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another-boids-in-rust/vendor/pxfm/src/bessel/j1_coeffs.rs

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/*
* // Copyright (c) Radzivon Bartoshyk 7/2025. All rights reserved.
* //
* // Redistribution and use in source and binary forms, with or without modification,
* // are permitted provided that the following conditions are met:
* //
* // 1. Redistributions of source code must retain the above copyright notice, this
* // list of conditions and the following disclaimer.
* //
* // 2. Redistributions in binary form must reproduce the above copyright notice,
* // this list of conditions and the following disclaimer in the documentation
* // and/or other materials provided with the distribution.
* //
* // 3. Neither the name of the copyright holder nor the names of its
* // contributors may be used to endorse or promote products derived from
* // this software without specific prior written permission.
* //
* // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
* // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
* // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
* // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
* // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
* // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
* // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
/**
J1 zeros and extremums on [-76;76]
Generated in SageMath:
```python
from mpmath import mp, mpf, findroot, j1
from sage.all import *
import struct
DR = RealField(52)
DD = RealField(190)
def double_to_hex(f):
packed = struct.pack('>d', float(f))
return '0x' + packed.hex()
def split_double_double(x):
x_hi = DR(x) # convert to f64
x_lo = x - DD(x_hi)
return (x_lo,x_hi)
def print_double_double(mark, x):
splat = split_double_double(x)
print(f"{mark}({double_to_hex(splat[0])}, {double_to_hex(splat[1])}),")
zeros = []
# Step size to detect sign changes
mp.prec = 150
step = mpf("0.001")
epsilon = mpf("1e-35")
x = mpf("0.0")
previous_zero = R120(0)
j1_zeros = []
while x < mpf("76.0"):
f1 = besselj(1, x)
f2 = besselj(1, x + step)
if f1 * f2 < 0:
zero = findroot(lambda t: j1(t), (x, x + step), solver='bisect', tol=mp.mpf("1e-41"))
previous_zero = zero
j1_zeros.append(zero)
if previous_zero is not None and abs(x - mpf(f'{round(x)}')) < epsilon:
zeros.append(previous_zero)
x += step
j1_extrema = []
x = mpf("0.0")
while x < mpf("76.0"):
d1 = mp.diff(lambda t: j1(t), x)
d2 = mp.diff(lambda t: j1(t), x + step)
if d1 * d2 < 0:
extremum = findroot(lambda t: mp.diff(lambda u: j1(u), t), (x, x + step), solver='bisect', tol=mp.mpf("1e-41"))
j1_extrema.append(extremum)
x += step
j1_zeros.extend(j1_extrema)
j1_zeros = sorted(j1_zeros)
print("static J1_ZEROS: [(u64, u64); 46] = [")
for z in j1_zeros:
k = split_double_double(DD(z))
hi = double_to_hex(k[1])
lo = double_to_hex(k[0])
print(f"({lo}, {hi}),")
print("];")
```
See notes/bessel_j1_taylor.ipynb
**/
pub(crate) static J1_ZEROS: [(u64, u64); 48] = [
(0x0, 0x0),
(0x3c5616d820cfdaeb, 0x3ffd757d1fec8a3a),
(0xbca60155a9d1b257, 0x400ea75575af6f09),
(0x3ca5c646a75d7539, 0x40155365bc032467),
(0xbc9b226d9d243828, 0x401c0ff5f3b47250),
(0xbca63e17ec20a31d, 0x402112980f0b88a1),
(0x3cc02610a51562b6, 0x402458d0d0bdfc29),
(0x3cc9a84d3a5fedc1, 0x40276979797ee5ac),
(0x3cb2bce7fd18e693, 0x402aa5baf310e5a2),
(0xbcc6932b987094f1, 0x402dba284a17ac59),
(0xbcdd2a68e88ab318, 0x4030787b360508c5),
(0xbca022f6b2b54db9, 0x403203f9a24e6527),
(0xbcd21830197e9e86, 0x40339da8e7416ca4),
(0x3cdeaafeaf8ec1af, 0x40352a1424a1a9fa),
(0xbcc1bf33afef88f2, 0x4036c294e3d4d8ac),
(0xbcb2d773b50cf8b9, 0x40384fb31dee1635),
(0x3cc1a2686480d882, 0x4039e7570dcea106),
(0x3cd0bdee27293d79, 0x403b75014427514d),
(0xbcb42ce39ec976fb, 0x403d0bfcf471fccc),
(0x3cbda49c2c143483, 0x403e9a179fba532a),
(0xbcdbe3a1cd066b67, 0x404018476e6b2bf0),
(0xbce6b00c1279ef0a, 0x4040df82ebd54e32),
(0xbced5fbbff045068, 0x4041aa890dc5e97c),
(0x3cd7d864bbf17a30, 0x404271eb1b80430e),
(0x3cc9eafeca0ca4fc, 0x40433cc523d5cb69),
(0xbce5cecac300a9a1, 0x40440447e50db184),
(0x3cc489bd556e5109, 0x4044cefcf1734b62),
(0x3cdd0fd96f29c211, 0x4045969bc7271083),
(0x3ce4f716f3179d90, 0x404661315d6b133f),
(0xbce158b763edd0e8, 0x404728e892a88fc9),
(0xbcef3950a842db79, 0x4047f36312028ad6),
(0x3ce97656bbc2396e, 0x4048bb2fa2037de3),
(0x3ce85d7bdb30baf1, 0x404985928f96d51e),
(0xbce71f8560ac9f18, 0x404a4d71fcb56f8c),
(0x3ce3d41e041caa68, 0x404b17c038c2018c),
(0xbcde6d04716d8d21, 0x404bdfb06eb790aa),
(0x3cda139ce2cd08ac, 0x404ca9ec5a82324b),
(0x3cc8b5cc7b4501c1, 0x404d71eb98682f07),
(0xbcb12e6ef2e594e2, 0x404e3c1731d64f1e),
(0x3cb399bfca430021, 0x404f0423f99b4b53),
(0x3cdfd1ee8286358a, 0x404fce40efb1156e),
(0x3c800660b51502f0, 0x40504b2cfcbb084d),
(0x3ced3cacfc720418, 0x4050b034dde75b42),
(0x3cc4b877d4f6d900, 0x40511446f60f1458),
(0xbcee669304bfe748, 0x40517948db63675c),
(0x3cfad20ca758a714, 0x4051dd600b743a9b),
(0x3cf8eb4a94b63936, 0x4052425c7dcacdf6),
(0xbcfa196892f68386, 0x4052a67859bc641e),
];
/**
Precomputed values in exact Bessel J1 zero.
Generated by MPFR:
```text
let mut arr = vec![];
for zeros in J1_ZEROS.iter() {
let mpfr = Float::with_val(107, f64::from_bits(zeros.1)).j1();
arr.push(mpfr.to_f64().to_bits());
}
println!(
"arr: [{}]",
arr.iter()
.map(|x| format!("0x{:016X}", x))
.collect::<Vec<_>>()
.join(", ")
);
```
**/
pub(crate) static J1_ZEROS_VALUE: [u64; 48] = [
0x0000000000000000,
0x3FE29EA3D19F035F,
0xBC91B9C1C3FB286F,
0xBFD626EE83500BF2,
0x3C8049770CE74C2E,
0x3FD17DBF09D40D25,
0x3CA0212F4E592523,
0xBFCDDCEB4CE1BF4A,
0xBC905DCC62D0D222,
0x3FCA7F63FEA81F26,
0xBCB6EB905BA2ABFA,
0xBFC810F50225B04B,
0x3CAA10B2F7B4E69D,
0x3FC633E7F7F05301,
0xBC97BC6D5A660382,
0xBFC4B71D4CA2CC69,
0xBC961C29FAC28FDF,
0x3FC37DFA8F5A550A,
0xBC87E3B01386785F,
0xBFC2768D29C69936,
0x3CAF5EFD41F756B6,
0x3FC194EBA75B32F9,
0xBCBF89DCEDB3EA9B,
0xBFC0D0D36473E98C,
0xBC9AAAF726A29E97,
0x3FC02455675AB6D2,
0x3C9451B6225ACBFB,
0xBFBF161D0C28B48C,
0xBCB40032091A4E00,
0x3FBE0357C158B119,
0xBCBCCB5A05A6E4AA,
0xBFBD0B36E5737458,
0xBCB5C457E4A6A2F1,
0x3FBC29AE8400A320,
0x3CB13169F65EFC7C,
0xBFBB5B8273B75055,
0xBCA5FB7DBD93E256,
0x3FBA9E13A0DB6429,
0xBC7C3482175F80D7,
0xBFB9EF3BB2213B0B,
0xBCA977092852774B,
0x3FB94D3276914E51,
0x3CB6D73591BFEB5D,
0xBFB8B67A2481077D,
0x3CB735BC851F7831,
0x3FB829D06FEE9266,
0x3CC29C7C75EEB12F,
0xBFB7A62320798175,
];
/**
Following search for J1 (see [J1_ZEROS]) zeros and extremums:
at each zero and extremum we're doing Taylor series expansion
one that should be enough to cover whole interval between zero or peak
which is PI/4
Generated in SageMath and Sollya:
```python
def compute_intervals(zeros):
intervals = []
for i in range(0, len(zeros)):
if i == 0:
a = (zeros[i]) / 2 - 0.05 - zeros[i]
b = (zeros[i] + zeros[i + 1]) / 2 + 0.05 - zeros[i]
intervals.append((RealField(18)(a), RealField(18)(b), RealField(110)(zeros[i])))
elif i + 1 > len(zeros) - 1:
a = (zeros[i - 1] + zeros[i]) / 2 - 0.05 - zeros[i]
b = (zeros[i]) + 0.83 + 0.05 - zeros[i]
intervals.append((RealField(18)(a), RealField(18)(b), RealField(110)(zeros[i])))
else:
a = (zeros[i - 1] + zeros[i]) / 2 - zeros[i] - 0.05
b = (zeros[i] + zeros[i + 1]) / 2 + 0.05 - zeros[i]
intervals.append((RealField(18)(a), RealField(18)(b), RealField(110)(zeros[i])))
return intervals
intervals = compute_intervals(j1_zeros)
# print(intervals)
def build_sollya_script(a, b, zero, deg):
return f"""
prec = 500;
bessel_j1 = library("./notes/bessel_sollya/cmake-build-release/libbessel_sollya.dylib");
f = bessel_j1(x + {zero});
d = [{a}, {b}];
pf = remez(f, {deg}, d);
for i from 0 to degree(pf) do {{
write(coeff(pf, i)) >> "coefficients.txt";
write("\\n") >> "coefficients.txt";
}};
"""
def load_coefficients(filename):
with open(filename, "r") as f:
return [RealField(500)(line.strip()) for line in f if line.strip()]
def call_sollya_on_interval(a, b, zero, degree=12):
sollya_script = build_sollya_script(a, b, zero, degree)
with open("tmp_interval.sollya", "w") as f:
f.write(sollya_script)
import subprocess
if os.path.exists("coefficients.txt"):
os.remove("coefficients.txt")
try:
result = subprocess.run(
["sollya", "tmp_interval.sollya"],
check=True,
capture_output=True,
text=True
)
except subprocess.CalledProcessError as e:
return
def print_remez_coeffs(poly):
print("J1TaylorExtendedSeries {")
print_double_double("a0: ", poly[0])
print_double_double("a1: ", poly[1])
print_double_double("a2: ", poly[2])
print_double_double("a3: ", poly[3])
print_double_double("a4: ", poly[4])
print_double_double("a5: ", poly[5])
print_double_double("a6: ", poly[6])
print_double_double("a7: ", poly[7])
print_double_double("a8: ", poly[8])
print_double_double("a9: ", poly[9])
print_double_double("a10: ", poly[10])
print_double_double("a11: ", poly[11])
print_double_double("a12: ", poly[12])
print_double_double("a13: ", poly[13])
print("c: [")
for i in range(14, len(poly)):
coeff = poly[i]
print(f"{double_to_hex(coeff)},")
print("],")
print("},")
degree = 23
print(f"pub(crate) static J1_COEFFS: [J1TaylorExtendedSeries; {len(intervals)}] = [")
for i in range(0, len(intervals)):
interval = intervals[i]
call_sollya_on_interval(interval[0], interval[1], interval[2], degree)
coeffs = load_coefficients(f"coefficients.txt")
print_remez_coeffs(coeffs)
print("];")
```
**/
pub(crate) static J1_COEFFS: [[(u64, u64); 24]; 47] = [
[
(0x3c61f1c324453b22, 0x3fe29ea3d19f035f),
(0x35fa0d84854cf50d, 0xb9a948d98c79544d),
(0xbc6e3631ae170265, 0xbfca41115c5df243),
(0xbc18acc50ffb929f, 0x3f78d1448e6fed48),
(0x3c0e85a7cffdb986, 0x3f8c441a2f9de22b),
(0x3bd94ca1957aa906, 0xbf386671c18b088a),
(0x3bd950ccbdd65736, 0xbf39e2504ddc7608),
(0xbb61b3ae884e8323, 0x3ee34ccbca0c75d1),
(0x3b7ab8f6e0e5328c, 0x3eda4973784d1087),
(0x3b1f40e6cc67a3d5, 0xbe81045322aaab45),
(0x3b06ea738f9ecf6a, 0xbe70fae0da6cdcef),
(0xbabbaad3785f91ec, 0x3e13546cef5ed00e),
(0x3a89c3da0f7d1467, 0x3dfe5ee82e6676f1),
(0xba262fd170d5dbad, 0xbd9ec80cc8b644d5),
(0xba23036bcf1b50d2, 0xbd83eb2e99627fad),
(0xb9cc7616af2b9f56, 0x3d222bfcdb211968),
(0xb9909fcb58b04d57, 0x3d03fb337676fc98),
(0x3946bee5d8936909, 0xbca0901290ec992d),
(0x391f5d570e3a4795, 0xbc7fa6a8aa2582e5),
(0xb8b23101eae00b2a, 0x3c18078252545e8a),
(0x386d13c8640abb7d, 0x3bf44b354a191eee),
(0xb827a697c66ddb50, 0xbb8c75d1f25f35fc),
(0x380d5a5801772dad, 0xbb6549fce4790a31),
(0x378246c0eb5ba71c, 0x3affdbc62c23326e),
],
[
(0x35f9dfa8e2931f8b, 0xb95730bb59760ac5),
(0x3c62de1143765a99, 0xbfd9c6cf582cbf7f),
(0xbc46b8d73329a70b, 0x3faae8a39f51ad04),
(0xbc47767d9698b3c4, 0x3fab589d1da13905),
(0x3c0e65e41f715973, 0xbf7537544c331da7),
(0x3bc117d1587f9833, 0xbf624b3409959064),
(0x3bb8c81b4b053820, 0x3f26e4c2d5354224),
(0x3b9859e946b98719, 0x3f083a06e30c4109),
(0x3b51fd9c02e20b11, 0xbec9799d4c9f2549),
(0xbb4ffc0b40a079e5, 0xbea33825cd2e2c16),
(0x3aeb5032c6339f4e, 0x3e617069233e916c),
(0x3a80c496c8db027f, 0x3e34569b22afc3c8),
(0xba9a374b068a8d41, 0xbdf03b9e9651056f),
(0x3a4c5798959dc28e, 0xbdbec62310af5fa2),
(0xba11dc4cca029308, 0x3d75ec84e47b7d9e),
(0x39d2cf2c8d7c612a, 0x3d417a40c942a964),
(0x3971ff6b4d2284f1, 0xbcf67cb1d01a03c7),
(0xb943da1ee0003f81, 0xbcbee7ff9372e125),
(0xb8f9e098c8ea9826, 0x3c721fb8d23ce65d),
(0x38d89765c4f16618, 0x3c35e053c1413792),
(0xb8575b3ab1c49578, 0xbbe79265a43bea11),
(0xb82c7bf56bef062f, 0xbba95d6cc94f810f),
(0xb7f5bad532a4cc34, 0x3b59332b7e688450),
(0x37ac610ae03b7f77, 0x3b1a2c1db24ae655),
],
[
(0xbc782d627160714a, 0xbfd626ee83500bf2),
(0x35e215832cad36da, 0x39465d6b7a56edac),
(0x3c6ae8952e6f1d78, 0x3fc55f6bec9ef962),
(0x3c0d30f1a30544cb, 0xbf83d23336fd10e4),
(0x3c2695c3589d6e1d, 0xbf88c77a983a0814),
(0x3bea1200f42f4b09, 0x3f45cdc98db1cbe2),
(0x3bd343248aee865b, 0x3f373576ff46ee3b),
(0x3b8a8cd3c63aa60b, 0xbef2461447d7b423),
(0x3b6a8a9edacbe6fe, 0xbed7b853456b6eaa),
(0xbaff3625eaafdafc, 0x3e90abfc68274a98),
(0xbaea64c013116245, 0x3e6ea7a1ee26124d),
(0x3acef44f1141cf97, 0xbe235c0413e01419),
(0xba7020655e19f397, 0xbdfb5c5d512fbafe),
(0x3a2522345bdc9b62, 0x3daf4c5e26fd6f49),
(0xba1a9c1d9fde6bb0, 0x3d81e4c43397bb8d),
(0x39aead7c3865323c, 0xbd32addefc4e427f),
(0x399ef826261673f0, 0xbd01e4fadc073884),
(0xb9584f8fae96d4c1, 0x3cb12a0b89e91563),
(0x3911cf3d91af7eef, 0x3c7c4246886f561c),
(0xb8932b17cb520173, 0xbc290ee1bd0b60e5),
(0x388f8ba7ff012875, 0xbbf21116fef61a90),
(0x38093d7080102807, 0x3b9dcbd61b5fcfcc),
(0xb7ddad88c94d63e2, 0x3b62f7833d3a6e0a),
(0xb7a7e960c2811c48, 0xbb0f75a3322d6adb),
],
[
(0x35d923fac9298498, 0xb943ab00450c21a5),
(0x3c7af22d033ee0a8, 0x3fd33518b3874e8a),
(0xbc23b4d62984701e, 0xbf95e70dc60362bf),
(0xbc476d8715df734e, 0xbfa80c83bdeee5b0),
(0x3c03dac20aaf9d3e, 0x3f69a4b292e3de42),
(0x3bd887eed7bf2f4a, 0x3f613fbc7d698217),
(0xbbac52a914699396, 0xbf207358bbdbff91),
(0xbb64764498326fe8, 0xbf0796a751f89051),
(0xbb6df66e2a5b80ce, 0x3ec4255b015aded4),
(0xbb3ae5bbd6b6b7b2, 0x3ea3026e0ce97ab9),
(0xbaf53c3132b45416, 0xbe5d48dcdae92f2c),
(0x3ac329ab58cdf5c8, 0xbe344639d7eeb0a6),
(0xba7227361e36b492, 0x3dec62ccb4a32eb2),
(0x3a436c156b981e3c, 0x3dbecae92e85448e),
(0xba15900bb4077790, 0xbd73bb6898d7381b),
(0x39ceeaf13516632f, 0xbd4183edbb8f3baa),
(0xb946b19e901a3f2a, 0x3cf4ae3e7e418192),
(0x395392c9d1b10508, 0x3cbefbb1e7e7f06a),
(0xb91d78a8cbca8fc8, 0xbc70f28d91f2bb8c),
(0xb8c1412abdb69981, 0xbc35ec327b001a36),
(0x3866c77527da5dfe, 0x3be65536194079a9),
(0x37d83c37616b601d, 0x3ba9644a2a48ce1f),
(0xb7fc2f10d93b3bda, 0xbb58069a9c48e50d),
(0x37b28c16423dadcd, 0xbb18f8666a0fbd52),
],
[
(0xbc7d48dbfa0ea1a4, 0x3fd17dbf09d40d25),
(0xb5f12f0e1a0ea25f, 0xb95b2c846748c809),
(0xbc61eb914d33c2b5, 0xbfc1404bf647c28f),
(0x3c098a23a393f866, 0x3f74f4df2769f830),
(0x3c24ae93dcb9bf99, 0x3f85c6285429b66d),
(0x3bdddb259c934d19, 0xbf3d68ab722881bd),
(0xbbd8cd731b94002b, 0xbf356acb6452d860),
(0x3b60d2c0a7987aa7, 0x3eec10b47cf7ef69),
(0xbb3cc8bfc9cf9d54, 0x3ed67eaae97bbc86),
(0x3b021d7ef8f06ed1, 0xbe8bb6530c63f2df),
(0xbaf7d003a6f8a38a, 0xbe6d87201e450edd),
(0xba5df682e9056c25, 0x3e20f47c83ec550b),
(0x3a784cc2fbcd5b3c, 0x3dfa98331f6ea797),
(0x3a3af02acc5d445a, 0xbdac70414a236ade),
(0xba1ac139214727ff, 0xbd817c057a5fc937),
(0x39abeb4e52ce6eb7, 0x3d316fea14d29892),
(0x39a6a5fe18553faf, 0x3d0189bd3eb9f3c6),
(0x3946aa341fe1b254, 0xbcb05af44bb23c5f),
(0x39010e85f5ece2bd, 0xbc7bbdb45d92e4fc),
(0xb89b4dc8ed25771b, 0x3c284360a40ed89d),
(0x38779d6e01f88769, 0x3bf1bfb967530933),
(0xb82198845d1335e3, 0xbb9d37a860a66a8e),
(0xb806974ac174343c, 0xbb62a0bbacab19c8),
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(0xb7bf6daddcc30386, 0xbb110e1251b8eda4),
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[
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[
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[
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],
];
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