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Diffstat (limited to 'library/compiler-builtins/libm/src/math/j0.rs')
| -rw-r--r-- | library/compiler-builtins/libm/src/math/j0.rs | 426 |
1 files changed, 426 insertions, 0 deletions
diff --git a/library/compiler-builtins/libm/src/math/j0.rs b/library/compiler-builtins/libm/src/math/j0.rs new file mode 100644 index 00000000000..99d656f0d08 --- /dev/null +++ b/library/compiler-builtins/libm/src/math/j0.rs @@ -0,0 +1,426 @@ +/* origin: FreeBSD /usr/src/lib/msun/src/e_j0.c */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ +/* j0(x), y0(x) + * Bessel function of the first and second kinds of order zero. + * Method -- j0(x): + * 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ... + * 2. Reduce x to |x| since j0(x)=j0(-x), and + * for x in (0,2) + * j0(x) = 1-z/4+ z^2*R0/S0, where z = x*x; + * (precision: |j0-1+z/4-z^2R0/S0 |<2**-63.67 ) + * for x in (2,inf) + * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0)) + * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) + * as follow: + * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) + * = 1/sqrt(2) * (cos(x) + sin(x)) + * sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/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 + * j0(nan)= nan + * j0(0) = 1 + * j0(inf) = 0 + * + * Method -- y0(x): + * 1. For x<2. + * Since + * y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...) + * therefore y0(x)-2/pi*j0(x)*ln(x) is an even function. + * We use the following function to approximate y0, + * y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2 + * where + * U(z) = u00 + u01*z + ... + u06*z^6 + * V(z) = 1 + v01*z + ... + v04*z^4 + * with absolute approximation error bounded by 2**-72. + * Note: For tiny x, U/V = u0 and j0(x)~1, hence + * y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27) + * 2. For x>=2. + * y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0)) + * where x0 = x-pi/4. It is better to compute sin(x0),cos(x0) + * by the method mentioned above. + * 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0. + */ + +use super::{cos, fabs, get_high_word, get_low_word, log, sin, sqrt}; +const INVSQRTPI: f64 = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */ +const TPI: f64 = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */ + +/* common method when |x|>=2 */ +fn common(ix: u32, x: f64, y0: bool) -> f64 { + let s: f64; + let mut c: f64; + let mut ss: f64; + let mut cc: f64; + let z: f64; + + /* + * j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x-pi/4)-q0(x)*sin(x-pi/4)) + * y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x-pi/4)+q0(x)*cos(x-pi/4)) + * + * sin(x-pi/4) = (sin(x) - cos(x))/sqrt(2) + * cos(x-pi/4) = (sin(x) + cos(x))/sqrt(2) + * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) + */ + s = sin(x); + c = cos(x); + if y0 { + c = -c; + } + cc = s + c; + /* avoid overflow in 2*x, big ulp error when x>=0x1p1023 */ + if ix < 0x7fe00000 { + ss = s - c; + z = -cos(2.0 * x); + if s * c < 0.0 { + cc = z / ss; + } else { + ss = z / cc; + } + if ix < 0x48000000 { + if y0 { + ss = -ss; + } + cc = pzero(x) * cc - qzero(x) * ss; + } + } + return INVSQRTPI * cc / sqrt(x); +} + +/* R0/S0 on [0, 2.00] */ +const R02: f64 = 1.56249999999999947958e-02; /* 0x3F8FFFFF, 0xFFFFFFFD */ +const R03: f64 = -1.89979294238854721751e-04; /* 0xBF28E6A5, 0xB61AC6E9 */ +const R04: f64 = 1.82954049532700665670e-06; /* 0x3EBEB1D1, 0x0C503919 */ +const R05: f64 = -4.61832688532103189199e-09; /* 0xBE33D5E7, 0x73D63FCE */ +const S01: f64 = 1.56191029464890010492e-02; /* 0x3F8FFCE8, 0x82C8C2A4 */ +const S02: f64 = 1.16926784663337450260e-04; /* 0x3F1EA6D2, 0xDD57DBF4 */ +const S03: f64 = 5.13546550207318111446e-07; /* 0x3EA13B54, 0xCE84D5A9 */ +const S04: f64 = 1.16614003333790000205e-09; /* 0x3E1408BC, 0xF4745D8F */ + +/// Zeroth order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the first kind (f64). +#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)] +pub fn j0(mut x: f64) -> f64 { + let z: f64; + let r: f64; + let s: f64; + let mut ix: u32; + + ix = get_high_word(x); + ix &= 0x7fffffff; + + /* j0(+-inf)=0, j0(nan)=nan */ + if ix >= 0x7ff00000 { + return 1.0 / (x * x); + } + x = fabs(x); + + if ix >= 0x40000000 { + /* |x| >= 2 */ + /* large ulp error near zeros: 2.4, 5.52, 8.6537,.. */ + return common(ix, x, false); + } + + /* 1 - x*x/4 + x*x*R(x^2)/S(x^2) */ + if ix >= 0x3f200000 { + /* |x| >= 2**-13 */ + /* up to 4ulp error close to 2 */ + z = x * x; + r = z * (R02 + z * (R03 + z * (R04 + z * R05))); + s = 1.0 + z * (S01 + z * (S02 + z * (S03 + z * S04))); + return (1.0 + x / 2.0) * (1.0 - x / 2.0) + z * (r / s); + } + + /* 1 - x*x/4 */ + /* prevent underflow */ + /* inexact should be raised when x!=0, this is not done correctly */ + if ix >= 0x38000000 { + /* |x| >= 2**-127 */ + x = 0.25 * x * x; + } + return 1.0 - x; +} + +const U00: f64 = -7.38042951086872317523e-02; /* 0xBFB2E4D6, 0x99CBD01F */ +const U01: f64 = 1.76666452509181115538e-01; /* 0x3FC69D01, 0x9DE9E3FC */ +const U02: f64 = -1.38185671945596898896e-02; /* 0xBF8C4CE8, 0xB16CFA97 */ +const U03: f64 = 3.47453432093683650238e-04; /* 0x3F36C54D, 0x20B29B6B */ +const U04: f64 = -3.81407053724364161125e-06; /* 0xBECFFEA7, 0x73D25CAD */ +const U05: f64 = 1.95590137035022920206e-08; /* 0x3E550057, 0x3B4EABD4 */ +const U06: f64 = -3.98205194132103398453e-11; /* 0xBDC5E43D, 0x693FB3C8 */ +const V01: f64 = 1.27304834834123699328e-02; /* 0x3F8A1270, 0x91C9C71A */ +const V02: f64 = 7.60068627350353253702e-05; /* 0x3F13ECBB, 0xF578C6C1 */ +const V03: f64 = 2.59150851840457805467e-07; /* 0x3E91642D, 0x7FF202FD */ +const V04: f64 = 4.41110311332675467403e-10; /* 0x3DFE5018, 0x3BD6D9EF */ + +/// Zeroth order of the [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the second kind (f64). +#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)] +pub fn y0(x: f64) -> f64 { + let z: f64; + let u: f64; + let v: f64; + let ix: u32; + let lx: u32; + + ix = get_high_word(x); + lx = get_low_word(x); + + /* y0(nan)=nan, y0(<0)=nan, y0(0)=-inf, y0(inf)=0 */ + if ((ix << 1) | lx) == 0 { + return -1.0 / 0.0; + } + if (ix >> 31) != 0 { + return 0.0 / 0.0; + } + if ix >= 0x7ff00000 { + return 1.0 / x; + } + + if ix >= 0x40000000 { + /* x >= 2 */ + /* large ulp errors near zeros: 3.958, 7.086,.. */ + return common(ix, x, true); + } + + /* U(x^2)/V(x^2) + (2/pi)*j0(x)*log(x) */ + if ix >= 0x3e400000 { + /* x >= 2**-27 */ + /* large ulp error near the first zero, x ~= 0.89 */ + z = x * x; + u = U00 + z * (U01 + z * (U02 + z * (U03 + z * (U04 + z * (U05 + z * U06))))); + v = 1.0 + z * (V01 + z * (V02 + z * (V03 + z * V04))); + return u / v + TPI * (j0(x) * log(x)); + } + return U00 + TPI * log(x); +} + +/* The asymptotic expansions of pzero is + * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. + * For x >= 2, We approximate pzero by + * pzero(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 + * | pzero(x)-1-R/S | <= 2 ** ( -60.26) + */ +const PR8: [f64; 6] = [ + /* for x in [inf, 8]=1/[0,0.125] */ + 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ + -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */ + -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */ + -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */ + -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */ + -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */ +]; +const PS8: [f64; 5] = [ + 1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */ + 3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */ + 4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */ + 1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */ + 4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */ +]; + +const PR5: [f64; 6] = [ + /* for x in [8,4.5454]=1/[0.125,0.22001] */ + -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */ + -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */ + -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */ + -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */ + -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */ + -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */ +]; +const PS5: [f64; 5] = [ + 6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */ + 1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */ + 5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */ + 9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */ + 2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */ +]; + +const PR3: [f64; 6] = [ + /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ + -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */ + -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */ + -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */ + -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */ + -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */ + -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */ +]; +const PS3: [f64; 5] = [ + 3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */ + 3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */ + 1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */ + 1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */ + 1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */ +]; + +const PR2: [f64; 6] = [ + /* for x in [2.8570,2]=1/[0.3499,0.5] */ + -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */ + -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */ + -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */ + -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */ + -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */ + -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */ +]; +const PS2: [f64; 5] = [ + 2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */ + 1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */ + 2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */ + 1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */ + 1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */ +]; + +fn pzero(x: f64) -> f64 { + let p: &[f64; 6]; + let q: &[f64; 5]; + let z: f64; + let r: f64; + let s: f64; + let mut ix: u32; + + ix = get_high_word(x); + ix &= 0x7fffffff; + if ix >= 0x40200000 { + p = &PR8; + q = &PS8; + } else if ix >= 0x40122E8B { + p = &PR5; + q = &PS5; + } else if ix >= 0x4006DB6D { + p = &PR3; + q = &PS3; + } else + /*ix >= 0x40000000*/ + { + p = &PR2; + q = &PS2; + } + z = 1.0 / (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 1.0 + r / s; +} + +/* For x >= 8, the asymptotic expansions of qzero is + * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. + * We approximate pzero by + * qzero(x) = s*(-1.25 + (R/S)) + * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 + * S = 1 + qS0*s^2 + ... + qS5*s^12 + * and + * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) + */ +const QR8: [f64; 6] = [ + /* for x in [inf, 8]=1/[0,0.125] */ + 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ + 7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */ + 1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */ + 5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */ + 8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */ + 3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */ +]; +const QS8: [f64; 6] = [ + 1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */ + 8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */ + 1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */ + 8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */ + 8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */ + -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */ +]; + +const QR5: [f64; 6] = [ + /* for x in [8,4.5454]=1/[0.125,0.22001] */ + 1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */ + 7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */ + 5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */ + 1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */ + 1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */ + 1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */ +]; +const QS5: [f64; 6] = [ + 8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */ + 2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */ + 1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */ + 5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */ + 3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */ + -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */ +]; + +const QR3: [f64; 6] = [ + /* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ + 4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */ + 7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */ + 3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */ + 4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */ + 1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */ + 1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */ +]; +const QS3: [f64; 6] = [ + 4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */ + 7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */ + 3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */ + 6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */ + 2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */ + -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */ +]; + +const QR2: [f64; 6] = [ + /* for x in [2.8570,2]=1/[0.3499,0.5] */ + 1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */ + 7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */ + 1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */ + 1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */ + 3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */ + 1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */ +]; +const QS2: [f64; 6] = [ + 3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */ + 2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */ + 8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */ + 8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */ + 2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */ + -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */ +]; + +fn qzero(x: f64) -> f64 { + let p: &[f64; 6]; + let q: &[f64; 6]; + let s: f64; + let r: f64; + let z: f64; + let mut ix: u32; + + ix = get_high_word(x); + ix &= 0x7fffffff; + if ix >= 0x40200000 { + p = &QR8; + q = &QS8; + } else if ix >= 0x40122E8B { + p = &QR5; + q = &QS5; + } else if ix >= 0x4006DB6D { + p = &QR3; + q = &QS3; + } else + /*ix >= 0x40000000*/ + { + p = &QR2; + q = &QS2; + } + z = 1.0 / (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] + z * q[5]))))); + return (-0.125 + r / s) / x; +} |