puruspe 0.4.4

//! This module provides implementations of the error function and related functions.
//!
//! It includes the following main functions:
//! - `erf`: Calculates the error function.
//! - `erfc`: Calculates the complementary error function.
//! - `erfcx`: Calculates the scaled complementary error function.
//! - `inverf`: Calculates the inverse of the error function.
//! - `inverfc`: Calculates the inverse of the complementary error function.

use core::f64::consts::FRAC_2_SQRT_PI;

use crate::{utils::frexp, INV_SQRT_PI};

/// Calculates the error function.
///
/// The error function is defined as:
///
/// $$ \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt $$
///
/// # Arguments
///
/// * `x` - The input value
///
/// # Returns
///
/// The value of the error function at `x`
pub fn erf(x: f64) -> f64 {
    if x >= 0f64 {
        1.0 - erfccheb(x)
    } else {
        erfccheb(-x) - 1f64
    }
}

/// Calculates the complementary error function.
///
/// The complementary error function is defined as:
///
/// $$ \text{erfc}(x) = 1 - \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} dt $$
///
/// # Arguments
///
/// * `x` - The input value
///
/// # Returns
///
/// The value of the complementary error function at `x`
pub fn erfc(x: f64) -> f64 {
    if x >= 0f64 {
        erfccheb(x)
    } else {
        2f64 - erfccheb(-x)
    }
}

/// Calculates the inverse of the complementary error function.
///
/// This function finds x such that:
///
/// $$ p = \text{erfc}(x) = 1 - \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} dt $$
///
/// # Arguments
///
/// * `p` - The probability value (between 0 and 2)
///
/// # Returns
///
/// The value of $x$ for which $\text{erfc}(x) = p$
pub fn inverfc(p: f64) -> f64 {
    // Return arbitrary large pos or neg value
    if p >= 2f64 {
        return -100f64;
    } else if p <= 0f64 {
        return 100f64;
    }

    let pp = if p < 1f64 { p } else { 2f64 - p };
    let t = (-2f64 * (pp / 2f64).ln()).sqrt();
    let mut x = -std::f64::consts::FRAC_1_SQRT_2
        * ((2.30753 + t * 0.27061) / (1f64 + t * (0.99229 + t * 0.04481)) - t);
    for _j in 0..2 {
        let err = erfc(x) - pp;
        x += err / (std::f64::consts::FRAC_2_SQRT_PI * (-x.powi(2)).exp() - x * err);
    }
    if p < 1f64 {
        x
    } else {
        -x
    }
}

/// Calculates the inverse of the error function.
///
/// This function finds x such that:
///
/// $$ p = \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt $$
///
/// # Arguments
///
/// * `p` - The probability value (between -1 and 1)
///
/// # Returns
///
/// The value of $x$ for which $\text{erf}(x) = p$
pub fn inverf(p: f64) -> f64 {
    inverfc(1f64 - p)
}

/// Helper function for erfcx(x).
fn cheb_interpolant(x: f64) -> f64 {
    // Application-specific constants:
    const M: i32 = 6; // 2^M subranges
    let j0 = -2; // first octave runs from 2^(j0-1) to 2^j0
    let l0 = 0; // index of x_min in full first octave
    let loff = (j0 + 1) * (1 << M) + l0; // precomputed offset

    // For given x, obtain mantissa xm and exponent je:
    let (xm, je) = frexp(x); // sets xm and je

    // Integer arithmetics to obtain reduced coordinate t:
    let ip = ((1 << (M + 1)) as f64 * xm) as i32; // index in octave + 2^M
    let lij = je * (1 << M) + ip - loff; // index in lookup table
    let t = ((1 << (M + 2)) as f64 * xm) - (1 + 2 * ip) as f64;

    let p_idx = lij * 8;
    let q_idx = lij * 2;

    assert!(
        p_idx >= 0 && q_idx >= 0,
        "Negative indices not permitted in Rust."
    );

    // hard-coded for N1cheb=10
    ((((((((CHEBYSHEV_COEFFS_1[(p_idx + 7) as usize] * t
        + CHEBYSHEV_COEFFS_1[(p_idx + 6) as usize])
        * t
        + CHEBYSHEV_COEFFS_1[(p_idx + 5) as usize])
        * t
        + CHEBYSHEV_COEFFS_1[(p_idx + 4) as usize])
        * t
        + CHEBYSHEV_COEFFS_1[(p_idx + 3) as usize])
        * t
        + CHEBYSHEV_COEFFS_1[(p_idx + 2) as usize])
        * t
        + CHEBYSHEV_COEFFS_1[(p_idx + 1) as usize])
        * t
        + CHEBYSHEV_COEFFS_1[p_idx as usize])
        * t
        + CHEBYSHEV_COEFFS_0[(q_idx + 1) as usize])
        * t
        + CHEBYSHEV_COEFFS_0[q_idx as usize]
}

/// The scaled complementary error function, or erfcx(x).
///
/// ### Definition:
/// erfcx(x) = erfc(x) / exp(-x^2) = (1 - erf(x)) / exp(-x^2)
///
/// ### Input:
/// - x: A real number
///
/// ### Output:
/// The result of the computation, expected to match `libcerf` to at least 1e-15.
///
/// ### Notes on Implementation:
/// This is a Rust port of the erfcx routine found in [`libcerf`](https://jugit.fz-juelich.de/mlz/libcerf),
/// which is under the MIT License.
///
/// It uses the following methods:
/// - Asymptotic expansion for large positive x,
/// - Chebyshev polynomials for medium positive x,
/// - Maclaurin series for small |x|,
/// - Asymptote 2exp(x^2) for large negative x,
/// - 2*exp(x^2)-erfcx(-x) for medium negative x.
pub fn erfcx(x: f64) -> f64 {
    let ax = x.abs();

    if ax < 0.125 {
        /* **************************************************************** */
        /* Use the Maclaurin series.                                        */
        /* **************************************************************** */
        return (((((((((((((1.984_126_984_126_984e-4 * x - 5.344_009_079_373_427e-4)
            * x
            + 1.388_888_888_888_889e-3)
            * x
            - 3.473_605_901_592_727_4e-3)
            * x
            + 8.333_333_333_333_333e-3)
            * x
            - 1.910_483_245_876e-2)
            * x
            + 4.166_666_666_666_666_4e-2)
            * x
            - 8.597_174_606_442e-2)
            * x
            + 1.666_666_666_666_666_6e-1)
            * x
            - 3.009_011_112_254_700_3e-1)
            * x
            + 5e-1)
            * x
            - 7.522_527_780_636_751e-1)
            * x
            + 1.0)
            * x
            - FRAC_2_SQRT_PI)
            * x
            + 1.0;
    }

    if x < 0. {
        if x < -26.7 {
            return f64::INFINITY;
        }
        if x < -6.1 {
            return 2. * (x * x).exp();
        }
        return 2. * (x * x).exp() - cheb_interpolant(-x);
    }

    if x <= 12. {
        return cheb_interpolant(x);
    }

    /* **************************************************************** */
    /*     Use asymptotic expansion.                                    */
    /* Coefficients are a_0 = 1 / sqrt(PI),                             */
    /*                  a_N = (2N-1)!! / 2^N / sqrt(PI)                 */
    /* **************************************************************** */
    let r = 1. / x;

    if x < 150. {
        if x < 23.2 {
            return ((((((((((3.607_337_150_008_376e5 * (r * r)
                - 3.797_197_000_008_816_4e4)
                * (r * r)
                + 4.467_290_588_245_667e3)
                * (r * r)
                - 5.956_387_450_994_222e2)
                * (r * r)
                + 9.163_673_001_529_573e1)
                * (r * r)
                - 1.666_122_363_914_467_6e1)
                * (r * r)
                + 3.702_494_142_032_150_7)
                * (r * r)
                - 1.057_855_469_152_043)
                * (r * r)
                + 4.231_421_876_608_172_4e-1)
                * (r * r)
                - 2.820_947_917_738_781_4e-1)
                * (r * r)
                + 5.641_895_835_477_563e-1)
                * r;
        }
        return ((((((9.163_673_001_529_573e1 * (r * r) - 1.666_122_363_914_467_6e1) * (r * r)
            + 3.702_494_142_032_150_7)
            * (r * r)
            - 1.057_855_469_152_043)
            * (r * r)
            + 4.231_421_876_608_172_4e-1)
            * (r * r)
            - 2.820_947_917_738_781_4e-1)
            * (r * r)
            + 5.641_895_835_477_563e-1)
            * r;
    }
    if x < 6.9e7 {
        return ((((-1.057_855_469_152_043) * (r * r) + 4.231_421_876_608_172_4e-1) * (r * r)
            - 2.820_947_917_738_781_4e-1)
            * (r * r)
            + 5.641_895_835_477_563e-1)
            * r;
    }
    // 1-term expansion, important to avoid overflow
    INV_SQRT_PI / x
}

// =============================================================================
// Chebyshev coefficients
// =============================================================================
const NCOEF: usize = 28;
const COF: [f64; 28] = [
    -1.3026537197817094,
    6.419_697_923_564_902e-1,
    1.9476473204185836e-2,
    -9.561_514_786_808_63e-3,
    -9.46595344482036e-4,
    3.66839497852761e-4,
    4.2523324806907e-5,
    -2.0278578112534e-5,
    -1.624290004647e-6,
    1.303655835580e-6,
    1.5626441722e-8,
    -8.5238095915e-8,
    6.529054439e-9,
    5.059343495e-9,
    -9.91364156e-10,
    -2.27365122e-10,
    9.6467911e-11,
    2.394038e-12,
    -6.886027e-12,
    8.94487e-13,
    3.13092e-13,
    -1.12708e-13,
    3.81e-16,
    7.106e-15,
    -1.523e-15,
    -9.4e-17,
    1.21e-16,
    -2.8e-17,
];

/// Chebyshev coefficients
///
/// # Panics
///
/// Panics if `z` < 0.
fn erfccheb(z: f64) -> f64 {
    let mut d = 0f64;
    let mut dd = 0f64;

    assert!(z >= 0f64, "erfccheb requires nonnegative argument");
    let t = 2f64 / (2f64 + z);
    let ty = 4f64 * t - 2f64;
    for j in (1..NCOEF - 1).rev() {
        let tmp = d;
        d = ty * d - dd + COF[j];
        dd = tmp;
    }
    t * (-z.powi(2) + 0.5 * (COF[0] + ty * d) - dd).exp()
}

// Literally just a whole bunch of coefficients for the Chebyshev expansion of erfcx().
//
// The `libcerf` code has these given in hexadecimal floating-point literals, which Rust
// does not currently support in the stable release. I parsed them using the following
// code:
// ```Rust
// use std::io::*;
// use hexf_parse::*;
//
// fn main() {
//     for line in stdin().lock().lines() {
//         println!("{}", hex_to_decimal(line.unwrap().trim()));
//     }
// }
//
// fn hex_to_decimal(hex: &str) -> f64 {
//     parse_hexf64(hex, false).unwrap()
// }
// ```
//
// Make sure to add the `hexf_parse` crate, and then pipe the hexadecimal numbers
// to the program. Assuming you have them in a file, with all whitespace and commas
// removed, and each number on a separate line, this works:
// ```bash
// cargo b --release
// cat coeffs.txt | ./target/release/hexfloat > results.txt
// ```

const CHEBYSHEV_COEFFS_0: [f64; 448 * 2] = [
    0.8723338250030206,
    -0.0008872968103933578,
    0.8705621184027384,
    -0.0008844118042796015,
    0.8687961697483176,
    -0.0008815388545539402,
    0.8670359549867388,
    -0.0008786779015153745,
    0.8652814501840489,
    -0.0008758288857988123,
    0.8635326315246904,
    -0.0008729917483729752,
    0.8617894753108368,
    -0.0008701664305383194,
    0.8600519579617303,
    -0.0008673528739249706,
    0.8583200560130249,
    -0.0008645510204906727,
    0.8565937461161331,
    -0.0008617608125187508,
    0.8548730050375775,
    -0.0008589821926160877,
    0.8531578096583441,
    -0.0008562151037111138,
    0.851448136973243,
    -0.0008534594890518116,
    0.8497439640902698,
    -0.0008507152922037327,
    0.8480452682299734,
    -0.0008479824570480282,
    0.8463520267248265,
    -0.0008452609277794933,
    0.8446642170185998,
    -0.0008425506489046247,
    0.8429818166657415,
    -0.0008398515652396899,
    0.8413048033307585,
    -0.000837163621908812,
    0.8396331547876036,
    -0.0008344867643420651,
    0.8379668489190648,
    -0.0008318209382735833,
    0.8363058637161596,
    -0.0008291660897396833,
    0.8346501772775319,
    -0.0008265221650769985,
    0.8329997678088539,
    -0.0008238891109206256,
    0.8313546136222306,
    -0.000821266874202285,
    0.8297146931356083,
    -0.0008186554021484919,
    0.828079984872187,
    -0.000816054642278741,
    0.8264504674598366,
    -0.0008134645424037024,
    0.8248261196305154,
    -0.0008108850506234304,
    0.8232069202196942,
    -0.0008083161153255836,
    0.8215928481657822,
    -0.0008057576851836578,
    0.8199838825095569,
    -0.0008032097091552296,
    0.8183800023935985,
    -0.000800672136480213,
    0.8167811870617263,
    -0.0007981449166791264,
    0.8151874158584393,
    -0.0007956279995513723,
    0.8135986682283604,
    -0.0007931213351735281,
    0.8120149237156833,
    -0.000790624873897648,
    0.8104361619636239,
    -0.0007881385663495765,
    0.8088623627138738,
    -0.0007856623634272733,
    0.807293505806058,
    -0.0007831962162991493,
    0.8057295711771956,
    -0.0007807400764024134,
    0.8041705388611637,
    -0.0007782938954414307,
    0.8026163889881652,
    -0.0007758576253860916,
    0.8010671017841989,
    -0.0007734312184701913,
    0.7995226575705333,
    -0.0007710146271898207,
    0.7979830367631838,
    -0.0007686078043017676,
    0.7964482198723931,
    -0.000766210702821929,
    0.7949181875021141,
    -0.0007638232760237331,
    0.7933929203494965,
    -0.0007614454774365724,
    0.7918723992043768,
    -0.0007590772608442476,
    0.7903566049487708,
    -0.0007567185802834204,
    0.7888455185563691,
    -0.0007543693900420783,
    0.7873391210920364,
    -0.0007520296446580084,
    0.7858373937113136,
    -0.0007496992989172816,
    0.784340317659922,
    -0.0007473783078527468,
    0.7828478742732718,
    -0.0007450666267425354,
    0.781360044975973,
    -0.0007427642111085756,
    0.7798768112813492,
    -0.0007404710167151157,
    0.7783981547909542,
    -0.0007381869995672581,
    0.7769240571940921,
    -0.0007359121159095033,
    0.7754545002673398,
    -0.0007336463222243019,
    0.7739894658740724,
    -0.000731389575230618,
    0.7725289359639921,
    -0.0007291418318825005,
    0.7710728925726587,
    -0.0007269030493676652,
    0.7688972006211272,
    -0.0014471231681543412,
    0.7660118146744753,
    -0.0014382745289161297,
    0.7631440557264844,
    -0.0014294960583864501,
    0.7602937841039766,
    -0.0014207870935242794,
    0.7574608614526956,
    -0.0014121469784349968,
    0.754645150723099,
    -0.001403575064284824,
    0.7518465161563217,
    -0.0013950707092163837,
    0.7490648232703065,
    -0.0013866332782653608,
    0.7462999388461005,
    -0.0013782621432782487,
    0.7435517309143146,
    -0.0013699566828311689,
    0.7408200687417447,
    -0.001361716282149744,
    0.738104822818152,
    -0.001353540333030016,
    0.7354058648432005,
    -0.0013454282337603876,
    0.7327230677135489,
    -0.0013373793890445796,
    0.7300563055100968,
    -0.0013293932099255829,
    0.7274054534853814,
    -0.0013214691137105985,
    0.7247703880511228,
    -0.0013136065238969455,
    0.7221509867659175,
    -0.0013058048700989276,
    0.7195471283230767,
    -0.0012980635879756436,
    0.7169586925386079,
    -0.0012903821191597266,
    0.7143855603393386,
    -0.001282759911187004,
    0.7118276137511794,
    -0.0012751964174270592,
    0.7092847358875253,
    -0.001267691097014689,
    0.7067568109377925,
    -0.0012602434147822382,
    0.7042437241560913,
    -0.0012528528411928027,
    0.7017453618500296,
    -0.0012455188522742912,
    0.6992616113696493,
    -0.0012382409295543266,
    0.6967923610964906,
    -0.0012310185599959825,
    0.6943375004327855,
    -0.0012238512359343406,
    0.6918969197907757,
    -0.001216738455013856,
    0.6894705105821566,
    -0.001209679720126523,
    0.6870581652076425,
    -0.0012026745393508263,
    0.6846597770466549,
    -0.0011957224258914703,
    0.6822752404471294,
    -0.0011888228980198744,
    0.6799044507154407,
    -0.001181975479015422,
    0.6775473041064454,
    -0.0011751796971074568,
    0.6752036978136393,
    -0.0011684350854180123,
    0.6728735299594285,
    -0.0011617411819052675,
    0.6705566995855131,
    -0.0011550975293077165,
    0.6682531066433821,
    -0.0011485036750890434,
    0.6659626519849173,
    -0.0011419591713836936,
    0.6636852373531056,
    -0.00113546357494313,
    0.6614207653728582,
    -0.0011290164470827667,
    0.6591691395419355,
    -0.0011226173536295705,
    0.6569302642219753,
    -0.0011162658648703199,
    0.6547040446296247,
    -0.001109961555500514,
    0.6524903868277735,
    -0.0011037040045739231,
    0.6502891977168868,
    -0.0010974927954527699,
    0.6481003850264381,
    -0.0010913275157585346,
    0.6459238573064391,
    -0.001085207757323376,
    0.6437595239190667,
    -0.0010791331161421579,
    0.6416072950303848,
    -0.0010731031923250757,
    0.6394670816021605,
    -0.0010671175900508712,
    0.6373387953837737,
    -0.0010611759175206327,
    0.6352223489042175,
    -0.0010552777869121677,
    0.6331176554641905,
    -0.001049422814334943,
    0.631024629128278,
    -0.0010436106197855852,
    0.6289431847172209,
    -0.0010378408271039317,
    0.6268732378002742,
    -0.0010321130639296269,
    0.624814704687649,
    -0.0010264269616592553,
    0.6227675024230419,
    -0.001020782155404005,
    0.6207315487762476,
    -0.0010151782839478536,
    0.618706762235854,
    -0.0010096149897062695,
    0.6166930620020209,
    -0.0010040919186854239,
    0.6136931234582421,
    -0.0019917639301181753,
    0.6097312657440812,
    -0.001970145800041228,
    0.6058123339966982,
    -0.0019488370656330794,
    0.6019357147752371,
    -0.0019278323942129086,
    0.5981008051981913,
    -0.00190712655871806,
    0.5943070127345129,
    -0.001886714435367685,
    0.5905537549993384,
    -0.001866591001383023,
    0.5868404595542178,
    -0.0018467513327628312,
    0.58316656371174,
    -0.0018271906021125274,
    0.5795315143444458,
    -0.0018079040765256369,
    0.575934767697928,
    -0.0017888871155161844,
    0.5723757892080168,
    -0.001770135169000704,
    0.5688540533219519,
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