/rust/registry/src/index.crates.io-1949cf8c6b5b557f/pxfm-0.1.25/src/gamma/trigamma.rs

Source
/*
 * // Copyright (c) Radzivon Bartoshyk 8/2025. 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.
 */
use crate::common::is_integer;
use crate::double_double::DoubleDouble;
use crate::sincospi::f_fast_sinpi_dd;

// Generated in Wolfram Mathematica:
// <<FunctionApproximations`
// ClearAll["Global`*"]
// f[x_]:=PolyGamma[1, x+1]
// {err0,approx}=MiniMaxApproximation[f[z],{z,{-0.99999999,0},11,11},WorkingPrecision->75,MaxIterations->100]
// num=Numerator[approx][[1]];
// poly=num;
// coeffs=CoefficientList[poly,z];
// TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
static P_1: [(u64, u64); 12] = [
    (0x3c81873d89121fec, 0x3ffa51a6625307d3),
    (0x3cb78bf7af507504, 0x4016a65cbac476ca),
    (0xbc94179c65e021c2, 0x40218234a0a79582),
    (0x3cb842a8ab5e0994, 0x401fde32175f8515),
    (0xbc8768b33f5776b7, 0x4012de6bde49abff),
    (0x3c8e06354a27f081, 0x3ffe5d6ef3a2eac6),
    (0xbc8b391d09fed17a, 0x3fe0d186688252cf),
    (0xbc59a5c46bb8b8cc, 0x3fb958f9a0f156b7),
    (0x3c2ac44c6a197244, 0x3f88e605f24e1a89),
    (0x3bd2d05fa8be27f2, 0x3f4cd369f5d68104),
    (0x3b84ad0a748fdd22, 0x3efde955ebb17874),
    (0xb96aa8b9a65e0899, 0xbce053d04459ead7),
];

// Generated by Wolfram Mathematica:
// <<FunctionApproximations`
// ClearAll["Global`*"]
// f[x_]:=PolyGamma[1, x+1]
// {err0,approx}=MiniMaxApproximation[f[z],{z,{0,3},11,11},WorkingPrecision->75,MaxIterations->100]
// num=Numerator[approx][[1]];
// poly=num;
// coeffs=CoefficientList[poly,z];
// TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
static P_2: [(u64, u64); 12] = [
    (0x3c81873d8912236c, 0x3ffa51a6625307d3),
    (0xbcbc731a62342288, 0x40176c23f7ea51e6),
    (0xbcc45a6fd00e67a8, 0x4022cb5ae67657ef),
    (0x3cc0876fde7fe4e6, 0x4021d766062b9550),
    (0xbcaec4a4859cba1d, 0x401629f91cd4f291),
    (0x3c76184014e4d7e3, 0x4002d43da3352004),
    (0x3c812c7609483e0e, 0x3fe62e3266eef8c7),
    (0xbc5f991047f52d2b, 0x3fc1eacb910b951c),
    (0x3c28b9f38d603f2f, 0x3f930960a301df34),
    (0x3bf9b620eb930504, 0x3f5814f8e057b14b),
    (0xbb990860b88b54e4, 0x3f0b9f67c71aa3bf),
    (0x38e5cb6acfbaab77, 0xbc4194b8c01afe9a),
];

// Generated by Wolfram Mathematica:
// <<FunctionApproximations`
// ClearAll["Global`*"]
// f[x_]:=PolyGamma[1, x+1]
// {err0,approx,err1}=MiniMaxApproximation[f[z],{z,{3,9},11,11},WorkingPrecision->75,MaxIterations->100]
// num=Numerator[approx];
// poly=num;
// coeffs=CoefficientList[poly,z];
// TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
static P_3: [(u64, u64); 12] = [
    (0x3c9cd56dbb295efc, 0x3ffa51a662556db9),
    (0x3c9f4ee74f5f9daf, 0x4018ff913088cb34),
    (0x3ccf08737350609c, 0x402593d55686b8b1),
    (0xbcc6cd4ed33afebb, 0x402641d10de4def5),
    (0xbcb24d1957c1303c, 0x401e682c37e8e2cf),
    (0x3ca30ac79162ceb2, 0x400ccfc7c4566f55),
    (0x3c9efea5ff293dc9, 0x3ff33eb2c6e89d0b),
    (0x3c74670a11068abc, 0x3fd1fbf456e5c6f0),
    (0x3c47b5dcdea19c36, 0x3fa6a6a2148c482c),
    (0xbc14642012a1cc1e, 0x3f71851e927f52e7),
    (0x3bc7db88a4ec5478, 0x3f29a45059a43475),
    (0xb7bc31e55271eab0, 0xbb375518529c52fb),
];

// Generated in Wolfram Mathematica:
// <<FunctionApproximations`
// ClearAll["Global`*"]
// f[x_]:=PolyGamma[1, x+1]
// {err0,approx}=MiniMaxApproximation[f[z],{z,{-0.99999999,0},11,11},WorkingPrecision->75,MaxIterations->100]
// den=Denominator[approx][[1]];
// poly=den;
// coeffs=CoefficientList[poly,z];
// TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
static Q_1: [(u64, u64); 12] = [
    (0x0000000000000000, 0x3ff0000000000000),
    (0xbcb84c43a11fc28a, 0x40139d9587da0fb5),
    (0x3ca1cf3dbcddbb57, 0x402507cb6225f0f0),
    (0x3cb01aa6ddcc3cfd, 0x402a1b416d0ed4e6),
    (0xbcbc31c216b5ff66, 0x4024ec8829e535d4),
    (0xbcb335c23022f43e, 0x4016d2ba6d1a18e6),
    (0x3cafbfffc03ad28a, 0x400158c4611ed51f),
    (0xbc8d1fb10a031a27, 0x3fe26f15bb52f89b),
    (0x3c56a9fea160eecb, 0x3fbaec13f663049d),
    (0xbc2f4ee869ba9364, 0x3f89cde0500cd68f),
    (0x3be3f23afc9398b6, 0x3f4d4b0f4dcf3eb8),
    (0x3b67a3ed4795d33e, 0x3efde955eafac9c2),
];

// Generated by Wolfram Mathematica:
// <<FunctionApproximations`
// ClearAll["Global`*"]
// f[x_]:=PolyGamma[1, x+1]
// {err0,approx}=MiniMaxApproximation[f[z],{z,{0,3},11,11},WorkingPrecision->75,MaxIterations->100]
// den=Denominator[approx][[1]];
// poly=den;
// coeffs=CoefficientList[poly,z];
// TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
static Q_2: [(u64, u64); 12] = [
    (0x0000000000000000, 0x3ff0000000000000),
    (0xbc81a3e4e026b7b1, 0x401415d1a20a9339),
    (0x3cc279576dfe3ec9, 0x402627c1a95d33d2),
    (0x3c9a94b5cf0cae88, 0x402c724dc5cf4577),
    (0xbc8aa1fa0c3820a8, 0x4027b7a332bb07f4),
    (0xbc96968367088d66, 0x401b14376177bdd7),
    (0x3ca2d3dfa5847f4d, 0x4005b0511cd98f2c),
    (0xbc8cfad394d41dd1, 0x3fe877bc2d02c7f3),
    (0xbc51592b8ec81a92, 0x3fc31f52afc72b95),
    (0x3c2cbef277d587e9, 0x3f93cb2f0e574376),
    (0xbbfbb670fd94f6ba, 0x3f5883767f745a92),
    (0xbb931b04d74e5893, 0x3f0b9f67c71a60f3),
];

// Generated by Wolfram Mathematica:
// <<FunctionApproximations`
// ClearAll["Global`*"]
// f[x_]:=PolyGamma[1, x+1]
// {err0,approx,err1}=MiniMaxApproximation[f[z],{z,{3,9},11,11},WorkingPrecision->75,MaxIterations->100]
// den=Denominator[approx];
// poly=den;
// coeffs=CoefficientList[poly,z];
// TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
static Q_3: [(u64, u64); 12] = [
    (0x0000000000000000, 0x3ff0000000000000),
    (0x3cbafcb4b6d646d9, 0x40150b12a79fc9cf),
    (0x3c989ef814b8dd2a, 0x40288c1d26ffdca5),
    (0x3cb0282cfea9c473, 0x4030d737c893cd5f),
    (0x3cc955b8aaadb37d, 0x402e5c289b6de3e0),
    (0x3cb377161f8861d2, 0x4022fb66d87bd522),
    (0xbcb4b0e4cff46ad6, 0x4010e5d13c2a5907),
    (0xbc8824539e4b1bd6, 0x3ff58c8fe8f26fca),
    (0xbc7d34220d810ea0, 0x3fd36c1351f43e66),
    (0xbc4cbdbe85570017, 0x3fa7c1170466605e),
    (0xbc0c3afb98775c53, 0x3f71ebafd3e5e3b9),
    (0x3bc0b0b7f16afd0a, 0x3f29a45059a43475),
];

#[inline]
fn approx_trigamma(x: f64) -> DoubleDouble {
    if x <= 10. {
        let (p, q) = if x <= 1. {
            (&P_1, &Q_1)
        } else if x <= 4. {
            (&P_2, &Q_2)
        } else {
            (&P_3, &Q_3)
        };
        let x2 = DoubleDouble::from_exact_mult(x, x);
        let x4 = DoubleDouble::quick_mult(x2, x2);
        let x8 = DoubleDouble::quick_mult(x4, x4);

        let e0 = DoubleDouble::mul_f64_add(
            DoubleDouble::from_bit_pair(p[1]),
            x,
            DoubleDouble::from_bit_pair(p[0]),
        );
        let e1 = DoubleDouble::mul_f64_add(
            DoubleDouble::from_bit_pair(p[3]),
            x,
            DoubleDouble::from_bit_pair(p[2]),
        );
        let e2 = DoubleDouble::mul_f64_add(
            DoubleDouble::from_bit_pair(p[5]),
            x,
            DoubleDouble::from_bit_pair(p[4]),
        );
        let e3 = DoubleDouble::mul_f64_add(
            DoubleDouble::from_bit_pair(p[7]),
            x,
            DoubleDouble::from_bit_pair(p[6]),
        );
        let e4 = DoubleDouble::mul_f64_add(
            DoubleDouble::from_bit_pair(p[9]),
            x,
            DoubleDouble::from_bit_pair(p[8]),
        );
        let e5 = DoubleDouble::mul_f64_add(
            DoubleDouble::from_bit_pair(p[11]),
            x,
            DoubleDouble::from_bit_pair(p[10]),
        );

        let f0 = DoubleDouble::mul_add(x2, e1, e0);
        let f1 = DoubleDouble::mul_add(x2, e3, e2);
        let f2 = DoubleDouble::mul_add(x2, e5, e4);

        let g0 = DoubleDouble::mul_add(x4, f1, f0);

        let p_num = DoubleDouble::mul_add(x8, f2, g0);

        let rcp = DoubleDouble::from_quick_recip(x);
        let rcp2 = DoubleDouble::quick_mult(rcp, rcp);

        let e0 = DoubleDouble::mul_f64_add_f64(
            DoubleDouble::from_bit_pair(q[1]),
            x,
            f64::from_bits(0x3ff0000000000000),
        );
        let e1 = DoubleDouble::mul_f64_add(
            DoubleDouble::from_bit_pair(q[3]),
            x,
            DoubleDouble::from_bit_pair(q[2]),
        );
        let e2 = DoubleDouble::mul_f64_add(
            DoubleDouble::from_bit_pair(q[5]),
            x,
            DoubleDouble::from_bit_pair(q[4]),
        );
        let e3 = DoubleDouble::mul_f64_add(
            DoubleDouble::from_bit_pair(q[7]),
            x,
            DoubleDouble::from_bit_pair(q[6]),
        );
        let e4 = DoubleDouble::mul_f64_add(
            DoubleDouble::from_bit_pair(q[9]),
            x,
            DoubleDouble::from_bit_pair(q[8]),
        );
        let e5 = DoubleDouble::mul_f64_add(
            DoubleDouble::from_bit_pair(q[11]),
            x,
            DoubleDouble::from_bit_pair(q[10]),
        );

        let f0 = DoubleDouble::mul_add(x2, e1, e0);
        let f1 = DoubleDouble::mul_add(x2, e3, e2);
        let f2 = DoubleDouble::mul_add(x2, e5, e4);

        let g0 = DoubleDouble::mul_add(x4, f1, f0);

        let p_den = DoubleDouble::mul_add(x8, f2, g0);

        let q = DoubleDouble::div(p_num, p_den);
        let r = DoubleDouble::quick_dd_add(q, rcp2);
        return r;
    }
    // asymptotic expansion Trigamma[x] = 1/x + 1/x^2 + sum(Bernoulli(2*k)/x^(2*k + 1))
    // Generated in SageMath:
    // var('x')
    // def bernoulli_terms(x, N):
    //     S = 0
    //     for k in range(1, N+1):
    //         B = bernoulli(2*k)
    //         term = B*x**(-(2*k+1))
    //         S += term
    //     return S
    //
    // terms = bernoulli_terms(x, 10)
    // coeffs = [RealField(150)(terms.coefficient(x, n)) for n in range(0, terms.degree(x)+1, 1)]
    // for k in range(0, 14):
    //     c = terms.coefficient(x, -k)  # coefficient of x^(-k)
    //     if c == 0:
    //         continue
    //     print("f64::from_bits(" + double_to_hex(c) + "),")
    const C: [(u64, u64); 10] = [
        (0x3c65555555555555, 0x3fc5555555555555),
        (0xbc21111111111111, 0xbfa1111111111111),
        (0x3c38618618618618, 0x3f98618618618618),
        (0xbc21111111111111, 0xbfa1111111111111),
        (0xbc4364d9364d9365, 0x3fb364d9364d9365),
        (0xbc6981981981981a, 0xbfd0330330330330),
        (0xbc95555555555555, 0x3ff2aaaaaaaaaaab),
        (0xbcb7979797979798, 0xc01c5e5e5e5e5e5e),
        (0xbcac3b070ec1c3b0, 0x404b7c4f8f13e3c5),
        (0x3cc8d3018d3018d3, 0xc08088fe72cfe72d),
    ];

    let rcp = DoubleDouble::from_quick_recip(x);

    let q = DoubleDouble::quick_mult(rcp, rcp);

    let q2 = DoubleDouble::quick_mult(q, q);
    let q4 = q2 * q2;
    let q8 = q4 * q4;

    let e0 = DoubleDouble::quick_mul_add(
        DoubleDouble::from_bit_pair(C[1]),
        q,
        DoubleDouble::from_bit_pair(C[0]),
    );
    let e1 = DoubleDouble::quick_mul_add(
        DoubleDouble::from_bit_pair(C[3]),
        q,
        DoubleDouble::from_bit_pair(C[2]),
    );
    let e2 = DoubleDouble::quick_mul_add(
        DoubleDouble::from_bit_pair(C[5]),
        q,
        DoubleDouble::from_bit_pair(C[4]),
    );
    let e3 = DoubleDouble::quick_mul_add(
        DoubleDouble::from_bit_pair(C[7]),
        q,
        DoubleDouble::from_bit_pair(C[6]),
    );
    let e4 = DoubleDouble::quick_mul_add(
        DoubleDouble::from_bit_pair(C[9]),
        q,
        DoubleDouble::from_bit_pair(C[8]),
    );

    let q0 = DoubleDouble::quick_mul_add(q2, e1, e0);
    let q1 = DoubleDouble::quick_mul_add(q2, e3, e2);

    let r0 = DoubleDouble::quick_mul_add(q4, q1, q0);
    let mut p = DoubleDouble::quick_mul_add(q8, e4, r0);

    let q_over_2 = DoubleDouble::quick_mult_f64(q, 0.5);
    p = DoubleDouble::quick_mult(p, q);
    p = DoubleDouble::quick_mult(p, rcp);
    p = DoubleDouble::quick_dd_add(q_over_2, p);
    p = DoubleDouble::quick_dd_add(p, rcp);
    p
}

#[inline]
fn approx_trigamma_dd(x: DoubleDouble) -> DoubleDouble {
    if x.hi <= 10. {
        let (p, q) = if x.hi <= 1. {
            (&P_1, &Q_1)
        } else if x.hi <= 4. {
            (&P_2, &Q_2)
        } else {
            (&P_3, &Q_3)
        };
        let x2 = DoubleDouble::quick_mult(x, x);
        let x4 = DoubleDouble::quick_mult(x2, x2);
        let x8 = DoubleDouble::quick_mult(x4, x4);

        let e0 = DoubleDouble::mul_add(
            DoubleDouble::from_bit_pair(p[1]),
            x,
            DoubleDouble::from_bit_pair(p[0]),
        );
        let e1 = DoubleDouble::mul_add(
            DoubleDouble::from_bit_pair(p[3]),
            x,
            DoubleDouble::from_bit_pair(p[2]),
        );
        let e2 = DoubleDouble::mul_add(
            DoubleDouble::from_bit_pair(p[5]),
            x,
            DoubleDouble::from_bit_pair(p[4]),
        );
        let e3 = DoubleDouble::mul_add(
            DoubleDouble::from_bit_pair(p[7]),
            x,
            DoubleDouble::from_bit_pair(p[6]),
        );
        let e4 = DoubleDouble::mul_add(
            DoubleDouble::from_bit_pair(p[9]),
            x,
            DoubleDouble::from_bit_pair(p[8]),
        );
        let e5 = DoubleDouble::mul_add(
            DoubleDouble::from_bit_pair(p[11]),
            x,
            DoubleDouble::from_bit_pair(p[10]),
        );

        let f0 = DoubleDouble::mul_add(x2, e1, e0);
        let f1 = DoubleDouble::mul_add(x2, e3, e2);
        let f2 = DoubleDouble::mul_add(x2, e5, e4);

        let g0 = DoubleDouble::mul_add(x4, f1, f0);

        let p_num = DoubleDouble::mul_add(x8, f2, g0);

        let rcp = x.recip();
        let rcp2 = DoubleDouble::quick_mult(rcp, rcp);

        let e0 = DoubleDouble::mul_add_f64(
            DoubleDouble::from_bit_pair(q[1]),
            x,
            f64::from_bits(0x3ff0000000000000),
        );
        let e1 = DoubleDouble::mul_add(
            DoubleDouble::from_bit_pair(q[3]),
            x,
            DoubleDouble::from_bit_pair(q[2]),
        );
        let e2 = DoubleDouble::mul_add(
            DoubleDouble::from_bit_pair(q[5]),
            x,
            DoubleDouble::from_bit_pair(q[4]),
        );
        let e3 = DoubleDouble::mul_add(
            DoubleDouble::from_bit_pair(q[7]),
            x,
            DoubleDouble::from_bit_pair(q[6]),
        );
        let e4 = DoubleDouble::mul_add(
            DoubleDouble::from_bit_pair(q[9]),
            x,
            DoubleDouble::from_bit_pair(q[8]),
        );
        let e5 = DoubleDouble::mul_add(
            DoubleDouble::from_bit_pair(q[11]),
            x,
            DoubleDouble::from_bit_pair(q[10]),
        );

        let f0 = DoubleDouble::mul_add(x2, e1, e0);
        let f1 = DoubleDouble::mul_add(x2, e3, e2);
        let f2 = DoubleDouble::mul_add(x2, e5, e4);

        let g0 = DoubleDouble::mul_add(x4, f1, f0);

        let p_den = DoubleDouble::mul_add(x8, f2, g0);

        let q = DoubleDouble::div(p_num, p_den);
        let r = DoubleDouble::quick_dd_add(q, rcp2);
        return r;
    }
    // asymptotic expansion Trigamma[x] = 1/x + 1/x^2 + sum(Bernoulli(2*k)/x^(2*k + 1))
    // Generated in SageMath:
    // var('x')
    // def bernoulli_terms(x, N):
    //     S = 0
    //     for k in range(1, N+1):
    //         B = bernoulli(2*k)
    //         term = B*x**(-(2*k+1))
    //         S += term
    //     return S
    //
    // terms = bernoulli_terms(x, 10)
    // coeffs = [RealField(150)(terms.coefficient(x, n)) for n in range(0, terms.degree(x)+1, 1)]
    // for k in range(0, 14):
    //     c = terms.coefficient(x, -k)  # coefficient of x^(-k)
    //     if c == 0:
    //         continue
    //     print("f64::from_bits(" + double_to_hex(c) + "),")
    const C: [(u64, u64); 10] = [
        (0x3c65555555555555, 0x3fc5555555555555),
        (0xbc21111111111111, 0xbfa1111111111111),
        (0x3c38618618618618, 0x3f98618618618618),
        (0xbc21111111111111, 0xbfa1111111111111),
        (0xbc4364d9364d9365, 0x3fb364d9364d9365),
        (0xbc6981981981981a, 0xbfd0330330330330),
        (0xbc95555555555555, 0x3ff2aaaaaaaaaaab),
        (0xbcb7979797979798, 0xc01c5e5e5e5e5e5e),
        (0xbcac3b070ec1c3b0, 0x404b7c4f8f13e3c5),
        (0x3cc8d3018d3018d3, 0xc08088fe72cfe72d),
    ];

    let rcp = x.recip();

    let q = DoubleDouble::quick_mult(rcp, rcp);

    let q2 = DoubleDouble::quick_mult(q, q);
    let q4 = q2 * q2;
    let q8 = q4 * q4;

    let e0 = DoubleDouble::quick_mul_add(
        DoubleDouble::from_bit_pair(C[1]),
        q,
        DoubleDouble::from_bit_pair(C[0]),
    );
    let e1 = DoubleDouble::quick_mul_add(
        DoubleDouble::from_bit_pair(C[3]),
        q,
        DoubleDouble::from_bit_pair(C[2]),
    );
    let e2 = DoubleDouble::quick_mul_add(
        DoubleDouble::from_bit_pair(C[5]),
        q,
        DoubleDouble::from_bit_pair(C[4]),
    );
    let e3 = DoubleDouble::quick_mul_add(
        DoubleDouble::from_bit_pair(C[7]),
        q,
        DoubleDouble::from_bit_pair(C[6]),
    );
    let e4 = DoubleDouble::quick_mul_add(
        DoubleDouble::from_bit_pair(C[9]),
        q,
        DoubleDouble::from_bit_pair(C[8]),
    );

    let q0 = DoubleDouble::quick_mul_add(q2, e1, e0);
    let q1 = DoubleDouble::quick_mul_add(q2, e3, e2);

    let r0 = DoubleDouble::quick_mul_add(q4, q1, q0);
    let mut p = DoubleDouble::quick_mul_add(q8, e4, r0);

    let q_over_2 = DoubleDouble::quick_mult_f64(q, 0.5);
    p = DoubleDouble::quick_mult(p, q);
    p = DoubleDouble::quick_mult(p, rcp);
    p = DoubleDouble::quick_dd_add(q_over_2, p);
    p = DoubleDouble::quick_dd_add(p, rcp);
    p
}

/// Computes the trigamma function ψ₁(x).
///
/// The trigamma function is the second derivative of the logarithm of the gamma function.
pub fn f_trigamma(x: f64) -> f64 {
    let xb = x.to_bits();
    if !x.is_normal() {
        if x.is_infinite() {
            return if x.is_sign_negative() {
                f64::NEG_INFINITY
            } else {
                0.
            };
        }
        if x.is_nan() {
            return f64::NAN;
        }
        if xb == 0 {
            return f64::INFINITY;
        }
    }

    let x_e = (x.to_bits() >> 52) & 0x7ff;

    const E_BIAS: u64 = (1u64 << (11 - 1u64)) - 1u64;

    if x_e < E_BIAS - 52 {
        // |x| < 2^-52
        let dx = x;
        return 1. / (dx * dx);
    }

    if x < 0. {
        if is_integer(x) {
            return f64::INFINITY;
        }
        // reflection formula
        // Trigamma[1-x] + Trigamma[x] = PI^2 / sinpi^2(x)
        const SQR_PI: DoubleDouble =
            DoubleDouble::from_bit_pair((0x3cc692b71366cc05, 0x4023bd3cc9be45de)); // pi^2
        let sinpi_ax = f_fast_sinpi_dd(-x);
        let dx = DoubleDouble::from_full_exact_sub(1., x);
        let result = DoubleDouble::div(SQR_PI, DoubleDouble::quick_mult(sinpi_ax, sinpi_ax));
        let trigamma_x = approx_trigamma_dd(dx);
        return DoubleDouble::quick_dd_sub(result, trigamma_x).to_f64();
    }

    approx_trigamma(x).to_f64()
}

#[cfg(test)]
mod tests {

    use super::*;

    #[test]
    fn test_trigamma() {
        assert_eq!(f_trigamma(-27.058018), 300.35629698636757);
        assert_eq!(f_trigamma(27.058018), 0.037648965757704725);
        assert_eq!(f_trigamma(8.058018), 0.13211796975281037);
        assert_eq!(f_trigamma(-8.058018), 300.2758629255111);
        assert_eq!(f_trigamma(2.23432), 0.5621320243666134);
        assert_eq!(f_trigamma(-2.4653), 9.653674003034206);
        assert_eq!(f_trigamma(0.123541), 66.91128231455282);
        assert_eq!(f_trigamma(-0.54331), 9.154415950366596);
        assert_eq!(f_trigamma(-5.), f64::INFINITY);
        assert_eq!(f_trigamma(f64::INFINITY), 0.0);
        assert_eq!(f_trigamma(f64::NEG_INFINITY), f64::NEG_INFINITY);
        assert!(f_trigamma(f64::NAN).is_nan());
    }
}

Coverage Report

Created: 2025-11-11 07:15

Line	Count	Source
1		/*
2		* // Copyright (c) Radzivon Bartoshyk 8/2025. All rights reserved.
3		* //
4		* // Redistribution and use in source and binary forms, with or without modification,
5		* // are permitted provided that the following conditions are met:
6		* //
7		* // 1. Redistributions of source code must retain the above copyright notice, this
8		* // list of conditions and the following disclaimer.
9		* //
10		* // 2. Redistributions in binary form must reproduce the above copyright notice,
11		* // this list of conditions and the following disclaimer in the documentation
12		* // and/or other materials provided with the distribution.
13		* //
14		* // 3. Neither the name of the copyright holder nor the names of its
15		* // contributors may be used to endorse or promote products derived from
16		* // this software without specific prior written permission.
17		* //
18		* // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
19		* // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20		* // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
21		* // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
22		* // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23		* // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
24		* // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
25		* // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
26		* // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
27		* // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
28		*/
29		use crate::common::is_integer;
30		use crate::double_double::DoubleDouble;
31		use crate::sincospi::f_fast_sinpi_dd;
32
33		// Generated in Wolfram Mathematica:
34		// <<FunctionApproximations`
35		// ClearAll["Global`*"]
36		// f[x_]:=PolyGamma[1, x+1]
37		// {err0,approx}=MiniMaxApproximation[f[z],{z,{-0.99999999,0},11,11},WorkingPrecision->75,MaxIterations->100]
38		// num=Numerator[approx][[1]];
39		// poly=num;
40		// coeffs=CoefficientList[poly,z];
41		// TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
42		static P_1: [(u64, u64); 12] = [
43		(0x3c81873d89121fec, 0x3ffa51a6625307d3),
44		(0x3cb78bf7af507504, 0x4016a65cbac476ca),
45		(0xbc94179c65e021c2, 0x40218234a0a79582),
46		(0x3cb842a8ab5e0994, 0x401fde32175f8515),
47		(0xbc8768b33f5776b7, 0x4012de6bde49abff),
48		(0x3c8e06354a27f081, 0x3ffe5d6ef3a2eac6),
49		(0xbc8b391d09fed17a, 0x3fe0d186688252cf),
50		(0xbc59a5c46bb8b8cc, 0x3fb958f9a0f156b7),
51		(0x3c2ac44c6a197244, 0x3f88e605f24e1a89),
52		(0x3bd2d05fa8be27f2, 0x3f4cd369f5d68104),
53		(0x3b84ad0a748fdd22, 0x3efde955ebb17874),
54		(0xb96aa8b9a65e0899, 0xbce053d04459ead7),
55		];
56
57		// Generated by Wolfram Mathematica:
58		// <<FunctionApproximations`
59		// ClearAll["Global`*"]
60		// f[x_]:=PolyGamma[1, x+1]
61		// {err0,approx}=MiniMaxApproximation[f[z],{z,{0,3},11,11},WorkingPrecision->75,MaxIterations->100]
62		// num=Numerator[approx][[1]];
63		// poly=num;
64		// coeffs=CoefficientList[poly,z];
65		// TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
66		static P_2: [(u64, u64); 12] = [
67		(0x3c81873d8912236c, 0x3ffa51a6625307d3),
68		(0xbcbc731a62342288, 0x40176c23f7ea51e6),
69		(0xbcc45a6fd00e67a8, 0x4022cb5ae67657ef),
70		(0x3cc0876fde7fe4e6, 0x4021d766062b9550),
71		(0xbcaec4a4859cba1d, 0x401629f91cd4f291),
72		(0x3c76184014e4d7e3, 0x4002d43da3352004),
73		(0x3c812c7609483e0e, 0x3fe62e3266eef8c7),
74		(0xbc5f991047f52d2b, 0x3fc1eacb910b951c),
75		(0x3c28b9f38d603f2f, 0x3f930960a301df34),
76		(0x3bf9b620eb930504, 0x3f5814f8e057b14b),
77		(0xbb990860b88b54e4, 0x3f0b9f67c71aa3bf),
78		(0x38e5cb6acfbaab77, 0xbc4194b8c01afe9a),
79		];
80
81		// Generated by Wolfram Mathematica:
82		// <<FunctionApproximations`
83		// ClearAll["Global`*"]
84		// f[x_]:=PolyGamma[1, x+1]
85		// {err0,approx,err1}=MiniMaxApproximation[f[z],{z,{3,9},11,11},WorkingPrecision->75,MaxIterations->100]
86		// num=Numerator[approx];
87		// poly=num;
88		// coeffs=CoefficientList[poly,z];
89		// TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
90		static P_3: [(u64, u64); 12] = [
91		(0x3c9cd56dbb295efc, 0x3ffa51a662556db9),
92		(0x3c9f4ee74f5f9daf, 0x4018ff913088cb34),
93		(0x3ccf08737350609c, 0x402593d55686b8b1),
94		(0xbcc6cd4ed33afebb, 0x402641d10de4def5),
95		(0xbcb24d1957c1303c, 0x401e682c37e8e2cf),
96		(0x3ca30ac79162ceb2, 0x400ccfc7c4566f55),
97		(0x3c9efea5ff293dc9, 0x3ff33eb2c6e89d0b),
98		(0x3c74670a11068abc, 0x3fd1fbf456e5c6f0),
99		(0x3c47b5dcdea19c36, 0x3fa6a6a2148c482c),
100		(0xbc14642012a1cc1e, 0x3f71851e927f52e7),
101		(0x3bc7db88a4ec5478, 0x3f29a45059a43475),
102		(0xb7bc31e55271eab0, 0xbb375518529c52fb),
103		];
104
105		// Generated in Wolfram Mathematica:
106		// <<FunctionApproximations`
107		// ClearAll["Global`*"]
108		// f[x_]:=PolyGamma[1, x+1]
109		// {err0,approx}=MiniMaxApproximation[f[z],{z,{-0.99999999,0},11,11},WorkingPrecision->75,MaxIterations->100]
110		// den=Denominator[approx][[1]];
111		// poly=den;
112		// coeffs=CoefficientList[poly,z];
113		// TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
114		static Q_1: [(u64, u64); 12] = [
115		(0x0000000000000000, 0x3ff0000000000000),
116		(0xbcb84c43a11fc28a, 0x40139d9587da0fb5),
117		(0x3ca1cf3dbcddbb57, 0x402507cb6225f0f0),
118		(0x3cb01aa6ddcc3cfd, 0x402a1b416d0ed4e6),
119		(0xbcbc31c216b5ff66, 0x4024ec8829e535d4),
120		(0xbcb335c23022f43e, 0x4016d2ba6d1a18e6),
121		(0x3cafbfffc03ad28a, 0x400158c4611ed51f),
122		(0xbc8d1fb10a031a27, 0x3fe26f15bb52f89b),
123		(0x3c56a9fea160eecb, 0x3fbaec13f663049d),
124		(0xbc2f4ee869ba9364, 0x3f89cde0500cd68f),
125		(0x3be3f23afc9398b6, 0x3f4d4b0f4dcf3eb8),
126		(0x3b67a3ed4795d33e, 0x3efde955eafac9c2),
127		];
128
129		// Generated by Wolfram Mathematica:
130		// <<FunctionApproximations`
131		// ClearAll["Global`*"]
132		// f[x_]:=PolyGamma[1, x+1]
133		// {err0,approx}=MiniMaxApproximation[f[z],{z,{0,3},11,11},WorkingPrecision->75,MaxIterations->100]
134		// den=Denominator[approx][[1]];
135		// poly=den;
136		// coeffs=CoefficientList[poly,z];
137		// TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
138		static Q_2: [(u64, u64); 12] = [
139		(0x0000000000000000, 0x3ff0000000000000),
140		(0xbc81a3e4e026b7b1, 0x401415d1a20a9339),
141		(0x3cc279576dfe3ec9, 0x402627c1a95d33d2),
142		(0x3c9a94b5cf0cae88, 0x402c724dc5cf4577),
143		(0xbc8aa1fa0c3820a8, 0x4027b7a332bb07f4),
144		(0xbc96968367088d66, 0x401b14376177bdd7),
145		(0x3ca2d3dfa5847f4d, 0x4005b0511cd98f2c),
146		(0xbc8cfad394d41dd1, 0x3fe877bc2d02c7f3),
147		(0xbc51592b8ec81a92, 0x3fc31f52afc72b95),
148		(0x3c2cbef277d587e9, 0x3f93cb2f0e574376),
149		(0xbbfbb670fd94f6ba, 0x3f5883767f745a92),
150		(0xbb931b04d74e5893, 0x3f0b9f67c71a60f3),
151		];
152
153		// Generated by Wolfram Mathematica:
154		// <<FunctionApproximations`
155		// ClearAll["Global`*"]
156		// f[x_]:=PolyGamma[1, x+1]
157		// {err0,approx,err1}=MiniMaxApproximation[f[z],{z,{3,9},11,11},WorkingPrecision->75,MaxIterations->100]
158		// den=Denominator[approx];
159		// poly=den;
160		// coeffs=CoefficientList[poly,z];
161		// TableForm[Table[Row[{"'",NumberForm[coeffs[[i+1]],{50,50},ExponentFunction->(Null&)],"',"}],{i,0,Length[coeffs]-1}]]
162		static Q_3: [(u64, u64); 12] = [
163		(0x0000000000000000, 0x3ff0000000000000),
164		(0x3cbafcb4b6d646d9, 0x40150b12a79fc9cf),
165		(0x3c989ef814b8dd2a, 0x40288c1d26ffdca5),
166		(0x3cb0282cfea9c473, 0x4030d737c893cd5f),
167		(0x3cc955b8aaadb37d, 0x402e5c289b6de3e0),
168		(0x3cb377161f8861d2, 0x4022fb66d87bd522),
169		(0xbcb4b0e4cff46ad6, 0x4010e5d13c2a5907),
170		(0xbc8824539e4b1bd6, 0x3ff58c8fe8f26fca),
171		(0xbc7d34220d810ea0, 0x3fd36c1351f43e66),
172		(0xbc4cbdbe85570017, 0x3fa7c1170466605e),
173		(0xbc0c3afb98775c53, 0x3f71ebafd3e5e3b9),
174		(0x3bc0b0b7f16afd0a, 0x3f29a45059a43475),
175		];
176
177		#[inline]
178	0	fn approx_trigamma(x: f64) -> DoubleDouble {
179	0	if x <= 10. {
180	0	let (p, q) = if x <= 1. {
181	0	(&P_1, &Q_1)
182	0	} else if x <= 4. {
183	0	(&P_2, &Q_2)
184		} else {
185	0	(&P_3, &Q_3)
186		};
187	0	let x2 = DoubleDouble::from_exact_mult(x, x);
188	0	let x4 = DoubleDouble::quick_mult(x2, x2);
189	0	let x8 = DoubleDouble::quick_mult(x4, x4);
190
191	0	let e0 = DoubleDouble::mul_f64_add(
192	0	DoubleDouble::from_bit_pair(p[1]),
193	0	x,
194	0	DoubleDouble::from_bit_pair(p[0]),
195		);
196	0	let e1 = DoubleDouble::mul_f64_add(
197	0	DoubleDouble::from_bit_pair(p[3]),
198	0	x,
199	0	DoubleDouble::from_bit_pair(p[2]),
200		);
201	0	let e2 = DoubleDouble::mul_f64_add(
202	0	DoubleDouble::from_bit_pair(p[5]),
203	0	x,
204	0	DoubleDouble::from_bit_pair(p[4]),
205		);
206	0	let e3 = DoubleDouble::mul_f64_add(
207	0	DoubleDouble::from_bit_pair(p[7]),
208	0	x,
209	0	DoubleDouble::from_bit_pair(p[6]),
210		);
211	0	let e4 = DoubleDouble::mul_f64_add(
212	0	DoubleDouble::from_bit_pair(p[9]),
213	0	x,
214	0	DoubleDouble::from_bit_pair(p[8]),
215		);
216	0	let e5 = DoubleDouble::mul_f64_add(
217	0	DoubleDouble::from_bit_pair(p[11]),
218	0	x,
219	0	DoubleDouble::from_bit_pair(p[10]),
220		);
221
222	0	let f0 = DoubleDouble::mul_add(x2, e1, e0);
223	0	let f1 = DoubleDouble::mul_add(x2, e3, e2);
224	0	let f2 = DoubleDouble::mul_add(x2, e5, e4);
225
226	0	let g0 = DoubleDouble::mul_add(x4, f1, f0);
227
228	0	let p_num = DoubleDouble::mul_add(x8, f2, g0);
229
230	0	let rcp = DoubleDouble::from_quick_recip(x);
231	0	let rcp2 = DoubleDouble::quick_mult(rcp, rcp);
232
233	0	let e0 = DoubleDouble::mul_f64_add_f64(
234	0	DoubleDouble::from_bit_pair(q[1]),
235	0	x,
236	0	f64::from_bits(0x3ff0000000000000),
237		);
238	0	let e1 = DoubleDouble::mul_f64_add(
239	0	DoubleDouble::from_bit_pair(q[3]),
240	0	x,
241	0	DoubleDouble::from_bit_pair(q[2]),
242		);
243	0	let e2 = DoubleDouble::mul_f64_add(
244	0	DoubleDouble::from_bit_pair(q[5]),
245	0	x,
246	0	DoubleDouble::from_bit_pair(q[4]),
247		);
248	0	let e3 = DoubleDouble::mul_f64_add(
249	0	DoubleDouble::from_bit_pair(q[7]),
250	0	x,
251	0	DoubleDouble::from_bit_pair(q[6]),
252		);
253	0	let e4 = DoubleDouble::mul_f64_add(
254	0	DoubleDouble::from_bit_pair(q[9]),
255	0	x,
256	0	DoubleDouble::from_bit_pair(q[8]),
257		);
258	0	let e5 = DoubleDouble::mul_f64_add(
259	0	DoubleDouble::from_bit_pair(q[11]),
260	0	x,
261	0	DoubleDouble::from_bit_pair(q[10]),
262		);
263
264	0	let f0 = DoubleDouble::mul_add(x2, e1, e0);
265	0	let f1 = DoubleDouble::mul_add(x2, e3, e2);
266	0	let f2 = DoubleDouble::mul_add(x2, e5, e4);
267
268	0	let g0 = DoubleDouble::mul_add(x4, f1, f0);
269
270	0	let p_den = DoubleDouble::mul_add(x8, f2, g0);
271
272	0	let q = DoubleDouble::div(p_num, p_den);
273	0	let r = DoubleDouble::quick_dd_add(q, rcp2);
274	0	return r;
275	0	}
276		// asymptotic expansion Trigamma[x] = 1/x + 1/x^2 + sum(Bernoulli(2k)/x^(2k + 1))
277		// Generated in SageMath:
278		// var('x')
279		// def bernoulli_terms(x, N):
280		// S = 0
281		// for k in range(1, N+1):
282		// B = bernoulli(2*k)
283		// term = Bx(-(2k+1))
284		// S += term
285		// return S
286		//
287		// terms = bernoulli_terms(x, 10)
288		// coeffs = [RealField(150)(terms.coefficient(x, n)) for n in range(0, terms.degree(x)+1, 1)]
289		// for k in range(0, 14):
290		// c = terms.coefficient(x, -k) # coefficient of x^(-k)
291		// if c == 0:
292		// continue
293		// print("f64::from_bits(" + double_to_hex(c) + "),")
294		const C: [(u64, u64); 10] = [
295		(0x3c65555555555555, 0x3fc5555555555555),
296		(0xbc21111111111111, 0xbfa1111111111111),
297		(0x3c38618618618618, 0x3f98618618618618),
298		(0xbc21111111111111, 0xbfa1111111111111),
299		(0xbc4364d9364d9365, 0x3fb364d9364d9365),
300		(0xbc6981981981981a, 0xbfd0330330330330),
301		(0xbc95555555555555, 0x3ff2aaaaaaaaaaab),
302		(0xbcb7979797979798, 0xc01c5e5e5e5e5e5e),
303		(0xbcac3b070ec1c3b0, 0x404b7c4f8f13e3c5),
304		(0x3cc8d3018d3018d3, 0xc08088fe72cfe72d),
305		];
306
307	0	let rcp = DoubleDouble::from_quick_recip(x);
308
309	0	let q = DoubleDouble::quick_mult(rcp, rcp);
310
311	0	let q2 = DoubleDouble::quick_mult(q, q);
312	0	let q4 = q2 * q2;
313	0	let q8 = q4 * q4;
314
315	0	let e0 = DoubleDouble::quick_mul_add(
316	0	DoubleDouble::from_bit_pair(C[1]),
317	0	q,
318	0	DoubleDouble::from_bit_pair(C[0]),
319		);
320	0	let e1 = DoubleDouble::quick_mul_add(
321	0	DoubleDouble::from_bit_pair(C[3]),
322	0	q,
323	0	DoubleDouble::from_bit_pair(C[2]),
324		);
325	0	let e2 = DoubleDouble::quick_mul_add(
326	0	DoubleDouble::from_bit_pair(C[5]),
327	0	q,
328	0	DoubleDouble::from_bit_pair(C[4]),
329		);
330	0	let e3 = DoubleDouble::quick_mul_add(
331	0	DoubleDouble::from_bit_pair(C[7]),
332	0	q,
333	0	DoubleDouble::from_bit_pair(C[6]),
334		);
335	0	let e4 = DoubleDouble::quick_mul_add(
336	0	DoubleDouble::from_bit_pair(C[9]),
337	0	q,
338	0	DoubleDouble::from_bit_pair(C[8]),
339		);
340
341	0	let q0 = DoubleDouble::quick_mul_add(q2, e1, e0);
342	0	let q1 = DoubleDouble::quick_mul_add(q2, e3, e2);
343
344	0	let r0 = DoubleDouble::quick_mul_add(q4, q1, q0);
345	0	let mut p = DoubleDouble::quick_mul_add(q8, e4, r0);
346
347	0	let q_over_2 = DoubleDouble::quick_mult_f64(q, 0.5);
348	0	p = DoubleDouble::quick_mult(p, q);
349	0	p = DoubleDouble::quick_mult(p, rcp);
350	0	p = DoubleDouble::quick_dd_add(q_over_2, p);
351	0	p = DoubleDouble::quick_dd_add(p, rcp);
352	0	p
353	0	}
354
355		#[inline]
356	0	fn approx_trigamma_dd(x: DoubleDouble) -> DoubleDouble {
357	0	if x.hi <= 10. {
358	0	let (p, q) = if x.hi <= 1. {
359	0	(&P_1, &Q_1)
360	0	} else if x.hi <= 4. {
361	0	(&P_2, &Q_2)
362		} else {
363	0	(&P_3, &Q_3)
364		};
365	0	let x2 = DoubleDouble::quick_mult(x, x);
366	0	let x4 = DoubleDouble::quick_mult(x2, x2);
367	0	let x8 = DoubleDouble::quick_mult(x4, x4);
368
369	0	let e0 = DoubleDouble::mul_add(
370	0	DoubleDouble::from_bit_pair(p[1]),
371	0	x,
372	0	DoubleDouble::from_bit_pair(p[0]),
373		);
374	0	let e1 = DoubleDouble::mul_add(
375	0	DoubleDouble::from_bit_pair(p[3]),
376	0	x,
377	0	DoubleDouble::from_bit_pair(p[2]),
378		);
379	0	let e2 = DoubleDouble::mul_add(
380	0	DoubleDouble::from_bit_pair(p[5]),
381	0	x,
382	0	DoubleDouble::from_bit_pair(p[4]),
383		);
384	0	let e3 = DoubleDouble::mul_add(
385	0	DoubleDouble::from_bit_pair(p[7]),
386	0	x,
387	0	DoubleDouble::from_bit_pair(p[6]),
388		);
389	0	let e4 = DoubleDouble::mul_add(
390	0	DoubleDouble::from_bit_pair(p[9]),
391	0	x,
392	0	DoubleDouble::from_bit_pair(p[8]),
393		);
394	0	let e5 = DoubleDouble::mul_add(
395	0	DoubleDouble::from_bit_pair(p[11]),
396	0	x,
397	0	DoubleDouble::from_bit_pair(p[10]),
398		);
399
400	0	let f0 = DoubleDouble::mul_add(x2, e1, e0);
401	0	let f1 = DoubleDouble::mul_add(x2, e3, e2);
402	0	let f2 = DoubleDouble::mul_add(x2, e5, e4);
403
404	0	let g0 = DoubleDouble::mul_add(x4, f1, f0);
405
406	0	let p_num = DoubleDouble::mul_add(x8, f2, g0);
407
408	0	let rcp = x.recip();
409	0	let rcp2 = DoubleDouble::quick_mult(rcp, rcp);
410
411	0	let e0 = DoubleDouble::mul_add_f64(
412	0	DoubleDouble::from_bit_pair(q[1]),
413	0	x,
414	0	f64::from_bits(0x3ff0000000000000),
415		);
416	0	let e1 = DoubleDouble::mul_add(
417	0	DoubleDouble::from_bit_pair(q[3]),
418	0	x,
419	0	DoubleDouble::from_bit_pair(q[2]),
420		);
421	0	let e2 = DoubleDouble::mul_add(
422	0	DoubleDouble::from_bit_pair(q[5]),
423	0	x,
424	0	DoubleDouble::from_bit_pair(q[4]),
425		);
426	0	let e3 = DoubleDouble::mul_add(
427	0	DoubleDouble::from_bit_pair(q[7]),
428	0	x,
429	0	DoubleDouble::from_bit_pair(q[6]),
430		);
431	0	let e4 = DoubleDouble::mul_add(
432	0	DoubleDouble::from_bit_pair(q[9]),
433	0	x,
434	0	DoubleDouble::from_bit_pair(q[8]),
435		);
436	0	let e5 = DoubleDouble::mul_add(
437	0	DoubleDouble::from_bit_pair(q[11]),
438	0	x,
439	0	DoubleDouble::from_bit_pair(q[10]),
440		);
441
442	0	let f0 = DoubleDouble::mul_add(x2, e1, e0);
443	0	let f1 = DoubleDouble::mul_add(x2, e3, e2);
444	0	let f2 = DoubleDouble::mul_add(x2, e5, e4);
445
446	0	let g0 = DoubleDouble::mul_add(x4, f1, f0);
447
448	0	let p_den = DoubleDouble::mul_add(x8, f2, g0);
449
450	0	let q = DoubleDouble::div(p_num, p_den);
451	0	let r = DoubleDouble::quick_dd_add(q, rcp2);
452	0	return r;
453	0	}
454		// asymptotic expansion Trigamma[x] = 1/x + 1/x^2 + sum(Bernoulli(2k)/x^(2k + 1))
455		// Generated in SageMath:
456		// var('x')
457		// def bernoulli_terms(x, N):
458		// S = 0
459		// for k in range(1, N+1):
460		// B = bernoulli(2*k)
461		// term = Bx(-(2k+1))
462		// S += term
463		// return S
464		//
465		// terms = bernoulli_terms(x, 10)
466		// coeffs = [RealField(150)(terms.coefficient(x, n)) for n in range(0, terms.degree(x)+1, 1)]
467		// for k in range(0, 14):
468		// c = terms.coefficient(x, -k) # coefficient of x^(-k)
469		// if c == 0:
470		// continue
471		// print("f64::from_bits(" + double_to_hex(c) + "),")
472		const C: [(u64, u64); 10] = [
473		(0x3c65555555555555, 0x3fc5555555555555),
474		(0xbc21111111111111, 0xbfa1111111111111),
475		(0x3c38618618618618, 0x3f98618618618618),
476		(0xbc21111111111111, 0xbfa1111111111111),
477		(0xbc4364d9364d9365, 0x3fb364d9364d9365),
478		(0xbc6981981981981a, 0xbfd0330330330330),
479		(0xbc95555555555555, 0x3ff2aaaaaaaaaaab),
480		(0xbcb7979797979798, 0xc01c5e5e5e5e5e5e),
481		(0xbcac3b070ec1c3b0, 0x404b7c4f8f13e3c5),
482		(0x3cc8d3018d3018d3, 0xc08088fe72cfe72d),
483		];
484
485	0	let rcp = x.recip();
486
487	0	let q = DoubleDouble::quick_mult(rcp, rcp);
488
489	0	let q2 = DoubleDouble::quick_mult(q, q);
490	0	let q4 = q2 * q2;
491	0	let q8 = q4 * q4;
492
493	0	let e0 = DoubleDouble::quick_mul_add(
494	0	DoubleDouble::from_bit_pair(C[1]),
495	0	q,
496	0	DoubleDouble::from_bit_pair(C[0]),
497		);
498	0	let e1 = DoubleDouble::quick_mul_add(
499	0	DoubleDouble::from_bit_pair(C[3]),
500	0	q,
501	0	DoubleDouble::from_bit_pair(C[2]),
502		);
503	0	let e2 = DoubleDouble::quick_mul_add(
504	0	DoubleDouble::from_bit_pair(C[5]),
505	0	q,
506	0	DoubleDouble::from_bit_pair(C[4]),
507		);
508	0	let e3 = DoubleDouble::quick_mul_add(
509	0	DoubleDouble::from_bit_pair(C[7]),
510	0	q,
511	0	DoubleDouble::from_bit_pair(C[6]),
512		);
513	0	let e4 = DoubleDouble::quick_mul_add(
514	0	DoubleDouble::from_bit_pair(C[9]),
515	0	q,
516	0	DoubleDouble::from_bit_pair(C[8]),
517		);
518
519	0	let q0 = DoubleDouble::quick_mul_add(q2, e1, e0);
520	0	let q1 = DoubleDouble::quick_mul_add(q2, e3, e2);
521
522	0	let r0 = DoubleDouble::quick_mul_add(q4, q1, q0);
523	0	let mut p = DoubleDouble::quick_mul_add(q8, e4, r0);
524
525	0	let q_over_2 = DoubleDouble::quick_mult_f64(q, 0.5);
526	0	p = DoubleDouble::quick_mult(p, q);
527	0	p = DoubleDouble::quick_mult(p, rcp);
528	0	p = DoubleDouble::quick_dd_add(q_over_2, p);
529	0	p = DoubleDouble::quick_dd_add(p, rcp);
530	0	p
531	0	}
532
533		/// Computes the trigamma function ψ₁(x).
534		///
535		/// The trigamma function is the second derivative of the logarithm of the gamma function.
536	0	pub fn f_trigamma(x: f64) -> f64 {
537	0	let xb = x.to_bits();
538	0	if !x.is_normal() {
539	0	if x.is_infinite() {
540	0	return if x.is_sign_negative() {
541	0	f64::NEG_INFINITY
542		} else {
543	0	0.
544		};
545	0	}
546	0	if x.is_nan() {
547	0	return f64::NAN;
548	0	}
549	0	if xb == 0 {
550	0	return f64::INFINITY;
551	0	}
552	0	}
553
554	0	let x_e = (x.to_bits() >> 52) & 0x7ff;
555
556		const E_BIAS: u64 = (1u64 << (11 - 1u64)) - 1u64;
557
558	0	if x_e < E_BIAS - 52 {
559		// \|x\| < 2^-52
560	0	let dx = x;
561	0	return 1. / (dx * dx);
562	0	}
563
564	0	if x < 0. {
565	0	if is_integer(x) {
566	0	return f64::INFINITY;
567	0	}
568		// reflection formula
569		// Trigamma[1-x] + Trigamma[x] = PI^2 / sinpi^2(x)
570		const SQR_PI: DoubleDouble =
571		DoubleDouble::from_bit_pair((0x3cc692b71366cc05, 0x4023bd3cc9be45de)); // pi^2
572	0	let sinpi_ax = f_fast_sinpi_dd(-x);
573	0	let dx = DoubleDouble::from_full_exact_sub(1., x);
574	0	let result = DoubleDouble::div(SQR_PI, DoubleDouble::quick_mult(sinpi_ax, sinpi_ax));
575	0	let trigamma_x = approx_trigamma_dd(dx);
576	0	return DoubleDouble::quick_dd_sub(result, trigamma_x).to_f64();
577	0	}
578
579	0	approx_trigamma(x).to_f64()
580	0	}
581
582		#[cfg(test)]
583		mod tests {
584
585		use super::*;
586
587		#[test]
588		fn test_trigamma() {
589		assert_eq!(f_trigamma(-27.058018), 300.35629698636757);
590		assert_eq!(f_trigamma(27.058018), 0.037648965757704725);
591		assert_eq!(f_trigamma(8.058018), 0.13211796975281037);
592		assert_eq!(f_trigamma(-8.058018), 300.2758629255111);
593		assert_eq!(f_trigamma(2.23432), 0.5621320243666134);
594		assert_eq!(f_trigamma(-2.4653), 9.653674003034206);
595		assert_eq!(f_trigamma(0.123541), 66.91128231455282);
596		assert_eq!(f_trigamma(-0.54331), 9.154415950366596);
597		assert_eq!(f_trigamma(-5.), f64::INFINITY);
598		assert_eq!(f_trigamma(f64::INFINITY), 0.0);
599		assert_eq!(f_trigamma(f64::NEG_INFINITY), f64::NEG_INFINITY);
600		assert!(f_trigamma(f64::NAN).is_nan());
601		}
602		}