1// Copyright 2010 The Go Authors. All rights reserved.
2// Use of this source code is governed by a BSD-style
3// license that can be found in the LICENSE file.
4
5package math
6
7/*
8	Bessel function of the first and second kinds of order one.
9*/
10
11// The original C code and the long comment below are
12// from FreeBSD's /usr/src/lib/msun/src/e_j1.c and
13// came with this notice. The go code is a simplified
14// version of the original C.
15//
16// ====================================================
17// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
18//
19// Developed at SunPro, a Sun Microsystems, Inc. business.
20// Permission to use, copy, modify, and distribute this
21// software is freely granted, provided that this notice
22// is preserved.
23// ====================================================
24//
25// __ieee754_j1(x), __ieee754_y1(x)
26// Bessel function of the first and second kinds of order one.
27// Method -- j1(x):
28//      1. For tiny x, we use j1(x) = x/2 - x**3/16 + x**5/384 - ...
29//      2. Reduce x to |x| since j1(x)=-j1(-x),  and
30//         for x in (0,2)
31//              j1(x) = x/2 + x*z*R0/S0,  where z = x*x;
32//         (precision:  |j1/x - 1/2 - R0/S0 |<2**-61.51 )
33//         for x in (2,inf)
34//              j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
35//              y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
36//         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
37//         as follow:
38//              cos(x1) =  cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
39//                      =  1/sqrt(2) * (sin(x) - cos(x))
40//              sin(x1) =  sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
41//                      = -1/sqrt(2) * (sin(x) + cos(x))
42//         (To avoid cancellation, use
43//              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
44//         to compute the worse one.)
45//
46//      3 Special cases
47//              j1(nan)= nan
48//              j1(0) = 0
49//              j1(inf) = 0
50//
51// Method -- y1(x):
52//      1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
53//      2. For x<2.
54//         Since
55//              y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x**3-...)
56//         therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
57//         We use the following function to approximate y1,
58//              y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x**2
59//         where for x in [0,2] (abs err less than 2**-65.89)
60//              U(z) = U0[0] + U0[1]*z + ... + U0[4]*z**4
61//              V(z) = 1  + v0[0]*z + ... + v0[4]*z**5
62//         Note: For tiny x, 1/x dominate y1 and hence
63//              y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
64//      3. For x>=2.
65//               y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
66//         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
67//         by method mentioned above.
68
69// J1 returns the order-one Bessel function of the first kind.
70//
71// Special cases are:
72//
73//	J1(±Inf) = 0
74//	J1(NaN) = NaN
75func J1(x float64) float64 {
76	const (
77		TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
78		Two129 = 1 << 129        // 2**129 0x4800000000000000
79		// R0/S0 on [0, 2]
80		R00 = -6.25000000000000000000e-02 // 0xBFB0000000000000
81		R01 = 1.40705666955189706048e-03  // 0x3F570D9F98472C61
82		R02 = -1.59955631084035597520e-05 // 0xBEF0C5C6BA169668
83		R03 = 4.96727999609584448412e-08  // 0x3E6AAAFA46CA0BD9
84		S01 = 1.91537599538363460805e-02  // 0x3F939D0B12637E53
85		S02 = 1.85946785588630915560e-04  // 0x3F285F56B9CDF664
86		S03 = 1.17718464042623683263e-06  // 0x3EB3BFF8333F8498
87		S04 = 5.04636257076217042715e-09  // 0x3E35AC88C97DFF2C
88		S05 = 1.23542274426137913908e-11  // 0x3DAB2ACFCFB97ED8
89	)
90	// special cases
91	switch {
92	case IsNaN(x):
93		return x
94	case IsInf(x, 0) || x == 0:
95		return 0
96	}
97
98	sign := false
99	if x < 0 {
100		x = -x
101		sign = true
102	}
103	if x >= 2 {
104		s, c := Sincos(x)
105		ss := -s - c
106		cc := s - c
107
108		// make sure x+x does not overflow
109		if x < MaxFloat64/2 {
110			z := Cos(x + x)
111			if s*c > 0 {
112				cc = z / ss
113			} else {
114				ss = z / cc
115			}
116		}
117
118		// j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
119		// y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
120
121		var z float64
122		if x > Two129 {
123			z = (1 / SqrtPi) * cc / Sqrt(x)
124		} else {
125			u := pone(x)
126			v := qone(x)
127			z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x)
128		}
129		if sign {
130			return -z
131		}
132		return z
133	}
134	if x < TwoM27 { // |x|<2**-27
135		return 0.5 * x // inexact if x!=0 necessary
136	}
137	z := x * x
138	r := z * (R00 + z*(R01+z*(R02+z*R03)))
139	s := 1.0 + z*(S01+z*(S02+z*(S03+z*(S04+z*S05))))
140	r *= x
141	z = 0.5*x + r/s
142	if sign {
143		return -z
144	}
145	return z
146}
147
148// Y1 returns the order-one Bessel function of the second kind.
149//
150// Special cases are:
151//
152//	Y1(+Inf) = 0
153//	Y1(0) = -Inf
154//	Y1(x < 0) = NaN
155//	Y1(NaN) = NaN
156func Y1(x float64) float64 {
157	const (
158		TwoM54 = 1.0 / (1 << 54)             // 2**-54 0x3c90000000000000
159		Two129 = 1 << 129                    // 2**129 0x4800000000000000
160		U00    = -1.96057090646238940668e-01 // 0xBFC91866143CBC8A
161		U01    = 5.04438716639811282616e-02  // 0x3FA9D3C776292CD1
162		U02    = -1.91256895875763547298e-03 // 0xBF5F55E54844F50F
163		U03    = 2.35252600561610495928e-05  // 0x3EF8AB038FA6B88E
164		U04    = -9.19099158039878874504e-08 // 0xBE78AC00569105B8
165		V00    = 1.99167318236649903973e-02  // 0x3F94650D3F4DA9F0
166		V01    = 2.02552581025135171496e-04  // 0x3F2A8C896C257764
167		V02    = 1.35608801097516229404e-06  // 0x3EB6C05A894E8CA6
168		V03    = 6.22741452364621501295e-09  // 0x3E3ABF1D5BA69A86
169		V04    = 1.66559246207992079114e-11  // 0x3DB25039DACA772A
170	)
171	// special cases
172	switch {
173	case x < 0 || IsNaN(x):
174		return NaN()
175	case IsInf(x, 1):
176		return 0
177	case x == 0:
178		return Inf(-1)
179	}
180
181	if x >= 2 {
182		s, c := Sincos(x)
183		ss := -s - c
184		cc := s - c
185
186		// make sure x+x does not overflow
187		if x < MaxFloat64/2 {
188			z := Cos(x + x)
189			if s*c > 0 {
190				cc = z / ss
191			} else {
192				ss = z / cc
193			}
194		}
195		// y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
196		// where x0 = x-3pi/4
197		//     Better formula:
198		//         cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
199		//                 =  1/sqrt(2) * (sin(x) - cos(x))
200		//         sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
201		//                 = -1/sqrt(2) * (cos(x) + sin(x))
202		// To avoid cancellation, use
203		//     sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
204		// to compute the worse one.
205
206		var z float64
207		if x > Two129 {
208			z = (1 / SqrtPi) * ss / Sqrt(x)
209		} else {
210			u := pone(x)
211			v := qone(x)
212			z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x)
213		}
214		return z
215	}
216	if x <= TwoM54 { // x < 2**-54
217		return -(2 / Pi) / x
218	}
219	z := x * x
220	u := U00 + z*(U01+z*(U02+z*(U03+z*U04)))
221	v := 1 + z*(V00+z*(V01+z*(V02+z*(V03+z*V04))))
222	return x*(u/v) + (2/Pi)*(J1(x)*Log(x)-1/x)
223}
224
225// For x >= 8, the asymptotic expansions of pone is
226//      1 + 15/128 s**2 - 4725/2**15 s**4 - ..., where s = 1/x.
227// We approximate pone by
228//      pone(x) = 1 + (R/S)
229// where R = pr0 + pr1*s**2 + pr2*s**4 + ... + pr5*s**10
230//       S = 1 + ps0*s**2 + ... + ps4*s**10
231// and
232//      | pone(x)-1-R/S | <= 2**(-60.06)
233
234// for x in [inf, 8]=1/[0,0.125]
235var p1R8 = [6]float64{
236	0.00000000000000000000e+00, // 0x0000000000000000
237	1.17187499999988647970e-01, // 0x3FBDFFFFFFFFFCCE
238	1.32394806593073575129e+01, // 0x402A7A9D357F7FCE
239	4.12051854307378562225e+02, // 0x4079C0D4652EA590
240	3.87474538913960532227e+03, // 0x40AE457DA3A532CC
241	7.91447954031891731574e+03, // 0x40BEEA7AC32782DD
242}
243var p1S8 = [5]float64{
244	1.14207370375678408436e+02, // 0x405C8D458E656CAC
245	3.65093083420853463394e+03, // 0x40AC85DC964D274F
246	3.69562060269033463555e+04, // 0x40E20B8697C5BB7F
247	9.76027935934950801311e+04, // 0x40F7D42CB28F17BB
248	3.08042720627888811578e+04, // 0x40DE1511697A0B2D
249}
250
251// for x in [8,4.5454] = 1/[0.125,0.22001]
252var p1R5 = [6]float64{
253	1.31990519556243522749e-11, // 0x3DAD0667DAE1CA7D
254	1.17187493190614097638e-01, // 0x3FBDFFFFE2C10043
255	6.80275127868432871736e+00, // 0x401B36046E6315E3
256	1.08308182990189109773e+02, // 0x405B13B9452602ED
257	5.17636139533199752805e+02, // 0x40802D16D052D649
258	5.28715201363337541807e+02, // 0x408085B8BB7E0CB7
259}
260var p1S5 = [5]float64{
261	5.92805987221131331921e+01, // 0x404DA3EAA8AF633D
262	9.91401418733614377743e+02, // 0x408EFB361B066701
263	5.35326695291487976647e+03, // 0x40B4E9445706B6FB
264	7.84469031749551231769e+03, // 0x40BEA4B0B8A5BB15
265	1.50404688810361062679e+03, // 0x40978030036F5E51
266}
267
268// for x in[4.5453,2.8571] = 1/[0.2199,0.35001]
269var p1R3 = [6]float64{
270	3.02503916137373618024e-09, // 0x3E29FC21A7AD9EDD
271	1.17186865567253592491e-01, // 0x3FBDFFF55B21D17B
272	3.93297750033315640650e+00, // 0x400F76BCE85EAD8A
273	3.51194035591636932736e+01, // 0x40418F489DA6D129
274	9.10550110750781271918e+01, // 0x4056C3854D2C1837
275	4.85590685197364919645e+01, // 0x4048478F8EA83EE5
276}
277var p1S3 = [5]float64{
278	3.47913095001251519989e+01, // 0x40416549A134069C
279	3.36762458747825746741e+02, // 0x40750C3307F1A75F
280	1.04687139975775130551e+03, // 0x40905B7C5037D523
281	8.90811346398256432622e+02, // 0x408BD67DA32E31E9
282	1.03787932439639277504e+02, // 0x4059F26D7C2EED53
283}
284
285// for x in [2.8570,2] = 1/[0.3499,0.5]
286var p1R2 = [6]float64{
287	1.07710830106873743082e-07, // 0x3E7CE9D4F65544F4
288	1.17176219462683348094e-01, // 0x3FBDFF42BE760D83
289	2.36851496667608785174e+00, // 0x4002F2B7F98FAEC0
290	1.22426109148261232917e+01, // 0x40287C377F71A964
291	1.76939711271687727390e+01, // 0x4031B1A8177F8EE2
292	5.07352312588818499250e+00, // 0x40144B49A574C1FE
293}
294var p1S2 = [5]float64{
295	2.14364859363821409488e+01, // 0x40356FBD8AD5ECDC
296	1.25290227168402751090e+02, // 0x405F529314F92CD5
297	2.32276469057162813669e+02, // 0x406D08D8D5A2DBD9
298	1.17679373287147100768e+02, // 0x405D6B7ADA1884A9
299	8.36463893371618283368e+00, // 0x4020BAB1F44E5192
300}
301
302func pone(x float64) float64 {
303	var p *[6]float64
304	var q *[5]float64
305	if x >= 8 {
306		p = &p1R8
307		q = &p1S8
308	} else if x >= 4.5454 {
309		p = &p1R5
310		q = &p1S5
311	} else if x >= 2.8571 {
312		p = &p1R3
313		q = &p1S3
314	} else if x >= 2 {
315		p = &p1R2
316		q = &p1S2
317	}
318	z := 1 / (x * x)
319	r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
320	s := 1.0 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))))
321	return 1 + r/s
322}
323
324// For x >= 8, the asymptotic expansions of qone is
325//      3/8 s - 105/1024 s**3 - ..., where s = 1/x.
326// We approximate qone by
327//      qone(x) = s*(0.375 + (R/S))
328// where R = qr1*s**2 + qr2*s**4 + ... + qr5*s**10
329//       S = 1 + qs1*s**2 + ... + qs6*s**12
330// and
331//      | qone(x)/s -0.375-R/S | <= 2**(-61.13)
332
333// for x in [inf, 8] = 1/[0,0.125]
334var q1R8 = [6]float64{
335	0.00000000000000000000e+00,  // 0x0000000000000000
336	-1.02539062499992714161e-01, // 0xBFBA3FFFFFFFFDF3
337	-1.62717534544589987888e+01, // 0xC0304591A26779F7
338	-7.59601722513950107896e+02, // 0xC087BCD053E4B576
339	-1.18498066702429587167e+04, // 0xC0C724E740F87415
340	-4.84385124285750353010e+04, // 0xC0E7A6D065D09C6A
341}
342var q1S8 = [6]float64{
343	1.61395369700722909556e+02,  // 0x40642CA6DE5BCDE5
344	7.82538599923348465381e+03,  // 0x40BE9162D0D88419
345	1.33875336287249578163e+05,  // 0x4100579AB0B75E98
346	7.19657723683240939863e+05,  // 0x4125F65372869C19
347	6.66601232617776375264e+05,  // 0x412457D27719AD5C
348	-2.94490264303834643215e+05, // 0xC111F9690EA5AA18
349}
350
351// for x in [8,4.5454] = 1/[0.125,0.22001]
352var q1R5 = [6]float64{
353	-2.08979931141764104297e-11, // 0xBDB6FA431AA1A098
354	-1.02539050241375426231e-01, // 0xBFBA3FFFCB597FEF
355	-8.05644828123936029840e+00, // 0xC0201CE6CA03AD4B
356	-1.83669607474888380239e+02, // 0xC066F56D6CA7B9B0
357	-1.37319376065508163265e+03, // 0xC09574C66931734F
358	-2.61244440453215656817e+03, // 0xC0A468E388FDA79D
359}
360var q1S5 = [6]float64{
361	8.12765501384335777857e+01,  // 0x405451B2FF5A11B2
362	1.99179873460485964642e+03,  // 0x409F1F31E77BF839
363	1.74684851924908907677e+04,  // 0x40D10F1F0D64CE29
364	4.98514270910352279316e+04,  // 0x40E8576DAABAD197
365	2.79480751638918118260e+04,  // 0x40DB4B04CF7C364B
366	-4.71918354795128470869e+03, // 0xC0B26F2EFCFFA004
367}
368
369// for x in [4.5454,2.8571] = 1/[0.2199,0.35001] ???
370var q1R3 = [6]float64{
371	-5.07831226461766561369e-09, // 0xBE35CFA9D38FC84F
372	-1.02537829820837089745e-01, // 0xBFBA3FEB51AEED54
373	-4.61011581139473403113e+00, // 0xC01270C23302D9FF
374	-5.78472216562783643212e+01, // 0xC04CEC71C25D16DA
375	-2.28244540737631695038e+02, // 0xC06C87D34718D55F
376	-2.19210128478909325622e+02, // 0xC06B66B95F5C1BF6
377}
378var q1S3 = [6]float64{
379	4.76651550323729509273e+01,  // 0x4047D523CCD367E4
380	6.73865112676699709482e+02,  // 0x40850EEBC031EE3E
381	3.38015286679526343505e+03,  // 0x40AA684E448E7C9A
382	5.54772909720722782367e+03,  // 0x40B5ABBAA61D54A6
383	1.90311919338810798763e+03,  // 0x409DBC7A0DD4DF4B
384	-1.35201191444307340817e+02, // 0xC060E670290A311F
385}
386
387// for x in [2.8570,2] = 1/[0.3499,0.5]
388var q1R2 = [6]float64{
389	-1.78381727510958865572e-07, // 0xBE87F12644C626D2
390	-1.02517042607985553460e-01, // 0xBFBA3E8E9148B010
391	-2.75220568278187460720e+00, // 0xC006048469BB4EDA
392	-1.96636162643703720221e+01, // 0xC033A9E2C168907F
393	-4.23253133372830490089e+01, // 0xC04529A3DE104AAA
394	-2.13719211703704061733e+01, // 0xC0355F3639CF6E52
395}
396var q1S2 = [6]float64{
397	2.95333629060523854548e+01,  // 0x403D888A78AE64FF
398	2.52981549982190529136e+02,  // 0x406F9F68DB821CBA
399	7.57502834868645436472e+02,  // 0x4087AC05CE49A0F7
400	7.39393205320467245656e+02,  // 0x40871B2548D4C029
401	1.55949003336666123687e+02,  // 0x40637E5E3C3ED8D4
402	-4.95949898822628210127e+00, // 0xC013D686E71BE86B
403}
404
405func qone(x float64) float64 {
406	var p, q *[6]float64
407	if x >= 8 {
408		p = &q1R8
409		q = &q1S8
410	} else if x >= 4.5454 {
411		p = &q1R5
412		q = &q1S5
413	} else if x >= 2.8571 {
414		p = &q1R3
415		q = &q1S3
416	} else if x >= 2 {
417		p = &q1R2
418		q = &q1S2
419	}
420	z := 1 / (x * x)
421	r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
422	s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))))
423	return (0.375 + r/s) / x
424}
425