9.3834586598354592860187267089e-01,
-2.093995902923148389186189429e-05,
}
+var j0 = []float64{
+ -1.8444682230601672018219338e-01,
+ 2.27353668906331975435892e-01,
+ 9.809259936157051116270273e-01,
+ -1.741170131426226587841181e-01,
+ -2.1389448451144143352039069e-01,
+ -2.340905848928038763337414e-01,
+ -1.0029099691890912094586326e-01,
+ -1.5466726714884328135358907e-01,
+ 3.252650187653420388714693e-01,
+ -8.72218484409407250005360235e-03,
+}
var lgamma = []fi{
fi{3.146492141244545774319734e+00, 1},
fi{8.003414490659126375852113e+00, 1},
1.0000000000000000e+00,
-8.0000000000000000e+00,
}
+var y0 = []float64{
+ -3.053399153780788357534855e-01,
+ 1.7437227649515231515503649e-01,
+ -8.6221781263678836910392572e-01,
+ -3.100664880987498407872839e-01,
+ 1.422200649300982280645377e-01,
+ 4.000004067997901144239363e-01,
+ -3.3340749753099352392332536e-01,
+ 4.5399790746668954555205502e-01,
+ 4.8290004112497761007536522e-01,
+ 2.7036697826604756229601611e-01,
+}
// arguments and expected results for special cases
var vfacoshSC = []float64{
MaxInt32,
}
+var vfj0SC = []float64{
+ Inf(-1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var j0SC = []float64{
+ 0,
+ 1,
+ 0,
+ NaN(),
+}
+
var vflgammaSC = []float64{
Inf(-1),
-3,
NaN(),
}
+var vfy0SC = []float64{
+ Inf(-1),
+ 0,
+ Inf(1),
+ NaN(),
+}
+var y0SC = []float64{
+ NaN(),
+ Inf(-1),
+ 0,
+ NaN(),
+}
+
func tolerance(a, b, e float64) bool {
d := a - b
if d < 0 {
func kindaclose(a, b float64) bool { return tolerance(a, b, 1e-8) }
func close(a, b float64) bool { return tolerance(a, b, 1e-14) }
func veryclose(a, b float64) bool { return tolerance(a, b, 4e-16) }
+func soclose(a, b, e float64) bool { return tolerance(a, b, e) }
func alike(a, b float64) bool {
switch {
case IsNaN(a) && IsNaN(b):
}
}
+func TestJ0(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := J0(vf[i]); !soclose(j0[i], f, 4e-14) {
+ t.Errorf("J0(%g) = %g, want %g\n", vf[i], f, j0[i])
+ }
+ }
+ for i := 0; i < len(vfj0SC); i++ {
+ if f := J0(vfj0SC[i]); !alike(j0SC[i], f) {
+ t.Errorf("J0(%g) = %g, want %g\n", vfj0SC[i], f, j0SC[i])
+ }
+ }
+}
+
func TestLdexp(t *testing.T) {
for i := 0; i < len(vf); i++ {
if f := Ldexp(frexp[i].f, frexp[i].i); !veryclose(vf[i], f) {
}
}
+func TestY0(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Fabs(vf[i])
+ if f := Y0(a); !close(y0[i], f) {
+ t.Errorf("Y0(%g) = %g, want %g\n", a, f, y0[i])
+ }
+ }
+ for i := 0; i < len(vfy0SC); i++ {
+ if f := Y0(vfy0SC[i]); !alike(y0SC[i], f) {
+ t.Errorf("Y0(%g) = %g, want %g\n", vfy0SC[i], f, y0SC[i])
+ }
+ }
+}
+
// Check that math functions of high angle values
// return similar results to low angle values
func TestLargeCos(t *testing.T) {
}
}
+func BenchmarkJ0(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ J0(2.5)
+ }
+}
+
func BenchmarkLdexp(b *testing.B) {
for i := 0; i < b.N; i++ {
Ldexp(.5, 2)
Trunc(.5)
}
}
+
+func BenchmarkY0(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Y0(2.5)
+ }
+}
--- /dev/null
+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Bessel function of the first and second kinds of order zero.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_j0.c and
+// came with this notice. The go code is a simplified
+// version of the original 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.
+// ====================================================
+//
+// __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.
+//
+
+// J0 returns the order-zero Bessel function of the first kind.
+//
+// Special cases are:
+// J0(Inf) = 0
+// J0(0) = 1
+// J0(-Inf) = 0
+// J0(NaN) = NaN
+func J0(x float64) float64 {
+ const (
+ Huge = 1e300
+ TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
+ TwoM13 = 1.0 / (1 << 13) // 2**-13 0x3f20000000000000
+ Two129 = 1 << 129 // 2**129 0x4800000000000000
+ // R0/S0 on [0, 2]
+ R02 = 1.56249999999999947958e-02 // 0x3F8FFFFFFFFFFFFD
+ R03 = -1.89979294238854721751e-04 // 0xBF28E6A5B61AC6E9
+ R04 = 1.82954049532700665670e-06 // 0x3EBEB1D10C503919
+ R05 = -4.61832688532103189199e-09 // 0xBE33D5E773D63FCE
+ S01 = 1.56191029464890010492e-02 // 0x3F8FFCE882C8C2A4
+ S02 = 1.16926784663337450260e-04 // 0x3F1EA6D2DD57DBF4
+ S03 = 5.13546550207318111446e-07 // 0x3EA13B54CE84D5A9
+ S04 = 1.16614003333790000205e-09 // 0x3E1408BCF4745D8F
+ )
+ // TODO(rsc): Remove manual inlining of IsNaN, IsInf
+ // when compiler does it for us
+ // special cases
+ switch {
+ case x != x: // IsNaN(x)
+ return x
+ case x < -MaxFloat64 || x > MaxFloat64: // IsInf(x, 0):
+ return 0
+ case x == 0:
+ return 1
+ }
+
+ if x < 0 {
+ x = -x
+ }
+ if x >= 2 {
+ s, c := Sincos(x)
+ ss := s - c
+ cc := s + c
+
+ // make sure x+x does not overflow
+ if x < MaxFloat64/2 {
+ z := -Cos(x + x)
+ if s*c < 0 {
+ 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)
+
+ var z float64
+ if x > Two129 { // |x| > ~6.8056e+38
+ z = (1 / SqrtPi) * cc / Sqrt(x)
+ } else {
+ u := pzero(x)
+ v := qzero(x)
+ z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x)
+ }
+ return z // |x| >= 2.0
+ }
+ if x < TwoM13 { // |x| < ~1.2207e-4
+ if x < TwoM27 {
+ return 1 // |x| < ~7.4506e-9
+ }
+ return 1 - 0.25*x*x // ~7.4506e-9 < |x| < ~1.2207e-4
+ }
+ z := x * x
+ r := z * (R02 + z*(R03+z*(R04+z*R05)))
+ s := 1 + z*(S01+z*(S02+z*(S03+z*S04)))
+ if x < 1 {
+ return 1 + z*(-0.25+(r/s)) // |x| < 1.00
+ }
+ u := 0.5 * x
+ return (1+u)*(1-u) + z*(r/s) // 1.0 < |x| < 2.0
+}
+
+// Y0 returns the order-zero Bessel function of the second kind.
+//
+// Special cases are:
+// Y0(+Inf) = 0
+// Y0(0) = -Inf
+// Y0(x < 0) = NaN
+// Y0(NaN) = NaN
+func Y0(x float64) float64 {
+ const (
+ TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
+ Two129 = 1 << 129 // 2**129 0x4800000000000000
+ U00 = -7.38042951086872317523e-02 // 0xBFB2E4D699CBD01F
+ U01 = 1.76666452509181115538e-01 // 0x3FC69D019DE9E3FC
+ U02 = -1.38185671945596898896e-02 // 0xBF8C4CE8B16CFA97
+ U03 = 3.47453432093683650238e-04 // 0x3F36C54D20B29B6B
+ U04 = -3.81407053724364161125e-06 // 0xBECFFEA773D25CAD
+ U05 = 1.95590137035022920206e-08 // 0x3E5500573B4EABD4
+ U06 = -3.98205194132103398453e-11 // 0xBDC5E43D693FB3C8
+ V01 = 1.27304834834123699328e-02 // 0x3F8A127091C9C71A
+ V02 = 7.60068627350353253702e-05 // 0x3F13ECBBF578C6C1
+ V03 = 2.59150851840457805467e-07 // 0x3E91642D7FF202FD
+ V04 = 4.41110311332675467403e-10 // 0x3DFE50183BD6D9EF
+ )
+ // TODO(rsc): Remove manual inlining of IsNaN, IsInf
+ // when compiler does it for us
+ // special cases
+ switch {
+ case x < 0 || x != x: // x < 0 || IsNaN(x):
+ return NaN()
+ case x < -MaxFloat64 || x > MaxFloat64: // IsInf(x, 0):
+ return 0
+ case x == 0:
+ return Inf(-1)
+ }
+
+ if x < 0 {
+ x = -x
+ }
+ if x >= 2 { // |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.
+
+ s, c := Sincos(x)
+ 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)
+
+ // make sure x+x does not overflow
+ if x < MaxFloat64/2 {
+ z := -Cos(x + x)
+ if s*c < 0 {
+ cc = z / ss
+ } else {
+ ss = z / cc
+ }
+ }
+ var z float64
+ if x > Two129 { // |x| > ~6.8056e+38
+ z = (1 / SqrtPi) * ss / Sqrt(x)
+ } else {
+ u := pzero(x)
+ v := qzero(x)
+ z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x)
+ }
+ return z // |x| >= 2.0
+ }
+ if x <= TwoM27 {
+ return U00 + (2/Pi)*Log(x) // |x| < ~7.4506e-9
+ }
+ z := x * x
+ u := U00 + z*(U01+z*(U02+z*(U03+z*(U04+z*(U05+z*U06)))))
+ v := 1 + z*(V01+z*(V02+z*(V03+z*V04)))
+ return u/v + (2/Pi)*J0(x)*Log(x) // ~7.4506e-9 < |x| < 2.0
+}
+
+// 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)
+
+// for x in [inf, 8]=1/[0,0.125]
+var pR8 = [6]float64{
+ 0.00000000000000000000e+00, // 0x0000000000000000
+ -7.03124999999900357484e-02, // 0xBFB1FFFFFFFFFD32
+ -8.08167041275349795626e+00, // 0xC02029D0B44FA779
+ -2.57063105679704847262e+02, // 0xC07011027B19E863
+ -2.48521641009428822144e+03, // 0xC0A36A6ECD4DCAFC
+ -5.25304380490729545272e+03, // 0xC0B4850B36CC643D
+}
+var pS8 = [5]float64{
+ 1.16534364619668181717e+02, // 0x405D223307A96751
+ 3.83374475364121826715e+03, // 0x40ADF37D50596938
+ 4.05978572648472545552e+04, // 0x40E3D2BB6EB6B05F
+ 1.16752972564375915681e+05, // 0x40FC810F8F9FA9BD
+ 4.76277284146730962675e+04, // 0x40E741774F2C49DC
+}
+
+// for x in [8,4.5454]=1/[0.125,0.22001]
+var pR5 = [6]float64{
+ -1.14125464691894502584e-11, // 0xBDA918B147E495CC
+ -7.03124940873599280078e-02, // 0xBFB1FFFFE69AFBC6
+ -4.15961064470587782438e+00, // 0xC010A370F90C6BBF
+ -6.76747652265167261021e+01, // 0xC050EB2F5A7D1783
+ -3.31231299649172967747e+02, // 0xC074B3B36742CC63
+ -3.46433388365604912451e+02, // 0xC075A6EF28A38BD7
+}
+var pS5 = [5]float64{
+ 6.07539382692300335975e+01, // 0x404E60810C98C5DE
+ 1.05125230595704579173e+03, // 0x40906D025C7E2864
+ 5.97897094333855784498e+03, // 0x40B75AF88FBE1D60
+ 9.62544514357774460223e+03, // 0x40C2CCB8FA76FA38
+ 2.40605815922939109441e+03, // 0x40A2CC1DC70BE864
+}
+
+// for x in [4.547,2.8571]=1/[0.2199,0.35001]
+var pR3 = [6]float64{
+ -2.54704601771951915620e-09, // 0xBE25E1036FE1AA86
+ -7.03119616381481654654e-02, // 0xBFB1FFF6F7C0E24B
+ -2.40903221549529611423e+00, // 0xC00345B2AEA48074
+ -2.19659774734883086467e+01, // 0xC035F74A4CB94E14
+ -5.80791704701737572236e+01, // 0xC04D0A22420A1A45
+ -3.14479470594888503854e+01, // 0xC03F72ACA892D80F
+}
+var pS3 = [5]float64{
+ 3.58560338055209726349e+01, // 0x4041ED9284077DD3
+ 3.61513983050303863820e+02, // 0x40769839464A7C0E
+ 1.19360783792111533330e+03, // 0x4092A66E6D1061D6
+ 1.12799679856907414432e+03, // 0x40919FFCB8C39B7E
+ 1.73580930813335754692e+02, // 0x4065B296FC379081
+}
+
+// for x in [2.8570,2]=1/[0.3499,0.5]
+var pR2 = [6]float64{
+ -8.87534333032526411254e-08, // 0xBE77D316E927026D
+ -7.03030995483624743247e-02, // 0xBFB1FF62495E1E42
+ -1.45073846780952986357e+00, // 0xBFF736398A24A843
+ -7.63569613823527770791e+00, // 0xC01E8AF3EDAFA7F3
+ -1.11931668860356747786e+01, // 0xC02662E6C5246303
+ -3.23364579351335335033e+00, // 0xC009DE81AF8FE70F
+}
+var pS2 = [5]float64{
+ 2.22202997532088808441e+01, // 0x40363865908B5959
+ 1.36206794218215208048e+02, // 0x4061069E0EE8878F
+ 2.70470278658083486789e+02, // 0x4070E78642EA079B
+ 1.53875394208320329881e+02, // 0x40633C033AB6FAFF
+ 1.46576176948256193810e+01, // 0x402D50B344391809
+}
+
+func pzero(x float64) float64 {
+ var p [6]float64
+ var q [5]float64
+ if x >= 8 {
+ p = pR8
+ q = pS8
+ } else if x >= 4.5454 {
+ p = pR5
+ q = pS5
+ } else if x >= 2.8571 {
+ p = pR3
+ q = pS3
+ } else if x >= 2 {
+ p = pR2
+ q = pS2
+ }
+ z := 1 / (x * x)
+ r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
+ s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))))
+ return 1 + 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)
+
+// for x in [inf, 8]=1/[0,0.125]
+var qR8 = [6]float64{
+ 0.00000000000000000000e+00, // 0x0000000000000000
+ 7.32421874999935051953e-02, // 0x3FB2BFFFFFFFFE2C
+ 1.17682064682252693899e+01, // 0x402789525BB334D6
+ 5.57673380256401856059e+02, // 0x40816D6315301825
+ 8.85919720756468632317e+03, // 0x40C14D993E18F46D
+ 3.70146267776887834771e+04, // 0x40E212D40E901566
+}
+var qS8 = [6]float64{
+ 1.63776026895689824414e+02, // 0x406478D5365B39BC
+ 8.09834494656449805916e+03, // 0x40BFA2584E6B0563
+ 1.42538291419120476348e+05, // 0x4101665254D38C3F
+ 8.03309257119514397345e+05, // 0x412883DA83A52B43
+ 8.40501579819060512818e+05, // 0x4129A66B28DE0B3D
+ -3.43899293537866615225e+05, // 0xC114FD6D2C9530C5
+}
+
+// for x in [8,4.5454]=1/[0.125,0.22001]
+var qR5 = [6]float64{
+ 1.84085963594515531381e-11, // 0x3DB43D8F29CC8CD9
+ 7.32421766612684765896e-02, // 0x3FB2BFFFD172B04C
+ 5.83563508962056953777e+00, // 0x401757B0B9953DD3
+ 1.35111577286449829671e+02, // 0x4060E3920A8788E9
+ 1.02724376596164097464e+03, // 0x40900CF99DC8C481
+ 1.98997785864605384631e+03, // 0x409F17E953C6E3A6
+}
+var qS5 = [6]float64{
+ 8.27766102236537761883e+01, // 0x4054B1B3FB5E1543
+ 2.07781416421392987104e+03, // 0x40A03BA0DA21C0CE
+ 1.88472887785718085070e+04, // 0x40D267D27B591E6D
+ 5.67511122894947329769e+04, // 0x40EBB5E397E02372
+ 3.59767538425114471465e+04, // 0x40E191181F7A54A0
+ -5.35434275601944773371e+03, // 0xC0B4EA57BEDBC609
+}
+
+// for x in [4.547,2.8571]=1/[0.2199,0.35001]
+var qR3 = [6]float64{
+ 4.37741014089738620906e-09, // 0x3E32CD036ADECB82
+ 7.32411180042911447163e-02, // 0x3FB2BFEE0E8D0842
+ 3.34423137516170720929e+00, // 0x400AC0FC61149CF5
+ 4.26218440745412650017e+01, // 0x40454F98962DAEDD
+ 1.70808091340565596283e+02, // 0x406559DBE25EFD1F
+ 1.66733948696651168575e+02, // 0x4064D77C81FA21E0
+}
+var qS3 = [6]float64{
+ 4.87588729724587182091e+01, // 0x40486122BFE343A6
+ 7.09689221056606015736e+02, // 0x40862D8386544EB3
+ 3.70414822620111362994e+03, // 0x40ACF04BE44DFC63
+ 6.46042516752568917582e+03, // 0x40B93C6CD7C76A28
+ 2.51633368920368957333e+03, // 0x40A3A8AAD94FB1C0
+ -1.49247451836156386662e+02, // 0xC062A7EB201CF40F
+}
+
+// for x in [2.8570,2]=1/[0.3499,0.5]
+var qR2 = [6]float64{
+ 1.50444444886983272379e-07, // 0x3E84313B54F76BDB
+ 7.32234265963079278272e-02, // 0x3FB2BEC53E883E34
+ 1.99819174093815998816e+00, // 0x3FFFF897E727779C
+ 1.44956029347885735348e+01, // 0x402CFDBFAAF96FE5
+ 3.16662317504781540833e+01, // 0x403FAA8E29FBDC4A
+ 1.62527075710929267416e+01, // 0x403040B171814BB4
+}
+var qS2 = [6]float64{
+ 3.03655848355219184498e+01, // 0x403E5D96F7C07AED
+ 2.69348118608049844624e+02, // 0x4070D591E4D14B40
+ 8.44783757595320139444e+02, // 0x408A664522B3BF22
+ 8.82935845112488550512e+02, // 0x408B977C9C5CC214
+ 2.12666388511798828631e+02, // 0x406A95530E001365
+ -5.31095493882666946917e+00, // 0xC0153E6AF8B32931
+}
+
+func qzero(x float64) float64 {
+ var p, q [6]float64
+ if x >= 8 {
+ p = qR8
+ q = qS8
+ } else if x >= 4.5454 {
+ p = qR5
+ q = qS5
+ } else if x >= 2.8571 {
+ p = qR3
+ q = qS3
+ } else if x >= 2 {
+ p = qR2
+ q = qS2
+ }
+ z := 1 / (x * x)
+ r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
+ s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))))
+ return (-0.125 + r/s) / x
+}