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

src/pkg/math/Makefile

@@ -57,6 +57,8 @@ ALLGOFILES=\
 	hypot.go\
 	hypot_port.go\
 	j0.go\
+	j1.go\
+	jn.go\
 	logb.go\
 	lgamma.go\
 	ldexp.go\

src/pkg/math/all_test.go

@@ -310,6 +310,42 @@ var j0 = []float64{
 	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 @@ var y0 = []float64{
 	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 @@ var j0SC = []float64{
 	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 @@ var y0SC = []float64{
 	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 @@ func alike(a, b float64) bool {
 	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 TestJ0(t *testing.T) {
 	}
 }
 
+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 TestY0(t *testing.T) {
 	}
 }
 
+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 BenchmarkJ0(b *testing.B) {
 	}
 }
 
+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 @@ func BenchmarkY0(b *testing.B) {
 		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)
+	}
+}

src/pkg/math/j0.go

@@ -70,9 +70,8 @@ package math
 // 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 @@ func Y0(x float64) float64 {
 	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 @@ func Y0(x float64) float64 {
 //      | 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 @@ var pR8 = [6]float64{
 	-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 @@ var pS8 = [5]float64{
 }
 
 // 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 @@ var pR5 = [6]float64{
 	-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 @@ var pS5 = [5]float64{
 }
 
 // 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 @@ var pR3 = [6]float64{
 	-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 @@ var pS3 = [5]float64{
 }
 
 // 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 @@ var pR2 = [6]float64{
 	-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 @@ func pzero(x float64) float64 {
 	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 @@ func pzero(x float64) float64 {
 //      | 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 @@ var qR8 = [6]float64{
 	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 @@ var qS8 = [6]float64{
 }
 
 // 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 @@ var qR5 = [6]float64{
 	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 @@ var qS5 = [6]float64{
 }
 
 // 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 @@ var qR3 = [6]float64{
 	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 @@ var qS3 = [6]float64{
 }
 
 // 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 @@ var qR2 = [6]float64{
 	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 @@ var qS2 = [6]float64{
 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]))))

src/pkg/math/jn.go

+// 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
+}