libgo/go/math/j1.go

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