1.25bb8p+7, 0x1.25bbdap+7, 0x1.25bc12p+7, -0x6.497dp-24,
0x1.0dabc8p-4, -0xe.a1d25p-16, -0x2.3378bp-8 }, /* 46 */
{ 0x1.2c04p+7, 0x1.2c046p+7, 0x1.2c04ap+7, 0x4.e4f338p-24,
-0x1.0ad512p-4, 0xe.52d84p-16, 0x4.3bfa08p-8 }, /* 47 */
{ 0x1.324cbp+7, 0x1.324ce6p+7, 0x1.324d4p+7, -0x1.287c58p-24,
0x1.0814d4p-4, -0xe.03a95p-16, 0x3.9930ap-12 }, /* 48 */
{ 0x1.3894f6p+7, 0x1.38956cp+7, 0x1.3895ap+7, -0x4.b594ep-24,
-0x1.0569fp-4, 0xd.6787ep-16, 0x4.0a5148p-8 }, /* 49 */
{ 0x1.3edd98p+7, 0x1.3eddfp+7, 0x1.3ede2ap+7, -0x3.a8f164p-24,
0x1.02d354p-4, -0xd.0309dp-16, -0x3.2ebfb4p-8 }, /* 50 */
{ 0x1.452638p+7, 0x1.452676p+7, 0x1.4526b4p+7, -0x6.12505p-24,
-0x1.005004p-4, 0xc.a0045p-16, 0x4.87c67p-8 }, /* 51 */
{ 0x1.4b6e8p+7, 0x1.4b6efap+7, 0x1.4b6f34p+7, 0x1.8acf4ep-24,
0xf.ddf16p-8, -0xc.2d207p-16, -0x1.da6c36p-8 }, /* 52 */
{ 0x1.51b742p+7, 0x1.51b77ep+7, 0x1.51b7b2p+7, 0x1.39cf86p-24,
-0xf.b7faep-8, 0xb.db598p-16, -0x8.945b1p-12 }, /* 53 */
{ 0x1.57ffc4p+7, 0x1.580002p+7, 0x1.58003cp+7, -0x2.5f8de8p-24,
0xf.930fep-8, -0xb.91889p-16, -0xa.30df9p-12 }, /* 54 */
{ 0x1.5e483p+7, 0x1.5e4886p+7, 0x1.5e48c8p+7, 0x2.073d64p-24,
-0xf.6f245p-8, 0xb.4085fp-16, 0x2.128188p-8 }, /* 55 */
{ 0x1.64908cp+7, 0x1.64910ap+7, 0x1.64912ap+7, -0x4.ed26ep-28,
0xf.4c2cep-8, -0xa.fe719p-16, -0x2.9374b8p-8 }, /* 56 */
{ 0x1.6ad91ep+7, 0x1.6ad98ep+7, 0x1.6ad9cep+7, -0x2.ae5204p-24,
-0xf.2a1efp-8, 0xa.aa585p-16, 0x2.1c0834p-8 }, /* 57 */
{ 0x1.7121cep+7, 0x1.712212p+7, 0x1.712238p+7, 0x6.d72168p-24,
0xf.08f09p-8, -0xa.7da49p-16, -0x3.4f5f1cp-8 }, /* 58 */
{ 0x1.776a0cp+7, 0x1.776a94p+7, 0x1.776accp+7, 0x2.d3f294p-24,
-0xe.e8986p-8, 0xa.23ccdp-16, 0x2.2a6678p-8 }, /* 59 */
{ 0x1.7db2e8p+7, 0x1.7db318p+7, 0x1.7db35ap+7, 0x3.88c0fp-24,
0xe.c90d7p-8, -0x9.eaeap-16, -0x2.86438cp-8 }, /* 60 */
{ 0x1.83fb56p+7, 0x1.83fb9ap+7, 0x1.83fbep+7, 0x3.d94d34p-24,
-0xe.aa478p-8, 0x9.abac7p-16, 0x1.ac2d84p-8 }, /* 61 */
{ 0x1.8a43e8p+7, 0x1.8a441ep+7, 0x1.8a446p+7, 0x4.66b7ep-24,
0xe.8c3e9p-8, -0x9.87682p-16, -0x7.9ab4a8p-12 }, /* 62 */
{ 0x1.908c6p+7, 0x1.908cap+7, 0x1.908ce6p+7, 0xf.f7ac9p-28,
-0xe.6eeb6p-8, 0x9.4423p-16, 0x4.54c4d8p-8 }, /* 63 */
};
/* Formula page 5 of https://www.cl.cam.ac.uk/~jrh13/papers/bessel.pdf:
y1f(x) ~ sqrt(2/(pi*x))*beta1(x)*sin(x-3pi/4-alpha1(x))
where beta1(x) = 1 + 3/(16*x^2) - 99/(512*x^4)
and alpha1(x) = -3/(8*x) + 21/(128*x^3) - 1899/(5120*x^5). */
static float
y1f_asympt (float x)
{
float cst = 0xc.c422ap-4; /* sqrt(2/pi) rounded to nearest */
double y = 1.0 / (double) x;
double y2 = y * y;
double beta1 = 1.0f + y2 * (0x3p-4 - 0x3.18p-4 * y2);
double alpha1;
alpha1 = y * (-0x6p-4 + y2 * (0x2.ap-4 - 0x5.ef33333333334p-4 * y2));
double h;
int n;
h = reduce_aux (x, &n, alpha1);
n--; /* Subtract pi/2. */
/* Now x - 3pi/4 - alpha1 = h + n*pi/2 mod (2*pi). */
float xr = (float) h;
n = n & 3;
float t = cst / sqrtf (x) * (float) beta1;
if (n == 0)
return t * __sinf (xr);
else if (n == 2) /* sin(x+pi) = -sin(x) */
return -t * __sinf (xr);
else if (n == 1) /* sin(x+pi/2) = cos(x) */
return t * __cosf (xr);
else /* sin(x+3pi/2) = -cos(x) */
return -t * __cosf (xr);
}
/* Special code for x near a root of y1.
z is the value computed by the generic code.
For small x, we use a polynomial approximating y1 around its root.
For large x, we use an asymptotic formula (y1f_asympt). */
static float
y1f_near_root (float x, float z)
{
float index_f;
int index;
index_f = roundf ((x - FIRST_ZERO_Y1) / M_PIf);
if (index_f >= SMALL_SIZE)
return y1f_asympt (x);
index = (int) index_f;
const float *p = Py[index];
float x0 = p[0];
float x1 = p[2];
/* If not in the interval [x0,x1] around xmid, return the value z. */
if (! (x0 <= x && x <= x1))
return z;
float xmid = p[1];
float y = x - xmid, p6;
if (index == 0)
p6 = p[6] + y * (-0x1.28043p-8 + y * 0x2.50e83p-8);
else if (index == 1)
p6 = p[6] + y * -0xf.ff6b8p-12;
else
p6 = p[6];
return p[3] + y * (p[4] + y * (p[5] + y * p6));
}
float
__ieee754_y1f(float x)
{
float z, s,c,ss,cc,u,v;
int32_t hx,ix;
GET_FLOAT_WORD(hx,x);
ix = 0x7fffffff&hx;
/* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
if(__builtin_expect(ix>=0x7f800000, 0)) return one/(x+x*x);
if(__builtin_expect(ix==0, 0))
return -1/zero; /* -inf and divide by zero exception. */
if(__builtin_expect(hx<0, 0)) return zero/(zero*x);
if (ix >= 0x3fe0dfbc) { /* |x| >= 0x1.c1bf78p+0 */
SET_RESTORE_ROUNDF (FE_TONEAREST);
__sincosf (x, &s, &c);
ss = -s-c;
cc = s-c;
if (ix >= 0x7f000000)
/* x >= 2^127: use asymptotic expansion. */
return y1f_asympt (x);
/* Now we are sure that x+x cannot overflow. */
z = __cosf(x+x);
if ((s*c)>zero) cc = z/ss;
else ss = z/cc;
/* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
* where x0 = x-3pi/4
* Better formula:
* cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
* = 1/sqrt(2) * (sin(x) - cos(x))
* sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
* = -1/sqrt(2) * (cos(x) + sin(x))
* To avoid cancellation, use
* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
* to compute the worse one.
*/
if (ix <= 0x5c000000)
{
u = ponef(x); v = qonef(x);
ss = u*ss+v*cc;
}
z = (invsqrtpi * ss) / sqrtf(x);
float threshold = 0x1.3e014cp-2;
/* The following threshold is optimal: for x=0x1.f7f16ap+0
and rounding upwards, |ss|=-0x1.3e014cp-2 and z is 11 ulps
far from the correctly rounded value. */
if (fabsf (ss) > threshold)
return z;
else
return y1f_near_root (x, z);
}
if(__builtin_expect(ix<=0x33000000, 0)) { /* x < 2**-25 */
z = -tpi / x;
if (isinf (z))
__set_errno (ERANGE);
return z;
}
/* Now 2**-25 <= x < 0x1.c1bf78p+0. */
z = x*x;
u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
return(x*(u/v) + tpi*(__ieee754_j1f(x)*__ieee754_logf(x)-one/x));
}
libm_alias_finite (__ieee754_y1f, __y1f)
/* For x >= 8, the asymptotic expansion of pone is
* 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
* We approximate pone by
* pone(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
* | pone(x)-1-R/S | <= 2 ** ( -60.06)
*/
static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
0.0000000000e+00, /* 0x00000000 */
1.1718750000e-01, /* 0x3df00000 */
1.3239480972e+01, /* 0x4153d4ea */
4.1205184937e+02, /* 0x43ce06a3 */
3.8747453613e+03, /* 0x45722bed */
7.9144794922e+03, /* 0x45f753d6 */
};
static const float ps8[5] = {
1.1420736694e+02, /* 0x42e46a2c */
3.6509309082e+03, /* 0x45642ee5 */
3.6956207031e+04, /* 0x47105c35 */
9.7602796875e+04, /* 0x47bea166 */
3.0804271484e+04, /* 0x46f0a88b */
};
static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
1.3199052094e-11, /* 0x2d68333f */
1.1718749255e-01, /* 0x3defffff */
6.8027510643e+00, /* 0x40d9b023 */
1.0830818176e+02, /* 0x42d89dca */
5.1763616943e+02, /* 0x440168b7 */
5.2871520996e+02, /* 0x44042dc6 */
};
static const float ps5[5] = {
5.9280597687e+01, /* 0x426d1f55 */
9.9140142822e+02, /* 0x4477d9b1 */
5.3532670898e+03, /* 0x45a74a23 */
7.8446904297e+03, /* 0x45f52586 */
1.5040468750e+03, /* 0x44bc0180 */
};
static const float pr3[6] = {
3.0250391081e-09, /* 0x314fe10d */
1.1718686670e-01, /* 0x3defffab */
3.9329774380e+00, /* 0x407bb5e7 */
3.5119403839e+01, /* 0x420c7a45 */
9.1055007935e+01, /* 0x42b61c2a */
4.8559066772e+01, /* 0x42423c7c */
};
static const float ps3[5] = {
3.4791309357e+01, /* 0x420b2a4d */
3.3676245117e+02, /* 0x43a86198 */
1.0468714600e+03, /* 0x4482dbe3 */
8.9081134033e+02, /* 0x445eb3ed */
1.0378793335e+02, /* 0x42cf936c */
};
static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
1.0771083225e-07, /* 0x33e74ea8 */
1.1717621982e-01, /* 0x3deffa16 */
2.3685150146e+00, /* 0x401795c0 */
1.2242610931e+01, /* 0x4143e1bc */
1.7693971634e+01, /* 0x418d8d41 */
5.0735230446e+00, /* 0x40a25a4d */
};
static const float ps2[5] = {
2.1436485291e+01, /* 0x41ab7dec */
1.2529022980e+02, /* 0x42fa9499 */
2.3227647400e+02, /* 0x436846c7 */
1.1767937469e+02, /* 0x42eb5bd7 */
8.3646392822e+00, /* 0x4105d590 */
};
static float
ponef(float x)
{
const float *p,*q;
float z,r,s;
int32_t ix;
GET_FLOAT_WORD(ix,x);
ix &= 0x7fffffff;
/* ix >= 0x40000000 for all calls to this function. */
if(ix>=0x41000000) {p = pr8; q= ps8;}
else if(ix>=0x40f71c58){p = pr5; q= ps5;}
else if(ix>=0x4036db68){p = pr3; q= ps3;}
else {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 expansion of qone is
* 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
* We approximate pone by
* qone(x) = s*(0.375 + (R/S))
* where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
* S = 1 + qs1*s^2 + ... + qs6*s^12
* and
* | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
*/
static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
0.0000000000e+00, /* 0x00000000 */
-1.0253906250e-01, /* 0xbdd20000 */
-1.6271753311e+01, /* 0xc1822c8d */
-7.5960174561e+02, /* 0xc43de683 */
-1.1849806641e+04, /* 0xc639273a */
-4.8438511719e+04, /* 0xc73d3683 */
};
static const float qs8[6] = {
1.6139537048e+02, /* 0x43216537 */
7.8253862305e+03, /* 0x45f48b17 */
1.3387534375e+05, /* 0x4802bcd6 */
7.1965775000e+05, /* 0x492fb29c */
6.6660125000e+05, /* 0x4922be94 */
-2.9449025000e+05, /* 0xc88fcb48 */
};
static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
-2.0897993405e-11, /* 0xadb7d219 */
-1.0253904760e-01, /* 0xbdd1fffe */
-8.0564479828e+00, /* 0xc100e736 */
-1.8366960144e+02, /* 0xc337ab6b */
-1.3731937256e+03, /* 0xc4aba633 */
-2.6124443359e+03, /* 0xc523471c */
};
static const float qs5[6] = {
8.1276550293e+01, /* 0x42a28d98 */
1.9917987061e+03, /* 0x44f8f98f */
1.7468484375e+04, /* 0x468878f8 */
4.9851425781e+04, /* 0x4742bb6d */
2.7948074219e+04, /* 0x46da5826 */
-4.7191835938e+03, /* 0xc5937978 */
};
static const float qr3[6] = {
-5.0783124372e-09, /* 0xb1ae7d4f */
-1.0253783315e-01, /* 0xbdd1ff5b */
-4.6101160049e+00, /* 0xc0938612 */
-5.7847221375e+01, /* 0xc267638e */
-2.2824453735e+02, /* 0xc3643e9a */
-2.1921012878e+02, /* 0xc35b35cb */
};
static const float qs3[6] = {
4.7665153503e+01, /* 0x423ea91e */
6.7386511230e+02, /* 0x4428775e */
3.3801528320e+03, /* 0x45534272 */
5.5477290039e+03, /* 0x45ad5dd5 */
1.9031191406e+03, /* 0x44ede3d0 */
-1.3520118713e+02, /* 0xc3073381 */
};
static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
-1.7838172539e-07, /* 0xb43f8932 */
-1.0251704603e-01, /* 0xbdd1f475 */
-2.7522056103e+00, /* 0xc0302423 */
-1.9663616180e+01, /* 0xc19d4f16 */
-4.2325313568e+01, /* 0xc2294d1f */
-2.1371921539e+01, /* 0xc1aaf9b2 */
};
static const float qs2[6] = {
2.9533363342e+01, /* 0x41ec4454 */
2.5298155212e+02, /* 0x437cfb47 */
7.5750280762e+02, /* 0x443d602e */
7.3939318848e+02, /* 0x4438d92a */
1.5594900513e+02, /* 0x431bf2f2 */
-4.9594988823e+00, /* 0xc09eb437 */
};
static float
qonef(float x)
{
const float *p,*q;
float s,r,z;
int32_t ix;
GET_FLOAT_WORD(ix,x);
ix &= 0x7fffffff;
/* ix >= 0x40000000 for all calls to this function. */
if(ix>=0x41000000) {p = qr8; q= qs8;} /* x >= 8 */
else if(ix>=0x40f71c58){p = qr5; q= qs5;} /* x >= 7.722209930e+00 */
else if(ix>=0x4036db68){p = qr3; q= qs3;} /* x >= 2.857141495e+00 */
else {p = qr2; q= qs2;} /* x >= 2 */
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).375 + r/s)/x;
}
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