summaryrefslogtreecommitdiffstats
path: root/src/math/j1.go
blob: cc19e75b95002a6ce3de86d17bb1e73639de8d9b (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
// 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
	)
	// 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
}