pxfm 0.1.28

/*
 * // 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.
 */

/**
J0 zeros and extremums.

Generated by SageMath:
```python
# searching for zeros and extremums
R120 = RealField(120)

zeros = []

mp.prec = 150

step = mpf("0.001")
epsilon = mpf("1e-35")
x = mpf("0.0")

def j0_prime(x):
    return diff(lambda t: besselj(0, t), x)

previous_zero = R120(0)
j0_zeros = []

while x < mpf("76.0"):
    f1 = besselj(0, x)
    f2 = besselj(0, x + step)
    if f1 * f2 < 0:
        zero = findroot(lambda t: j0(t), (x, x + step), solver='bisect', tol=mp.mpf("1e-41"))
        previous_zero = zero
        j0_zeros.append(zero)
    if previous_zero is not None and abs(x - mpf(f'{round(x)}')) < epsilon:
        zeros.append(previous_zero)
    x += step

j0_extrema = []

x = mpf("0.0")
while x < mpf("76.0"):
    d1 = mp.diff(lambda t: j0(t), x)
    d2 = mp.diff(lambda t: j0(t), x + step)
    if d1 * d2 < 0:
        extremum = findroot(lambda t: mp.diff(lambda u: j0(u), t), (x, x + step), solver='bisect', tol=mp.mpf("1e-41"))
        j0_extrema.append(extremum)
    x += step

# Print results
for i, z in enumerate(j0_zeros):
    print(f"Zero {i+1}: x ≈ {z}")

print("Extrema (peaks/valleys) of J0(x):")
for e in j0_extrema:
    print(f"nExtrema: {e}")

j0_zeros.extend(j0_extrema)

j0_zeros = sorted(j0_zeros)

# Print results
for i, z in enumerate(j0_zeros):
    print(f"Peak or zero {i+1}: x ≈ {z}")

print("")

print("pub(crate) static J0_ZEROS: [(u64, u64); 48] = [")
print(f"(0x0, 0x0),")
for z in j0_zeros:
    k = split_double_double(z)
    hi = double_to_hex(k[1])
    lo = double_to_hex(k[0])
    print(f"({lo}, {hi}),")

print("];")
```
**/
pub(crate) static J0_ZEROS: [(u64, u64); 48] = [
    (0x0, 0x0),
    (0xbca0f539d7da258e, 0x40033d152e971b40),
    (0xbca60155a9d1b256, 0x400ea75575af6f09),
    (0x3c975054cd60a517, 0x4016148f5b2c2e45),
    (0xbc9b226d9d243827, 0x401c0ff5f3b47250),
    (0xbcb51970714c7c25, 0x40214eb56cccdeca),
    (0x3cc02610a51562b6, 0x402458d0d0bdfc29),
    (0x3cb444fd5821d5b1, 0x40279544008272b6),
    (0x3cb2bce7fd18e693, 0x402aa5baf310e5a2),
    (0xbcc9796609364e85, 0x402ddca13ef271d2),
    (0xbcdd2a68e88ab317, 0x4030787b360508c5),
    (0xbcd165fd108f46ff, 0x403212313f8a19f6),
    (0xbcd21830197e9e86, 0x40339da8e7416ca4),
    (0x3cc1d2dfa1c3b5a8, 0x4035362dd173f792),
    (0xbcc1bf33afef88f1, 0x4036c294e3d4d8ac),
    (0x3cd0847c620015e0, 0x40385a3b930156dd),
    (0x3cc1a2686480d882, 0x4039e7570dcea106),
    (0x3cdd2b3714972b28, 0x403b7e54a5fd5f11),
    (0xbcb42ce39ec976fb, 0x403d0bfcf471fccc),
    (0xbcc36bbabc1c9f31, 0x403ea27591cbbed2),
    (0xbcdbe3a1cd066b66, 0x404018476e6b2bf0),
    (0x3cda326cf4307839, 0x4040e34e13a66fe6),
    (0xbced5fbbff045068, 0x4041aa890dc5e97c),
    (0xbcd0b6068f861c6f, 0x404275637a9619ec),
    (0x3cc9eafeca0ca4fd, 0x40433cc523d5cb69),
    (0xbcb34c86f4e27936, 0x4044077a7ed6293a),
    (0x3cc489bd556e510a, 0x4044cefcf1734b62),
    (0x3cced48fe99f45ef, 0x40459992c65d0d8d),
    (0x3ce4f716f3179d90, 0x404661315d6b133f),
    (0xbcd05a7a0525058f, 0x40472bac0f810810),
    (0xbcef3950a842db79, 0x4047f36312028ad6),
    (0x3ce575dc7f8a031a, 0x4048bdc6293f0657),
    (0x3ce85d7bdb30baf1, 0x404985928f96d51e),
    (0xbcdfa16a338bbaee, 0x404a4fe0ee444c7b),
    (0x3ce3d41e041caa68, 0x404b17c038c2018c),
    (0xbce43e4a90356acf, 0x404be1fc41a4c607),
    (0x3cda139ce2cd08ac, 0x404ca9ec5a82324b),
    (0x3ce0f4b1c9544480, 0x404d74180c9e41f6),
    (0xbcb12e6ef2e594e1, 0x404e3c1731d64f1e),
    (0x3ceff10a69607aab, 0x404f06343d0971d4),
    (0x3cdfd1ee8286358a, 0x404fce40efb1156e),
    (0xbcf8991ca07c84c0, 0x40504c28621f11e6),
    (0x3ced3cacfc720419, 0x4050b034dde75b42),
    (0xbcee90a52cffc26f, 0x40511536cb22d72b),
    (0xbcee669304bfe748, 0x40517948db63675c),
    (0xbcf5328276c045f2, 0x4051de4554a1c2dd),
    (0x3cf8eb4a94b63936, 0x4052425c7dcacdf6),
    (0x3cff05f585843675, 0x4052a753fa820480),
];

/**
Roots and values for extremums and J0 zeros.

Generated by MPFR:
```text
let mut arr = vec![];
for zeros in J0_ZEROS.iter() {
    let mpfr = Float::with_val(107, f64::from_bits(zeros.1)).j0();
    arr.push(mpfr.to_f64().to_bits());
}
println!(
    "arr: [{}]",
    arr.iter()
        .map(|x| format!("0x{:016x}", x))
        .collect::<Vec<_>>()
        .join(", ")
);
```
**/
pub(crate) static J0_ZEROS_VALUE: [u64; 48] = [
    0x3ff0000000000000,
    0xbc919b7921f03c8e,
    0xbfd9c6cf582cbf7f,
    0xbc7fbb40985f6e34,
    0x3fd33518b3874e8a,
    0xbc96e8eeb22e5818,
    0xbfcff654544ebcd1,
    0xbc92d8ed368e0843,
    0x3fcbf3337873a7d8,
    0xbca50be2ef09843e,
    0xbfc925c6fca08f55,
    0x3caa2122af76659e,
    0x3fc70c511227d5aa,
    0x3c98b4b912da2218,
    0xbfc5664e13b70622,
    0xbca55e059345b430,
    0x3fc40f90793605bb,
    0x3cb1c17abe35eaae,
    0xbfc2f2072e638cf4,
    0x3c966608ac164dbd,
    0x3fc1ff5eec6a01cd,
    0x3cacc62c8409daa7,
    0xbfc12dd57bf18ada,
    0x3ca18e7b1f5c77ae,
    0x3fc076826cc2c191,
    0xbc8376cc57901722,
    0xbfbfa8b41711c83a,
    0xbc9df12b66b26573,
    0x3fbe8727daa3daed,
    0xbc9eab3b2bf5813d,
    0xbfbd8293aa55d18f,
    0xbcb3795064667200,
    0x3fbc96700bf039e2,
    0xbcabd526bce38ac7,
    0xbfbbbf246914235f,
    0x3cb14dbc2a692434,
    0x3fbaf9cb49c4f935,
    0x3cac341b9f831174,
    0xbfba4407e04298d1,
    0xbcb9e23f9c9ca91e,
    0x3fb99be744018c90,
    0xbcc3725f95922088,
    0xbfb8ffc9bd24fe08,
    0x3cb79a2187735244,
    0x3fb86e51be0a9153,
    0xbcc0012df1d3ef73,
    0xbfb7e656efb009ae,
    0xbcc6ecd27843db59,
];

/**
Taylor coefficients for J0 [0; 74.62]

Generated by SageMath with 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(j0_zeros)
# print(intervals)

def build_sollya_script(a, b, zero, deg):
    return f"""
prec = 500;
bessel_j0 = library("./notes/bessel_sollya/cmake-build-release/libbessel_sollya.dylib");
f = bessel_j0(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 [RR(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

degree = 13

print(f"pub(crate) static J0F_COEFFS: [[u64;{degree + 1}]; {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("[")
    for c in coeffs:
        print(double_to_hex(c) + ",")
    print("],")
print("];")
```
**/
pub(crate) static J0F_COEFFS: [[u64; 14]; 47] = [
    [
        0xbca69de51bcc0120,
        0xbfe09cdb3655127d,
        0x3fbba1deea029925,
        0x3facfae8643687e7,
        0xbf81bb1cbe1caeb6,
        0xbf61f992590d3f89,
        0x3f315382bbf1a06b,
        0x3f06ed3ba591e6da,
        0xbed232c8d8aba5cc,
        0xbea1cceb52db45ee,
        0x3e68006b6b5682ca,
        0x3e329cc1da0793f0,
        0xbdf5dbb0053da32b,
        0xbdc04e14e5d96539,
    ],
    [
        0xbfd9c6cf582cbf7f,
        0x3cb1d608597b5d8f,
        0x3fc9c6cf582cbf55,
        0xbf91f06d14e12b29,
        0xbf8b589d1da0f7e4,
        0x3f50f9103d00ae93,
        0x3f38644561ce9290,
        0xbefa2a034c749192,
        0xbed83a0686fd1dd2,
        0x3e96a5085d7844b5,
        0x3e6ebfcaacd49b87,
        0xbe2964b9932b57f3,
        0xbdfad73cf4701511,
        0x3db5acbea609d3e5,
    ],
    [
        0xbc6a1105f9dbcdd5,
        0x3fd5c6e60a097826,
        0xbf9f8f72e7a8471e,
        0xbfab2150cb41ece4,
        0x3f72f7ffe901b59c,
        0x3f627e31fe9ddafb,
        0xbf26f641f369bbbd,
        0xbf0863f485fc7e89,
        0x3ecad77cbbb32bcd,
        0x3ea32e705af41964,
        0xbe62d9d2bb447e35,
        0xbe341f0e4f4dc44a,
        0x3df198b51eee9749,
        0x3dbec8f104a030ca,
    ],
    [
        0x3fd33518b3874e8a,
        0xbca7f56ee546a569,
        0xbfc33518b3874e3b,
        0x3f7d34125d59fb96,
        0x3f880c83bdee71b4,
        0xbf4483c20f2abbd1,
        0xbf36ffa5fc0ccc0f,
        0x3ef2ccf7bbc1d620,
        0x3ed796a6caa245fc,
        0xbe91e85738cfdd42,
        0xbe6e69b581df4be3,
        0x3e2550287122730e,
        0x3dfab73cc2e87ac7,
        0xbdb250809ed630ba,
    ],
    [
        0x3c684670459725e1,
        0xbfd15f7977a772d5,
        0x3f900f7fcf183c65,
        0x3fa68b984ec64b1d,
        0xbf648e63600c54a1,
        0xbf60e0d6038721a4,
        0x3f1d7960512fb682,
        0x3f07800bc7410c66,
        0xbec3324799a7e0fb,
        0xbea30e8df2325ca1,
        0x3e5cecefe3d75abb,
        0x3e3458cbaa06ef8d,
        0xbdec4dd6cf551d07,
        0xbdbefa0975b65713,
    ],
    [
        0xbfcff654544ebcd1,
        0x3ca0ffc636e7373a,
        0x3fbff654544ebc20,
        0xbf70c17ff72b1352,
        0xbf84b0c5d5d9e7c9,
        0x3f394154be849b76,
        0x3f34e12c2fdccc3a,
        0xbee9f32fcfbab067,
        0xbed63c53e4cc158b,
        0x3e8adbb88b7826d7,
        0x3e6d5f825ac22ee0,
        0xbe20f17a50351084,
        0xbdfa326458f60fce,
        0x3dae0e0c6fa040ee,
    ],
    [
        0xbc648b5c145d146f,
        0x3fcdc13e66ac2e78,
        0xbf842ff0cdc58194,
        0xbfa38d1dd8992ef5,
        0x3f5a55e9b344ea02,
        0x3f5e2e16f97e7f5f,
        0xbf13dfc37714f92e,
        0xbf05ce7f4a51d917,
        0x3ebbb177bc7895b4,
        0x3ea2346df84ab988,
        0xbe56125b44798bd5,
        0xbe33d7cc75fbdd4e,
        0x3de68bc84d67bb0a,
        0x3dbe749161c75b7f,
    ],
    [
        0x3fcbf3337873a7d8,
        0xbc999eb31028b75f,
        0xbfbbf3337873a725,
        0x3f66604d91f94ba7,
        0x3f8251858010ffd8,
        0xbf314bc11a41832e,
        0xbf32e7decc933381,
        0x3ee293b4d39e1d9e,
        0x3ed4a66fbc1f8e7c,
        0xbe843cb02661be36,
        0xbe6bdd61e41af9d3,
        0x3e1ab24db7abbdd7,
        0x3df940cc3ecafe33,
        0xbda870435d8d1863,
    ],
    [
        0x3c6110461d393f0c,
        0xbfca701d0f967501,
        0x3f7c54b930fef3e4,
        0x3fa17798aa09f194,
        0xbf52a21514062f70,
        0xbf5b541f829ca99f,
        0x3f0cc0bd9fd11443,
        0x3f041f3b06c79efa,
        0xbeb4b22fc3292c98,
        0xbea1223e5773d27c,
        0x3e5118e5cf14a591,
        0x3e32ffb6478c2281,
        0xbde20a2b34bd7e9f,
        0xbdbd77afc051bc6c,
    ],
    [
        0xbfc925c6fca08f54,
        0x3c941bba0a9908a3,
        0x3fb925c6fca08ea3,
        0xbf6049377403d97d,
        0xbf809463bbcfb70d,
        0x3f297b35471ddd4a,
        0x3f314dd43fc4bda8,
        0xbedbf665e54737fe,
        0xbed32caf8123dd12,
        0x3e7f510484fa37ba,
        0x3e6a42a3b8b1effd,
        0xbe1544b8d56d77a8,
        0xbdf81cfb72892ddb,
        0x3da3f5749b01c5c3,
    ],
    [
        0xbc5c64a3611bf4cb,
        0x3fc8077f56c9b782,
        0xbf75467eb535dac4,
        0xbf9fd7c3ad6f5a3e,
        0x3f4c1b47c806f9c6,
        0x3f59166c7d3ee8f7,
        0xbf05f01522f72552,
        0xbf02aa939ff2e34a,
        0x3eb0129642c0eedc,
        0x3ea01716d7274ce0,
        0xbe4b21ada69ec1ec,
        0xbe320ffbd6f367e8,
        0x3ddd441699eaa4df,
        0x3dbc46351a30e033,
    ],
    [
        0x3fc70c511227d5aa,
        0xbc903dd711e6b7a7,
        0xbfb70c511227d4fe,
        0x3f5910ebe1f1fa45,
        0x3f7e7dc08e6ff244,
        0xbf23bd7d15b0a88f,
        0xbf300357a101d595,
        0x3ed5ec733cb20d5b,
        0x3ed1e5d1fa4386ae,
        0xbe78f423a07fedeb,
        0xbe68c089aeb3a360,
        0x3e1143a2a87b28be,
        0x3df6f2d06548607b,
        0xbda07d7d6f49ddb9,
    ],
    [
        0x3c57d1baaf487c60,
        0xbfc62d93aa9d05bb,
        0x3f70ba9ce88926ac,
        0x3f9d7073daebb038,
        0xbf462813c7f32d76,
        0xbf574a948d055ffc,
        0x3f01695764ad01fd,
        0x3f0170ab5ed77aa3,
        0xbea9c8e2cba34328,
        0xbe9e4d88f9ebcadb,
        0x3e460ec3e9bbfca2,
        0x3e312751ad8cf2c9,
        0xbdd825b2017389c1,
        0xbdbb0b4f2fdaca95,
    ],
    [
        0xbfc5664e13b70621,
        0x3c8adfee580612ab,
        0x3fb5664e13b7057b,
        0xbf540ee3940b2f9d,
        0xbf7c5e1ad9fa40ad,
        0x3f1fb8a98f136a0c,
        0x3f2de9be57a255f8,
        0xbed1bec95e73b04b,
        0xbed0cf25bf302fd8,
        0x3e746784408bf531,
        0x3e676640f349bd8a,
        0xbe0c944164ad511b,
        0xbdf5d837e7a07aba,
        0x3d9ba3ab0edd6dab,
    ],
    [
        0xbc543722dc20f623,
        0x3fc4b2a2ebf61ecd,
        0xbf6b3297fdae7373,
        0xbf9b8105d59b1125,
        0x3f420a3f8c10a2c5,
        0x3f55d18d69de4109,
        0xbefc79db490d157b,
        0xbf00679c92947f5d,
        0x3ea53ac3d1de0eb7,
        0x3e9ca750128116cf,
        0xbe4252ac12ff61fa,
        0xbe3051067c8a6420,
        0x3dd4435be57e146d,
        0x3db9dcd0ca8a2a1d,
    ],
    [
        0x3fc40f90793605bb,
        0xbc86a24c27e99837,
        0xbfb40f907936051b,
        0x3f5085775a554a25,
        0x3f7aa0ce0420ebfd,
        0xbf1a32a28e65afde,
        0xbf2c26ebc916e22b,
        0x3ecd740098d21c07,
        0x3ecfc1baf78438dd,
        0xbe710c76235f5aa1,
        0xbe663455427d9c57,
        0x3e08141f4d843200,
        0x3df4d5216dec0df1,
        0xbd977e29b16d5fdb,
    ],
    [
        0x3c5158813dc387f0,
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