// 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.)
///|
/// Compute Bessel function of the first kind of order one.
///
/// # Examples
///
/// ```moonbit nocheck
/// assert_eq(j1(0), 0)
/// assert_eq(j1(1), 0.4400505857449335)
/// assert_eq(j1(2), 0.5767248077568733)
/// assert_eq(j1(1.542), 0.5634545029920421)
/// ```
///
/// # Special Cases
///
/// 1. `j1(x)` is NaN if `x` is NaN.
/// 2. `j1(x)` is 0 if `x` is ±∞.
/// 3. `j1(x)` is NaN if `x` is less than 0.
///
/// # Accuracy
///
/// 4 ulp
pub fn bessel_j1(x : Double) -> Double {
if isnan(x) {
return @double.not_a_number
}
if isinf(x) {
return 0.0
}
let huge = 1.0e300
let invsqrtpi = 5.64189583547756279280e-01 // 0x3FE20DD7, 0x50429B6D
let r00 = -6.25000000000000000000e-02 // 0xBFB00000, 0x00000000
let r01 = 1.40705666955189706048e-03 // 0x3F570D9F, 0x98472C61
let r02 = -1.59955631084035597520e-05 // 0xBEF0C5C6, 0xBA169668
let r03 = 4.96727999609584448412e-08 // 0x3E6AAAFA, 0x46CA0BD9
let s01 = 1.91537599538363460805e-02 // 0x3F939D0B, 0x12637E53
let s02 = 1.85946785588630915560e-04 // 0x3F285F56, 0xB9CDF664
let s03 = 1.17718464042623683263e-06 // 0x3EB3BFF8, 0x333F8498
let s04 = 5.04636257076217042715e-09 // 0x3E35AC88, 0xC97DFF2C
let s05 = 1.23542274426137913908e-11 // 0x3DAB2ACF, 0xCFB97ED8
let hx : Int = __hi(x).reinterpret_as_int()
let ix = hx & 0x7fffffff
let y = fabs(x)
if fabs(x) >= 2.0 {
let s = sin(y)
let c = cos(y)
let mut ss = -s - c
let mut cc = s - c
if ix < 0x7fe00000 {
let z = cos(y + y)
if s * c > 0.0 {
cc = z / ss
} else {
ss = z / cc
}
}
let z = if ix > 0x48000000 {
invsqrtpi * cc / sqrt(y)
} else {
let u = pone(y)
let v = qone(y)
invsqrtpi * (u * cc - v * ss) / sqrt(y)
}
if hx < 0 {
return -z
} else {
return z
}
}
if ix < 0x3e400000 {
if huge + x > 1.0 {
return 0.5 * x
}
}
let z = x * x
let r = z * (r00 + z * (r01 + z * (r02 + z * r03)))
let s = 1.0 + z * (s01 + z * (s02 + z * (s03 + z * (s04 + z * s05))))
let r = r * x
x * 0.5 + r / s
}
///|
/// `j1` is an alias of `bessel_j1`.
pub let j1 : (Double) -> Double = bessel_j1
// 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)
///|
fn pone(x : Double) -> Double {
fn pr_ps(i : Int) -> (Array[Double], Array[Double]) {
let (pr, ps) = match i {
2 => {
let pr = [
1.07710830106873743082e-07, // 0x3E7CE9D4, 0xF65544F4
1.17176219462683348094e-01, // 0x3FBDFF42, 0xBE760D83
2.36851496667608785174e+00, // 0x4002F2B7, 0xF98FAEC0
1.22426109148261232917e+01, // 0x40287C37, 0x7F71A964
1.76939711271687727390e+01, // 0x4031B1A8, 0x177F8EE2
5.07352312588818499250e+00, // 0x40144B49, 0xA574C1FE
]
let ps = [
2.14364859363821409488e+01, // 0x40356FBD, 0x8AD5ECDC
1.25290227168402751090e+02, // 0x405F5293, 0x14F92CD5
2.32276469057162813669e+02, // 0x406D08D8, 0xD5A2DBD9
1.17679373287147100768e+02, // 0x405D6B7A, 0xDA1884A9
8.36463893371618283368e+00, // 0x4020BAB1, 0xF44E5192
]
(pr, ps)
}
3 => {
let pr = [
3.02503916137373618024e-09, // 0x3E29FC21, 0xA7AD9EDD
1.17186865567253592491e-01, // 0x3FBDFFF5, 0x5B21D17B
3.93297750033315640650e+00, // 0x400F76BC, 0xE85EAD8A
3.51194035591636932736e+01, // 0x40418F48, 0x9DA6D129
9.10550110750781271918e+01, // 0x4056C385, 0x4D2C1837
4.85590685197364919645e+01, // 0x4048478F, 0x8EA83EE5
]
let ps = [
3.47913095001251519989e+01, // 0x40416549, 0xA134069C
3.36762458747825746741e+02, // 0x40750C33, 0x07F1A75F
1.04687139975775130551e+03, // 0x40905B7C, 0x5037D523
8.90811346398256432622e+02, // 0x408BD67D, 0xA32E31E9
1.03787932439639277504e+02, // 0x4059F26D, 0x7C2EED53
]
(pr, ps)
}
5 => {
let pr = [
1.31990519556243522749e-11, // 0x3DAD0667, 0xDAE1CA7D
1.17187493190614097638e-01, // 0x3FBDFFFF, 0xE2C10043
6.80275127868432871736e+00, // 0x401B3604, 0x6E6315E3
1.08308182990189109773e+02, // 0x405B13B9, 0x452602ED
5.17636139533199752805e+02, // 0x40802D16, 0xD052D649
5.28715201363337541807e+02, // 0x408085B8, 0xBB7E0CB7
]
let ps = [
5.92805987221131331921e+01, // 0x404DA3EA, 0xA8AF633D
9.91401418733614377743e+02, // 0x408EFB36, 0x1B066701
5.35326695291487976647e+03, // 0x40B4E944, 0x5706B6FB
7.84469031749551231769e+03, // 0x40BEA4B0, 0xB8A5BB15
1.50404688810361062679e+03, // 0x40978030, 0x036F5E51
]
(pr, ps)
}
8 => {
let pr = [
0.00000000000000000000e+00, // 0x00000000, 0x00000000
1.17187499999988647970e-01, // 0x3FBDFFFF, 0xFFFFFCCE
1.32394806593073575129e+01, // 0x402A7A9D, 0x357F7FCE
4.12051854307378562225e+02, // 0x4079C0D4, 0x652EA590
3.87474538913960532227e+03, // 0x40AE457D, 0xA3A532CC
7.91447954031891731574e+03, // 0x40BEEA7A, 0xC32782DD
]
let ps = [
1.14207370375678408436e+02, // 0x405C8D45, 0x8E656CAC
3.65093083420853463394e+03, // 0x40AC85DC, 0x964D274F
3.69562060269033463555e+04, // 0x40E20B86, 0x97C5BB7F
9.76027935934950801311e+04, // 0x40F7D42C, 0xB28F17BB
3.08042720627888811578e+04, // 0x40DE1511, 0x697A0B2D
]
(pr, ps)
}
_ => panic()
}
(pr, ps)
}
let ix = __hi(x) & 0x7fffffff
let ix = ix.reinterpret_as_int()
let (p, q) = if ix >= 0x40200000 {
pr_ps(8)
} else if ix >= 0x40122E8B {
pr_ps(5)
} else if ix >= 0x4006DB6D {
pr_ps(3)
} else if ix >= 0x40000000 {
pr_ps(2)
} else {
panic()
}
let z = 1.0 / (x * x)
let r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))))
let s = 1.0 + z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * q[4]))))
1.0 + r / s
}
// For x >= 8, the asymptotic expansions of qone is
// 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
// We approximate pone 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)
//
///|
fn qone(x : Double) -> Double {
fn qr_qs(i : Int) -> (Array[Double], Array[Double]) {
let (qr, qs) = match i {
2 => {
let qr = [
-1.78381727510958865572e-07, // 0xBE87F126, 0x44C626D2
-1.02517042607985553460e-01, // 0xBFBA3E8E, 0x9148B010
-2.75220568278187460720e+00, // 0xC0060484, 0x69BB4EDA
-1.96636162643703720221e+01, // 0xC033A9E2, 0xC168907F
-4.23253133372830490089e+01, // 0xC04529A3, 0xDE104AAA
-2.13719211703704061733e+01, // 0xC0355F36, 0x39CF6E52
]
let qs = [
2.95333629060523854548e+01, // 0x403D888A, 0x78AE64FF
2.52981549982190529136e+02, // 0x406F9F68, 0xDB821CBA
7.57502834868645436472e+02, // 0x4087AC05, 0xCE49A0F7
7.39393205320467245656e+02, // 0x40871B25, 0x48D4C029
1.55949003336666123687e+02, // 0x40637E5E, 0x3C3ED8D4
-4.95949898822628210127e+00, // 0xC013D686, 0xE71BE86B
]
(qr, qs)
}
3 => {
let qr = [
-5.07831226461766561369e-09, // 0xBE35CFA9, 0xD38FC84F
-1.02537829820837089745e-01, // 0xBFBA3FEB, 0x51AEED54
-4.61011581139473403113e+00, // 0xC01270C2, 0x3302D9FF
-5.78472216562783643212e+01, // 0xC04CEC71, 0xC25D16DA
-2.28244540737631695038e+02, // 0xC06C87D3, 0x4718D55F
-2.19210128478909325622e+02, // 0xC06B66B9, 0x5F5C1BF6
]
let qs = [
4.76651550323729509273e+01, // 0x4047D523, 0xCCD367E4
6.73865112676699709482e+02, // 0x40850EEB, 0xC031EE3E
3.38015286679526343505e+03, // 0x40AA684E, 0x448E7C9A
5.54772909720722782367e+03, // 0x40B5ABBA, 0xA61D54A6
1.90311919338810798763e+03, // 0x409DBC7A, 0x0DD4DF4B
-1.35201191444307340817e+02, // 0xC060E670, 0x290A311F
]
(qr, qs)
}
5 => {
let qr = [
-2.08979931141764104297e-11, // 0xBDB6FA43, 0x1AA1A098
-1.02539050241375426231e-01, // 0xBFBA3FFF, 0xCB597FEF
-8.05644828123936029840e+00, // 0xC0201CE6, 0xCA03AD4B
-1.83669607474888380239e+02, // 0xC066F56D, 0x6CA7B9B0
-1.37319376065508163265e+03, // 0xC09574C6, 0x6931734F
-2.61244440453215656817e+03, // 0xC0A468E3, 0x88FDA79D
]
let qs = [
8.12765501384335777857e+01, // 0x405451B2, 0xFF5A11B2
1.99179873460485964642e+03, // 0x409F1F31, 0xE77BF839
1.74684851924908907677e+04, // 0x40D10F1F, 0x0D64CE29
4.98514270910352279316e+04, // 0x40E8576D, 0xAABAD197
2.79480751638918118260e+04, // 0x40DB4B04, 0xCF7C364B
-4.71918354795128470869e+03, // 0xC0B26F2E, 0xFCFFA004
]
(qr, qs)
}
8 => {
let qr = [
0.00000000000000000000e+00, // 0x00000000, 0x00000000
-1.02539062499992714161e-01, // 0xBFBA3FFF, 0xFFFFFDF3
-1.62717534544589987888e+01, // 0xC0304591, 0xA26779F7
-7.59601722513950107896e+02, // 0xC087BCD0, 0x53E4B576
-1.18498066702429587167e+04, // 0xC0C724E7, 0x40F87415
-4.84385124285750353010e+04, // 0xC0E7A6D0, 0x65D09C6A
]
let qs = [
1.61395369700722909556e+02, // 0x40642CA6, 0xDE5BCDE5
7.82538599923348465381e+03, // 0x40BE9162, 0xD0D88419
1.33875336287249578163e+05, // 0x4100579A, 0xB0B75E98
7.19657723683240939863e+05, // 0x4125F653, 0x72869C19
6.66601232617776375264e+05, // 0x412457D2, 0x7719AD5C
-2.94490264303834643215e+05, // 0xC111F969, 0x0EA5AA18
]
(qr, qs)
}
_ => panic()
}
(qr, qs)
}
let ix = __hi(x) & 0x7fffffff
let ix = ix.reinterpret_as_int()
let (p, q) = if ix >= 0x40200000 {
qr_qs(8)
} else if ix >= 0x40122E8B {
qr_qs(5)
} else if ix >= 0x4006DB6D {
qr_qs(3)
} else if ix >= 0x40000000 {
qr_qs(2)
} else {
panic()
}
let z = 1.0 / (x * x)
let r = p[0] + z * (p[1] + z * (p[2] + z * (p[3] + z * (p[4] + z * p[5]))))
let s = 1.0 +
z * (q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * q[5])))))
(0.375 + r / s) / x
}
///|
test "j1" {
fn assert_j1_ulp(input, expect) raise {
assert_ulp(expect, j1(input), J1_MAX_ULP)
}
assert_j1_ulp(-0.8, -0.36884204609417)
assert_j1_ulp(-0.7, -0.32899574154005895)
assert_j1_ulp(-0.6, -0.28670098806391575)
assert_j1_ulp(-0.5, -0.2422684576748739)
assert_j1_ulp(-0.4, -0.19602657795531875)
assert_j1_ulp(-0.3, -0.148318816273104)
assert_j1_ulp(-0.2, -0.099500832639236)
assert_j1_ulp(-0.1, -0.049937526036242)
assert_j1_ulp(-0, -0)
assert_j1_ulp(-3.141592653589793, -0.28461534317975284)
assert_j1_ulp(-1.5707963267948966, -0.5668240889058739)
assert_j1_ulp(-0.7853981633974483, -0.3631878383468673)
assert_j1_ulp(0, 0)
assert_j1_ulp(0.1, 0.049937526036242)
assert_j1_ulp(0.2, 0.099500832639236)
assert_j1_ulp(0.3, 0.148318816273104)
assert_j1_ulp(0.4, 0.19602657795531875)
assert_j1_ulp(0.5, 0.2422684576748739)
assert_j1_ulp(0.6, 0.28670098806391575)
assert_j1_ulp(0.7, 0.32899574154005895)
assert_j1_ulp(0.8, 0.36884204609417)
assert_j1_ulp(0.9, 0.4059495460788057)
assert_j1_ulp(1, 0.4400505857449335)
assert_j1_ulp(3.141592653589793, 0.28461534317975284)
assert_j1_ulp(1.5707963267948966, 0.5668240889058739)
assert_j1_ulp(0.7853981633974483, 0.3631878383468673)
assert_j1_ulp(-1, -0.4400505857449335)
assert_j1_ulp(-2, -0.5767248077568733)
assert_j1_ulp(-3, -0.33905895852593637)
assert_j1_ulp(-4, 0.06604332802354915)
assert_j1_ulp(-5, 0.32757913759146523)
assert_j1_ulp(-6, 0.27668385812756563)
assert_j1_ulp(-7, 0.004682823482345837)
assert_j1_ulp(-8, -0.2346363468539146)
assert_j1_ulp(-9, -0.24531178657332528)
assert_j1_ulp(1, 0.4400505857449335)
assert_j1_ulp(2, 0.5767248077568733)
assert_j1_ulp(3, 0.33905895852593637)
assert_j1_ulp(4, -0.06604332802354915)
assert_j1_ulp(5, -0.32757913759146523)
assert_j1_ulp(6, -0.27668385812756563)
assert_j1_ulp(7, -0.004682823482345837)
assert_j1_ulp(8, 0.2346363468539146)
assert_j1_ulp(9, 0.24531178657332528)
assert_j1_ulp(10, 0.04347274616886144)
assert_j1_ulp(100, -0.07714535201411217)
assert_j1_ulp(1000, 0.004728311907089523)
assert_j1_ulp(10000, 0.0036474507555295803)
assert_j1_ulp(2.5, 0.49709410246427393)
assert_j1_ulp(3.4, 0.17922585168150712)
assert_j1_ulp(5.3, -0.3459608338011862)
assert_j1_ulp(6.2, -0.23291656707322267)
assert_j1_ulp(7.1, 0.025153274253910344)
assert_j1_ulp(8.9, 0.2559023714439758)
assert_j1_ulp(9.8, 0.09284009111281007)
assert_j1_ulp(10.7, -0.12239942392717612)
assert_j1_ulp(101.6, 0.02243294985022214)
assert_j1_ulp(1.542, 0.5634545029920421)
assert_j1_ulp(2.846, 0.3942184920070648)
assert_j1_ulp(7.881, 0.21597108275134932)
assert_j1_ulp(3.772, 0.024223060780135384)
assert_j1_ulp(-1.542, -0.5634545029920421)
assert_j1_ulp(-2.846, -0.3942184920070648)
assert_j1_ulp(-7.881, -0.21597108275134932)
assert_j1_ulp(-3.772, -0.024223060780135384)
assert_j1_ulp(-1, -0.4400505857449335)
assert_j1_ulp(0, 0)
assert_j1_ulp(-0, -0)
assert_j1_ulp(@double.not_a_number, @double.not_a_number)
assert_j1_ulp(@double.infinity, 0)
assert_j1_ulp(@double.neg_infinity, 0)
}