Source: j1.go in package math


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

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

special cases

	switch {
	case IsNaN(x):
		return x
	case 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

special cases

	switch {
	case x < 0 || IsNaN(x):
		return NaN()
	case 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