math: add J1, Y1, Jn and Yn (Bessel functions)
Also amend j0.go (variable name conflict, small corrections).
R=rsc
CC=golang-dev
https://golang.org/cl/769041
diff --git a/src/pkg/math/Makefile b/src/pkg/math/Makefile
index e8c4a22..a29245f 100644
--- a/src/pkg/math/Makefile
+++ b/src/pkg/math/Makefile
@@ -57,6 +57,8 @@
hypot.go\
hypot_port.go\
j0.go\
+ j1.go\
+ jn.go\
logb.go\
lgamma.go\
ldexp.go\
diff --git a/src/pkg/math/all_test.go b/src/pkg/math/all_test.go
index 4b0aec6..2f133f1 100644
--- a/src/pkg/math/all_test.go
+++ b/src/pkg/math/all_test.go
@@ -310,6 +310,42 @@
3.252650187653420388714693e-01,
-8.72218484409407250005360235e-03,
}
+var j1 = []float64{
+ -3.251526395295203422162967e-01,
+ 1.893581711430515718062564e-01,
+ -1.3711761352467242914491514e-01,
+ 3.287486536269617297529617e-01,
+ 1.3133899188830978473849215e-01,
+ 3.660243417832986825301766e-01,
+ -3.4436769271848174665420672e-01,
+ 4.329481396640773768835036e-01,
+ 5.8181350531954794639333955e-01,
+ -2.7030574577733036112996607e-01,
+}
+var j2 = []float64{
+ 5.3837518920137802565192769e-02,
+ -1.7841678003393207281244667e-01,
+ 9.521746934916464142495821e-03,
+ 4.28958355470987397983072e-02,
+ 2.4115371837854494725492872e-01,
+ 4.842458532394520316844449e-01,
+ -3.142145220618633390125946e-02,
+ 4.720849184745124761189957e-01,
+ 3.122312022520957042957497e-01,
+ 7.096213118930231185707277e-02,
+}
+var jM3 = []float64{
+ -3.684042080996403091021151e-01,
+ 2.8157665936340887268092661e-01,
+ 4.401005480841948348343589e-04,
+ 3.629926999056814081597135e-01,
+ 3.123672198825455192489266e-02,
+ -2.958805510589623607540455e-01,
+ -3.2033177696533233403289416e-01,
+ -2.592737332129663376736604e-01,
+ -1.0241334641061485092351251e-01,
+ -2.3762660886100206491674503e-01,
+}
var lgamma = []fi{
fi{3.146492141244545774319734e+00, 1},
fi{8.003414490659126375852113e+00, 1},
@@ -514,6 +550,42 @@
4.8290004112497761007536522e-01,
2.7036697826604756229601611e-01,
}
+var y1 = []float64{
+ 0.15494213737457922210218611,
+ -0.2165955142081145245075746,
+ -2.4644949631241895201032829,
+ 0.1442740489541836405154505,
+ 0.2215379960518984777080163,
+ 0.3038800915160754150565448,
+ 0.0691107642452362383808547,
+ 0.2380116417809914424860165,
+ -0.20849492979459761009678934,
+ 0.0242503179793232308250804,
+}
+var y2 = []float64{
+ 0.3675780219390303613394936,
+ -0.23034826393250119879267257,
+ -16.939677983817727205631397,
+ 0.367653980523052152867791,
+ -0.0962401471767804440353136,
+ -0.1923169356184851105200523,
+ 0.35984072054267882391843766,
+ -0.2794987252299739821654982,
+ -0.7113490692587462579757954,
+ -0.2647831587821263302087457,
+}
+var yM3 = []float64{
+ -0.14035984421094849100895341,
+ -0.097535139617792072703973,
+ 242.25775994555580176377379,
+ -0.1492267014802818619511046,
+ 0.26148702629155918694500469,
+ 0.56675383593895176530394248,
+ -0.206150264009006981070575,
+ 0.64784284687568332737963658,
+ 1.3503631555901938037008443,
+ 0.1461869756579956803341844,
+}
// arguments and expected results for special cases
var vfacoshSC = []float64{
@@ -847,6 +919,24 @@
0,
NaN(),
}
+var j1SC = []float64{
+ 0,
+ 0,
+ 0,
+ NaN(),
+}
+var j2SC = []float64{
+ 0,
+ 0,
+ 0,
+ NaN(),
+}
+var jM3SC = []float64{
+ 0,
+ 0,
+ 0,
+ NaN(),
+}
var vflgammaSC = []float64{
Inf(-1),
@@ -1042,6 +1132,24 @@
0,
NaN(),
}
+var y1SC = []float64{
+ NaN(),
+ Inf(-1),
+ 0,
+ NaN(),
+}
+var y2SC = []float64{
+ NaN(),
+ Inf(-1),
+ 0,
+ NaN(),
+}
+var yM3SC = []float64{
+ NaN(),
+ Inf(1),
+ 0,
+ NaN(),
+}
func tolerance(a, b, e float64) bool {
d := a - b
@@ -1065,10 +1173,6 @@
switch {
case IsNaN(a) && IsNaN(b):
return true
- case IsInf(a, 1) && IsInf(b, 1):
- return true
- case IsInf(a, -1) && IsInf(b, -1):
- return true
case a == b:
return true
}
@@ -1409,6 +1513,38 @@
}
}
+func TestJ1(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := J1(vf[i]); !close(j1[i], f) {
+ t.Errorf("J1(%g) = %g, want %g\n", vf[i], f, j1[i])
+ }
+ }
+ for i := 0; i < len(vfj0SC); i++ {
+ if f := J1(vfj0SC[i]); !alike(j1SC[i], f) {
+ t.Errorf("J1(%g) = %g, want %g\n", vfj0SC[i], f, j1SC[i])
+ }
+ }
+}
+
+func TestJn(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ if f := Jn(2, vf[i]); !close(j2[i], f) {
+ t.Errorf("Jn(2, %g) = %g, want %g\n", vf[i], f, j2[i])
+ }
+ if f := Jn(-3, vf[i]); !close(jM3[i], f) {
+ t.Errorf("Jn(-3, %g) = %g, want %g\n", vf[i], f, jM3[i])
+ }
+ }
+ for i := 0; i < len(vfj0SC); i++ {
+ if f := Jn(2, vfj0SC[i]); !alike(j2SC[i], f) {
+ t.Errorf("Jn(2, %g) = %g, want %g\n", vfj0SC[i], f, j2SC[i])
+ }
+ if f := Jn(-3, vfj0SC[i]); !alike(jM3SC[i], f) {
+ t.Errorf("Jn(-3, %g) = %g, want %g\n", vfj0SC[i], f, jM3SC[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) {
@@ -1654,6 +1790,40 @@
}
}
+func TestY1(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Fabs(vf[i])
+ if f := Y1(a); !soclose(y1[i], f, 2e-14) {
+ t.Errorf("Y1(%g) = %g, want %g\n", a, f, y1[i])
+ }
+ }
+ for i := 0; i < len(vfy0SC); i++ {
+ if f := Y1(vfy0SC[i]); !alike(y1SC[i], f) {
+ t.Errorf("Y1(%g) = %g, want %g\n", vfy0SC[i], f, y1SC[i])
+ }
+ }
+}
+
+func TestYn(t *testing.T) {
+ for i := 0; i < len(vf); i++ {
+ a := Fabs(vf[i])
+ if f := Yn(2, a); !close(y2[i], f) {
+ t.Errorf("Yn(2, %g) = %g, want %g\n", a, f, y2[i])
+ }
+ if f := Yn(-3, a); !close(yM3[i], f) {
+ t.Errorf("Yn(-3, %g) = %g, want %g\n", a, f, yM3[i])
+ }
+ }
+ for i := 0; i < len(vfy0SC); i++ {
+ if f := Yn(2, vfy0SC[i]); !alike(y2SC[i], f) {
+ t.Errorf("Yn(2, %g) = %g, want %g\n", vfy0SC[i], f, y2SC[i])
+ }
+ if f := Yn(-3, vfy0SC[i]); !alike(yM3SC[i], f) {
+ t.Errorf("Yn(-3, %g) = %g, want %g\n", vfy0SC[i], f, yM3SC[i])
+ }
+ }
+}
+
// Check that math functions of high angle values
// return similar results to low angle values
func TestLargeCos(t *testing.T) {
@@ -1896,6 +2066,18 @@
}
}
+func BenchmarkJ1(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ J1(2.5)
+ }
+}
+
+func BenchmarkJn(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Jn(2, 2.5)
+ }
+}
+
func BenchmarkLdexp(b *testing.B) {
for i := 0; i < b.N; i++ {
Ldexp(.5, 2)
@@ -2020,3 +2202,15 @@
Y0(2.5)
}
}
+
+func BenchmarkY1(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Y1(2.5)
+ }
+}
+
+func BenchmarkYn(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ Yn(2, 2.5)
+ }
+}
diff --git a/src/pkg/math/j0.go b/src/pkg/math/j0.go
index 8f0b7fd..8a6db3b 100644
--- a/src/pkg/math/j0.go
+++ b/src/pkg/math/j0.go
@@ -70,9 +70,8 @@
// J0 returns the order-zero Bessel function of the first kind.
//
// Special cases are:
-// J0(Inf) = 0
+// J0(±Inf) = 0
// J0(0) = 1
-// J0(-Inf) = 0
// J0(NaN) = NaN
func J0(x float64) float64 {
const (
@@ -178,15 +177,12 @@
switch {
case x < 0 || x != x: // x < 0 || IsNaN(x):
return NaN()
- case x < -MaxFloat64 || x > MaxFloat64: // IsInf(x, 0):
+ case x > MaxFloat64: // IsInf(x, 1):
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))
@@ -245,7 +241,7 @@
// | pzero(x)-1-R/S | <= 2 ** ( -60.26)
// for x in [inf, 8]=1/[0,0.125]
-var pR8 = [6]float64{
+var p0R8 = [6]float64{
0.00000000000000000000e+00, // 0x0000000000000000
-7.03124999999900357484e-02, // 0xBFB1FFFFFFFFFD32
-8.08167041275349795626e+00, // 0xC02029D0B44FA779
@@ -253,7 +249,7 @@
-2.48521641009428822144e+03, // 0xC0A36A6ECD4DCAFC
-5.25304380490729545272e+03, // 0xC0B4850B36CC643D
}
-var pS8 = [5]float64{
+var p0S8 = [5]float64{
1.16534364619668181717e+02, // 0x405D223307A96751
3.83374475364121826715e+03, // 0x40ADF37D50596938
4.05978572648472545552e+04, // 0x40E3D2BB6EB6B05F
@@ -262,7 +258,7 @@
}
// for x in [8,4.5454]=1/[0.125,0.22001]
-var pR5 = [6]float64{
+var p0R5 = [6]float64{
-1.14125464691894502584e-11, // 0xBDA918B147E495CC
-7.03124940873599280078e-02, // 0xBFB1FFFFE69AFBC6
-4.15961064470587782438e+00, // 0xC010A370F90C6BBF
@@ -270,7 +266,7 @@
-3.31231299649172967747e+02, // 0xC074B3B36742CC63
-3.46433388365604912451e+02, // 0xC075A6EF28A38BD7
}
-var pS5 = [5]float64{
+var p0S5 = [5]float64{
6.07539382692300335975e+01, // 0x404E60810C98C5DE
1.05125230595704579173e+03, // 0x40906D025C7E2864
5.97897094333855784498e+03, // 0x40B75AF88FBE1D60
@@ -279,7 +275,7 @@
}
// for x in [4.547,2.8571]=1/[0.2199,0.35001]
-var pR3 = [6]float64{
+var p0R3 = [6]float64{
-2.54704601771951915620e-09, // 0xBE25E1036FE1AA86
-7.03119616381481654654e-02, // 0xBFB1FFF6F7C0E24B
-2.40903221549529611423e+00, // 0xC00345B2AEA48074
@@ -287,7 +283,7 @@
-5.80791704701737572236e+01, // 0xC04D0A22420A1A45
-3.14479470594888503854e+01, // 0xC03F72ACA892D80F
}
-var pS3 = [5]float64{
+var p0S3 = [5]float64{
3.58560338055209726349e+01, // 0x4041ED9284077DD3
3.61513983050303863820e+02, // 0x40769839464A7C0E
1.19360783792111533330e+03, // 0x4092A66E6D1061D6
@@ -296,7 +292,7 @@
}
// for x in [2.8570,2]=1/[0.3499,0.5]
-var pR2 = [6]float64{
+var p0R2 = [6]float64{
-8.87534333032526411254e-08, // 0xBE77D316E927026D
-7.03030995483624743247e-02, // 0xBFB1FF62495E1E42
-1.45073846780952986357e+00, // 0xBFF736398A24A843
@@ -304,7 +300,7 @@
-1.11931668860356747786e+01, // 0xC02662E6C5246303
-3.23364579351335335033e+00, // 0xC009DE81AF8FE70F
}
-var pS2 = [5]float64{
+var p0S2 = [5]float64{
2.22202997532088808441e+01, // 0x40363865908B5959
1.36206794218215208048e+02, // 0x4061069E0EE8878F
2.70470278658083486789e+02, // 0x4070E78642EA079B
@@ -316,17 +312,17 @@
var p [6]float64
var q [5]float64
if x >= 8 {
- p = pR8
- q = pS8
+ p = p0R8
+ q = p0S8
} else if x >= 4.5454 {
- p = pR5
- q = pS5
+ p = p0R5
+ q = p0S5
} else if x >= 2.8571 {
- p = pR3
- q = pS3
+ p = p0R3
+ q = p0S3
} else if x >= 2 {
- p = pR2
- q = pS2
+ p = p0R2
+ q = p0S2
}
z := 1 / (x * x)
r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
@@ -344,7 +340,7 @@
// | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
// for x in [inf, 8]=1/[0,0.125]
-var qR8 = [6]float64{
+var q0R8 = [6]float64{
0.00000000000000000000e+00, // 0x0000000000000000
7.32421874999935051953e-02, // 0x3FB2BFFFFFFFFE2C
1.17682064682252693899e+01, // 0x402789525BB334D6
@@ -352,7 +348,7 @@
8.85919720756468632317e+03, // 0x40C14D993E18F46D
3.70146267776887834771e+04, // 0x40E212D40E901566
}
-var qS8 = [6]float64{
+var q0S8 = [6]float64{
1.63776026895689824414e+02, // 0x406478D5365B39BC
8.09834494656449805916e+03, // 0x40BFA2584E6B0563
1.42538291419120476348e+05, // 0x4101665254D38C3F
@@ -362,7 +358,7 @@
}
// for x in [8,4.5454]=1/[0.125,0.22001]
-var qR5 = [6]float64{
+var q0R5 = [6]float64{
1.84085963594515531381e-11, // 0x3DB43D8F29CC8CD9
7.32421766612684765896e-02, // 0x3FB2BFFFD172B04C
5.83563508962056953777e+00, // 0x401757B0B9953DD3
@@ -370,7 +366,7 @@
1.02724376596164097464e+03, // 0x40900CF99DC8C481
1.98997785864605384631e+03, // 0x409F17E953C6E3A6
}
-var qS5 = [6]float64{
+var q0S5 = [6]float64{
8.27766102236537761883e+01, // 0x4054B1B3FB5E1543
2.07781416421392987104e+03, // 0x40A03BA0DA21C0CE
1.88472887785718085070e+04, // 0x40D267D27B591E6D
@@ -380,7 +376,7 @@
}
// for x in [4.547,2.8571]=1/[0.2199,0.35001]
-var qR3 = [6]float64{
+var q0R3 = [6]float64{
4.37741014089738620906e-09, // 0x3E32CD036ADECB82
7.32411180042911447163e-02, // 0x3FB2BFEE0E8D0842
3.34423137516170720929e+00, // 0x400AC0FC61149CF5
@@ -388,7 +384,7 @@
1.70808091340565596283e+02, // 0x406559DBE25EFD1F
1.66733948696651168575e+02, // 0x4064D77C81FA21E0
}
-var qS3 = [6]float64{
+var q0S3 = [6]float64{
4.87588729724587182091e+01, // 0x40486122BFE343A6
7.09689221056606015736e+02, // 0x40862D8386544EB3
3.70414822620111362994e+03, // 0x40ACF04BE44DFC63
@@ -398,7 +394,7 @@
}
// for x in [2.8570,2]=1/[0.3499,0.5]
-var qR2 = [6]float64{
+var q0R2 = [6]float64{
1.50444444886983272379e-07, // 0x3E84313B54F76BDB
7.32234265963079278272e-02, // 0x3FB2BEC53E883E34
1.99819174093815998816e+00, // 0x3FFFF897E727779C
@@ -406,7 +402,7 @@
3.16662317504781540833e+01, // 0x403FAA8E29FBDC4A
1.62527075710929267416e+01, // 0x403040B171814BB4
}
-var qS2 = [6]float64{
+var q0S2 = [6]float64{
3.03655848355219184498e+01, // 0x403E5D96F7C07AED
2.69348118608049844624e+02, // 0x4070D591E4D14B40
8.44783757595320139444e+02, // 0x408A664522B3BF22
@@ -418,17 +414,17 @@
func qzero(x float64) float64 {
var p, q [6]float64
if x >= 8 {
- p = qR8
- q = qS8
+ p = q0R8
+ q = q0S8
} else if x >= 4.5454 {
- p = qR5
- q = qS5
+ p = q0R5
+ q = q0S5
} else if x >= 2.8571 {
- p = qR3
- q = qS3
+ p = q0R3
+ q = q0S3
} else if x >= 2 {
- p = qR2
- q = qS2
+ p = q0R2
+ q = q0S2
}
z := 1 / (x * x)
r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
diff --git a/src/pkg/math/j1.go b/src/pkg/math/j1.go
new file mode 100644
index 0000000..5c7b799
--- /dev/null
+++ b/src/pkg/math/j1.go
@@ -0,0 +1,426 @@
+// 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 one.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_j1.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_j1(x), __ieee754_y1(x)
+// Bessel function of the first and second kinds of order one.
+// Method -- j1(x):
+// 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
+// 2. Reduce x to |x| since j1(x)=-j1(-x), and
+// for x in (0,2)
+// j1(x) = x/2 + x*z*R0/S0, where z = x*x;
+// (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 )
+// for x in (2,inf)
+// j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
+// y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
+// where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
+// as follow:
+// cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
+// = 1/sqrt(2) * (sin(x) - cos(x))
+// sin(x1) = 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.)
+//
+// 3 Special cases
+// j1(nan)= nan
+// j1(0) = 0
+// j1(inf) = 0
+//
+// Method -- y1(x):
+// 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
+// 2. For x<2.
+// Since
+// y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
+// therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
+// We use the following function to approximate y1,
+// y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
+// where for x in [0,2] (abs err less than 2**-65.89)
+// U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
+// V(z) = 1 + v0[0]*z + ... + v0[4]*z^5
+// Note: For tiny x, 1/x dominate y1 and hence
+// y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
+// 3. For x>=2.
+// y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
+// where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
+// by method mentioned above.
+
+// J1 returns the order-one Bessel function of the first kind.
+//
+// Special cases are:
+// J1(±Inf) = 0
+// J1(NaN) = NaN
+func J1(x float64) float64 {
+ const (
+ TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
+ Two129 = 1 << 129 // 2**129 0x4800000000000000
+ // R0/S0 on [0, 2]
+ R00 = -6.25000000000000000000e-02 // 0xBFB0000000000000
+ R01 = 1.40705666955189706048e-03 // 0x3F570D9F98472C61
+ R02 = -1.59955631084035597520e-05 // 0xBEF0C5C6BA169668
+ R03 = 4.96727999609584448412e-08 // 0x3E6AAAFA46CA0BD9
+ S01 = 1.91537599538363460805e-02 // 0x3F939D0B12637E53
+ S02 = 1.85946785588630915560e-04 // 0x3F285F56B9CDF664
+ S03 = 1.17718464042623683263e-06 // 0x3EB3BFF8333F8498
+ S04 = 5.04636257076217042715e-09 // 0x3E35AC88C97DFF2C
+ S05 = 1.23542274426137913908e-11 // 0x3DAB2ACFCFB97ED8
+ )
+ // 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 || x == 0: // IsInf(x, 0) || x == 0:
+ return 0
+ }
+
+ sign := false
+ if x < 0 {
+ x = -x
+ sign = true
+ }
+ 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
+ }
+ }
+
+ // 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)
+
+ var z float64
+ if x > Two129 {
+ z = (1 / SqrtPi) * cc / Sqrt(x)
+ } else {
+ u := pone(x)
+ v := qone(x)
+ z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x)
+ }
+ if sign {
+ return -z
+ }
+ return z
+ }
+ if x < TwoM27 { // |x|<2**-27
+ return 0.5 * x // inexact if x!=0 necessary
+ }
+ z := x * x
+ r := z * (R00 + z*(R01+z*(R02+z*R03)))
+ s := 1.0 + z*(S01+z*(S02+z*(S03+z*(S04+z*S05))))
+ r *= x
+ z = 0.5*x + r/s
+ if sign {
+ return -z
+ }
+ return z
+}
+
+// Y1 returns the order-one Bessel function of the second kind.
+//
+// Special cases are:
+// Y1(+Inf) = 0
+// Y1(0) = -Inf
+// Y1(x < 0) = NaN
+// Y1(NaN) = NaN
+func Y1(x float64) float64 {
+ const (
+ TwoM54 = 1.0 / (1 << 54) // 2**-54 0x3c90000000000000
+ Two129 = 1 << 129 // 2**129 0x4800000000000000
+ U00 = -1.96057090646238940668e-01 // 0xBFC91866143CBC8A
+ U01 = 5.04438716639811282616e-02 // 0x3FA9D3C776292CD1
+ U02 = -1.91256895875763547298e-03 // 0xBF5F55E54844F50F
+ U03 = 2.35252600561610495928e-05 // 0x3EF8AB038FA6B88E
+ U04 = -9.19099158039878874504e-08 // 0xBE78AC00569105B8
+ V00 = 1.99167318236649903973e-02 // 0x3F94650D3F4DA9F0
+ V01 = 2.02552581025135171496e-04 // 0x3F2A8C896C257764
+ V02 = 1.35608801097516229404e-06 // 0x3EB6C05A894E8CA6
+ V03 = 6.22741452364621501295e-09 // 0x3E3ABF1D5BA69A86
+ V04 = 1.66559246207992079114e-11 // 0x3DB25039DACA772A
+ )
+ // 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: // IsInf(x, 1):
+ return 0
+ case x == 0:
+ return Inf(-1)
+ }
+
+ 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
+ }
+ }
+ // 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.
+
+ var z float64
+ if x > Two129 {
+ z = (1 / SqrtPi) * ss / Sqrt(x)
+ } else {
+ u := pone(x)
+ v := qone(x)
+ z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x)
+ }
+ return z
+ }
+ if x <= TwoM54 { // x < 2**-54
+ return -(2 / Pi) / x
+ }
+ z := x * x
+ u := U00 + z*(U01+z*(U02+z*(U03+z*U04)))
+ v := 1 + z*(V00+z*(V01+z*(V02+z*(V03+z*V04))))
+ return x*(u/v) + (2/Pi)*(J1(x)*Log(x)-1/x)
+}
+
+// For x >= 8, the asymptotic expansions 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)
+
+// for x in [inf, 8]=1/[0,0.125]
+var p1R8 = [6]float64{
+ 0.00000000000000000000e+00, // 0x0000000000000000
+ 1.17187499999988647970e-01, // 0x3FBDFFFFFFFFFCCE
+ 1.32394806593073575129e+01, // 0x402A7A9D357F7FCE
+ 4.12051854307378562225e+02, // 0x4079C0D4652EA590
+ 3.87474538913960532227e+03, // 0x40AE457DA3A532CC
+ 7.91447954031891731574e+03, // 0x40BEEA7AC32782DD
+}
+var p1S8 = [5]float64{
+ 1.14207370375678408436e+02, // 0x405C8D458E656CAC
+ 3.65093083420853463394e+03, // 0x40AC85DC964D274F
+ 3.69562060269033463555e+04, // 0x40E20B8697C5BB7F
+ 9.76027935934950801311e+04, // 0x40F7D42CB28F17BB
+ 3.08042720627888811578e+04, // 0x40DE1511697A0B2D
+}
+
+// for x in [8,4.5454] = 1/[0.125,0.22001]
+var p1R5 = [6]float64{
+ 1.31990519556243522749e-11, // 0x3DAD0667DAE1CA7D
+ 1.17187493190614097638e-01, // 0x3FBDFFFFE2C10043
+ 6.80275127868432871736e+00, // 0x401B36046E6315E3
+ 1.08308182990189109773e+02, // 0x405B13B9452602ED
+ 5.17636139533199752805e+02, // 0x40802D16D052D649
+ 5.28715201363337541807e+02, // 0x408085B8BB7E0CB7
+}
+var p1S5 = [5]float64{
+ 5.92805987221131331921e+01, // 0x404DA3EAA8AF633D
+ 9.91401418733614377743e+02, // 0x408EFB361B066701
+ 5.35326695291487976647e+03, // 0x40B4E9445706B6FB
+ 7.84469031749551231769e+03, // 0x40BEA4B0B8A5BB15
+ 1.50404688810361062679e+03, // 0x40978030036F5E51
+}
+
+// for x in[4.5453,2.8571] = 1/[0.2199,0.35001]
+var p1R3 = [6]float64{
+ 3.02503916137373618024e-09, // 0x3E29FC21A7AD9EDD
+ 1.17186865567253592491e-01, // 0x3FBDFFF55B21D17B
+ 3.93297750033315640650e+00, // 0x400F76BCE85EAD8A
+ 3.51194035591636932736e+01, // 0x40418F489DA6D129
+ 9.10550110750781271918e+01, // 0x4056C3854D2C1837
+ 4.85590685197364919645e+01, // 0x4048478F8EA83EE5
+}
+var p1S3 = [5]float64{
+ 3.47913095001251519989e+01, // 0x40416549A134069C
+ 3.36762458747825746741e+02, // 0x40750C3307F1A75F
+ 1.04687139975775130551e+03, // 0x40905B7C5037D523
+ 8.90811346398256432622e+02, // 0x408BD67DA32E31E9
+ 1.03787932439639277504e+02, // 0x4059F26D7C2EED53
+}
+
+// for x in [2.8570,2] = 1/[0.3499,0.5]
+var p1R2 = [6]float64{
+ 1.07710830106873743082e-07, // 0x3E7CE9D4F65544F4
+ 1.17176219462683348094e-01, // 0x3FBDFF42BE760D83
+ 2.36851496667608785174e+00, // 0x4002F2B7F98FAEC0
+ 1.22426109148261232917e+01, // 0x40287C377F71A964
+ 1.76939711271687727390e+01, // 0x4031B1A8177F8EE2
+ 5.07352312588818499250e+00, // 0x40144B49A574C1FE
+}
+var p1S2 = [5]float64{
+ 2.14364859363821409488e+01, // 0x40356FBD8AD5ECDC
+ 1.25290227168402751090e+02, // 0x405F529314F92CD5
+ 2.32276469057162813669e+02, // 0x406D08D8D5A2DBD9
+ 1.17679373287147100768e+02, // 0x405D6B7ADA1884A9
+ 8.36463893371618283368e+00, // 0x4020BAB1F44E5192
+}
+
+func pone(x float64) float64 {
+ var p [6]float64
+ var q [5]float64
+ if x >= 8 {
+ p = p1R8
+ q = p1S8
+ } else if x >= 4.5454 {
+ p = p1R5
+ q = p1S5
+ } else if x >= 2.8571 {
+ p = p1R3
+ q = p1S3
+ } else if x >= 2 {
+ p = p1R2
+ q = p1S2
+ }
+ z := 1 / (x * x)
+ r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
+ s := 1.0 + 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 qone is
+// 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
+// We approximate qone 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)
+
+// for x in [inf, 8] = 1/[0,0.125]
+var q1R8 = [6]float64{
+ 0.00000000000000000000e+00, // 0x0000000000000000
+ -1.02539062499992714161e-01, // 0xBFBA3FFFFFFFFDF3
+ -1.62717534544589987888e+01, // 0xC0304591A26779F7
+ -7.59601722513950107896e+02, // 0xC087BCD053E4B576
+ -1.18498066702429587167e+04, // 0xC0C724E740F87415
+ -4.84385124285750353010e+04, // 0xC0E7A6D065D09C6A
+}
+var q1S8 = [6]float64{
+ 1.61395369700722909556e+02, // 0x40642CA6DE5BCDE5
+ 7.82538599923348465381e+03, // 0x40BE9162D0D88419
+ 1.33875336287249578163e+05, // 0x4100579AB0B75E98
+ 7.19657723683240939863e+05, // 0x4125F65372869C19
+ 6.66601232617776375264e+05, // 0x412457D27719AD5C
+ -2.94490264303834643215e+05, // 0xC111F9690EA5AA18
+}
+
+// for x in [8,4.5454] = 1/[0.125,0.22001]
+var q1R5 = [6]float64{
+ -2.08979931141764104297e-11, // 0xBDB6FA431AA1A098
+ -1.02539050241375426231e-01, // 0xBFBA3FFFCB597FEF
+ -8.05644828123936029840e+00, // 0xC0201CE6CA03AD4B
+ -1.83669607474888380239e+02, // 0xC066F56D6CA7B9B0
+ -1.37319376065508163265e+03, // 0xC09574C66931734F
+ -2.61244440453215656817e+03, // 0xC0A468E388FDA79D
+}
+var q1S5 = [6]float64{
+ 8.12765501384335777857e+01, // 0x405451B2FF5A11B2
+ 1.99179873460485964642e+03, // 0x409F1F31E77BF839
+ 1.74684851924908907677e+04, // 0x40D10F1F0D64CE29
+ 4.98514270910352279316e+04, // 0x40E8576DAABAD197
+ 2.79480751638918118260e+04, // 0x40DB4B04CF7C364B
+ -4.71918354795128470869e+03, // 0xC0B26F2EFCFFA004
+}
+
+// for x in [4.5454,2.8571] = 1/[0.2199,0.35001] ???
+var q1R3 = [6]float64{
+ -5.07831226461766561369e-09, // 0xBE35CFA9D38FC84F
+ -1.02537829820837089745e-01, // 0xBFBA3FEB51AEED54
+ -4.61011581139473403113e+00, // 0xC01270C23302D9FF
+ -5.78472216562783643212e+01, // 0xC04CEC71C25D16DA
+ -2.28244540737631695038e+02, // 0xC06C87D34718D55F
+ -2.19210128478909325622e+02, // 0xC06B66B95F5C1BF6
+}
+var q1S3 = [6]float64{
+ 4.76651550323729509273e+01, // 0x4047D523CCD367E4
+ 6.73865112676699709482e+02, // 0x40850EEBC031EE3E
+ 3.38015286679526343505e+03, // 0x40AA684E448E7C9A
+ 5.54772909720722782367e+03, // 0x40B5ABBAA61D54A6
+ 1.90311919338810798763e+03, // 0x409DBC7A0DD4DF4B
+ -1.35201191444307340817e+02, // 0xC060E670290A311F
+}
+
+// for x in [2.8570,2] = 1/[0.3499,0.5]
+var q1R2 = [6]float64{
+ -1.78381727510958865572e-07, // 0xBE87F12644C626D2
+ -1.02517042607985553460e-01, // 0xBFBA3E8E9148B010
+ -2.75220568278187460720e+00, // 0xC006048469BB4EDA
+ -1.96636162643703720221e+01, // 0xC033A9E2C168907F
+ -4.23253133372830490089e+01, // 0xC04529A3DE104AAA
+ -2.13719211703704061733e+01, // 0xC0355F3639CF6E52
+}
+var q1S2 = [6]float64{
+ 2.95333629060523854548e+01, // 0x403D888A78AE64FF
+ 2.52981549982190529136e+02, // 0x406F9F68DB821CBA
+ 7.57502834868645436472e+02, // 0x4087AC05CE49A0F7
+ 7.39393205320467245656e+02, // 0x40871B2548D4C029
+ 1.55949003336666123687e+02, // 0x40637E5E3C3ED8D4
+ -4.95949898822628210127e+00, // 0xC013D686E71BE86B
+}
+
+func qone(x float64) float64 {
+ var p, q [6]float64
+ if x >= 8 {
+ p = q1R8
+ q = q1S8
+ } else if x >= 4.5454 {
+ p = q1R5
+ q = q1S5
+ } else if x >= 2.8571 {
+ p = q1R3
+ q = q1S3
+ } else if x >= 2 {
+ p = q1R2
+ q = q1S2
+ }
+ 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 (0.375 + r/s) / x
+}
diff --git a/src/pkg/math/jn.go b/src/pkg/math/jn.go
new file mode 100644
index 0000000..ecd7ab6
--- /dev/null
+++ b/src/pkg/math/jn.go
@@ -0,0 +1,310 @@
+// 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 n.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_jn.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_jn(n, x), __ieee754_yn(n, x)
+// floating point Bessel's function of the 1st and 2nd kind
+// of order n
+//
+// Special cases:
+// y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
+// y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
+// Note 2. About jn(n,x), yn(n,x)
+// For n=0, j0(x) is called,
+// for n=1, j1(x) is called,
+// for n<x, forward recursion is used starting
+// from values of j0(x) and j1(x).
+// for n>x, a continued fraction approximation to
+// j(n,x)/j(n-1,x) is evaluated and then backward
+// recursion is used starting from a supposed value
+// for j(n,x). The resulting value of j(0,x) is
+// compared with the actual value to correct the
+// supposed value of j(n,x).
+//
+// yn(n,x) is similar in all respects, except
+// that forward recursion is used for all
+// values of n>1.
+
+// Jn returns the order-n Bessel function of the first kind.
+//
+// Special cases are:
+// Jn(n, ±Inf) = 0
+// Jn(n, NaN) = NaN
+func Jn(n int, x float64) float64 {
+ const (
+ TwoM29 = 1.0 / (1 << 29) // 2**-29 0x3e10000000000000
+ Two302 = 1 << 302 // 2**302 0x52D0000000000000
+ )
+ // 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
+ }
+ // J(-n, x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
+ // Thus, J(-n, x) = J(n, -x)
+
+ if n == 0 {
+ return J0(x)
+ }
+ if x == 0 {
+ return 0
+ }
+ if n < 0 {
+ n, x = -n, -x
+ }
+ if n == 1 {
+ return J1(x)
+ }
+ sign := false
+ if x < 0 {
+ x = -x
+ if n&1 == 1 {
+ sign = true // odd n and negative x
+ }
+ }
+ var b float64
+ if float64(n) <= x {
+ // Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x)
+ if x >= Two302 { // x > 2**302
+
+ // (x >> n**2)
+ // Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ // Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ // Let s=sin(x), c=cos(x),
+ // xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
+ //
+ // n sin(xn)*sqt2 cos(xn)*sqt2
+ // ----------------------------------
+ // 0 s-c c+s
+ // 1 -s-c -c+s
+ // 2 -s+c -c-s
+ // 3 s+c c-s
+
+ var temp float64
+ switch n & 3 {
+ case 0:
+ temp = Cos(x) + Sin(x)
+ case 1:
+ temp = -Cos(x) + Sin(x)
+ case 2:
+ temp = -Cos(x) - Sin(x)
+ case 3:
+ temp = Cos(x) - Sin(x)
+ }
+ b = (1 / SqrtPi) * temp / Sqrt(x)
+ } else {
+ b = J1(x)
+ for i, a := 1, J0(x); i < n; i++ {
+ a, b = b, b*(float64(i+i)/x)-a // avoid underflow
+ }
+ }
+ } else {
+ if x < TwoM29 { // x < 2**-29
+ // x is tiny, return the first Taylor expansion of J(n,x)
+ // J(n,x) = 1/n!*(x/2)^n - ...
+
+ if n > 33 { // underflow
+ b = 0
+ } else {
+ temp := x * 0.5
+ b = temp
+ a := float64(1)
+ for i := 2; i <= n; i++ {
+ a *= float64(i) // a = n!
+ b *= temp // b = (x/2)^n
+ }
+ b /= a
+ }
+ } else {
+ // use backward recurrence
+ // x x^2 x^2
+ // J(n,x)/J(n-1,x) = ---- ------ ------ .....
+ // 2n - 2(n+1) - 2(n+2)
+ //
+ // 1 1 1
+ // (for large x) = ---- ------ ------ .....
+ // 2n 2(n+1) 2(n+2)
+ // -- - ------ - ------ -
+ // x x x
+ //
+ // Let w = 2n/x and h=2/x, then the above quotient
+ // is equal to the continued fraction:
+ // 1
+ // = -----------------------
+ // 1
+ // w - -----------------
+ // 1
+ // w+h - ---------
+ // w+2h - ...
+ //
+ // To determine how many terms needed, let
+ // Q(0) = w, Q(1) = w(w+h) - 1,
+ // Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
+ // When Q(k) > 1e4 good for single
+ // When Q(k) > 1e9 good for double
+ // When Q(k) > 1e17 good for quadruple
+
+ // determine k
+ w := float64(n+n) / x
+ h := 2 / x
+ q0 := w
+ z := w + h
+ q1 := w*z - 1
+ k := 1
+ for q1 < 1e9 {
+ k += 1
+ z += h
+ q0, q1 = q1, z*q1-q0
+ }
+ m := n + n
+ t := float64(0)
+ for i := 2 * (n + k); i >= m; i -= 2 {
+ t = 1 / (float64(i)/x - t)
+ }
+ a := t
+ b = 1
+ // estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
+ // Hence, if n*(log(2n/x)) > ...
+ // single 8.8722839355e+01
+ // double 7.09782712893383973096e+02
+ // long double 1.1356523406294143949491931077970765006170e+04
+ // then recurrent value may overflow and the result is
+ // likely underflow to zero
+
+ tmp := float64(n)
+ v := 2 / x
+ tmp = tmp * Log(Fabs(v*tmp))
+ if tmp < 7.09782712893383973096e+02 {
+ for i := n - 1; i > 0; i-- {
+ di := float64(i + i)
+ a, b = b, b*di/x-a
+ di -= 2
+ }
+ } else {
+ for i := n - 1; i > 0; i-- {
+ di := float64(i + i)
+ a, b = b, b*di/x-a
+ di -= 2
+ // scale b to avoid spurious overflow
+ if b > 1e100 {
+ a /= b
+ t /= b
+ b = 1
+ }
+ }
+ }
+ b = t * J0(x) / b
+ }
+ }
+ if sign {
+ return -b
+ }
+ return b
+}
+
+// Yn returns the order-n Bessel function of the second kind.
+//
+// Special cases are:
+// Yn(n, +Inf) = 0
+// Yn(n > 0, 0) = -Inf
+// Yn(n < 0, 0) = +Inf if n is odd, -Inf if n is even
+// Y1(n, x < 0) = NaN
+// Y1(n, NaN) = NaN
+func Yn(n int, x float64) float64 {
+ const Two302 = 1 << 302 // 2**302 0x52D0000000000000
+ // 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: // IsInf(x, 1)
+ return 0
+ }
+
+ if n == 0 {
+ return Y0(x)
+ }
+ if x == 0 {
+ if n < 0 && n&1 == 1 {
+ return Inf(1)
+ }
+ return Inf(-1)
+ }
+ sign := false
+ if n < 0 {
+ n = -n
+ if n&1 == 1 {
+ sign = true // sign true if n < 0 && |n| odd
+ }
+ }
+ if n == 1 {
+ if sign {
+ return -Y1(x)
+ }
+ return Y1(x)
+ }
+ var b float64
+ if x >= Two302 { // x > 2**302
+ // (x >> n**2)
+ // Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ // Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
+ // Let s=sin(x), c=cos(x),
+ // xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
+ //
+ // n sin(xn)*sqt2 cos(xn)*sqt2
+ // ----------------------------------
+ // 0 s-c c+s
+ // 1 -s-c -c+s
+ // 2 -s+c -c-s
+ // 3 s+c c-s
+
+ var temp float64
+ switch n & 3 {
+ case 0:
+ temp = Sin(x) - Cos(x)
+ case 1:
+ temp = -Sin(x) - Cos(x)
+ case 2:
+ temp = -Sin(x) + Cos(x)
+ case 3:
+ temp = Sin(x) + Cos(x)
+ }
+ b = (1 / SqrtPi) * temp / Sqrt(x)
+ } else {
+ a := Y0(x)
+ b = Y1(x)
+ // quit if b is -inf
+ for i := 1; i < n && b >= -MaxFloat64; i++ { // for i := 1; i < n && !IsInf(b, -1); i++ {
+ a, b = b, (float64(i+i)/x)*b-a
+ }
+ }
+ if sign {
+ return -b
+ }
+ return b
+}
