//! The double exponential algorithm is naturally adaptive, it stops calling the integrand when the error is reduced to below the desired threshold.
//! It also does not allocate. No box, no vec, etc.
//! It has a hard coded maximum of approximately 350 function evaluations. This guarantees that the algorithm will return.
//! The error in the algorithm decreases exponentially in the number of function evaluations, specifically O(exp(-cN/log(N))). So if 350 function evaluations is not giving the desired accuracy than the programmer probably needs to give some guidance by splitting up the range at singularities or [other preparation techniques](http://www.johndcook.com/blog/2012/02/21/care-and-treatment-of-singularities/).
//!
//! This is a port of the [Fast Numerical Integration](https://www.codeproject.com/kb/recipes/fastnumericalintegration.aspx) from c++ to rust. The original code is by John D. Cook, and is licensed under the [BSD](https://opensource.org/licenses/bsd-license.php).

use num_complex::Complex64;

/// Integrate an analytic function over a finite interval.
/// f is the function to be integrated.
/// a is left limit of integration.
/// b is right limit of integration
/// target_absolute_error is the desired bound on error
///
/// # Examples
///
/// ```
/// use quadrature::integrate;
/// fn integrand(x: f64) -> f64 {
///     (-x / 5.0).exp() * x.powf(-1.0 / 3.0)
/// }
/// let o = integrate(integrand , 0.0, 10.0, 1e-6);
/// assert!((o.integral - 3.6798142583691758).abs() <= 1e-6);
/// ```
pub fn integrate<F>(f: F, a: f64, b: f64, target_absolute_error: f64) -> Complex64
where
    F: Fn(f64) -> Complex64,
{
    // Apply the linear change of variables x = ct + d
    // $$\int_a^b f(x) dx = c \int_{-1}^1 f( ct + d ) dt$$
    // c = (b-a)/2, d = (a+b)/2
    let c = 0.5 * (b - a);
    let d = 0.5 * (a + b);
    integrate_core(
        |x| {
            let out = f(c * x + d);
            if out.is_finite() {
                out
            } else {
                Complex64::new(0.0, 0.0)
            }
        },
        0.25 * target_absolute_error / c,
    )
    .scale(c)
}

/// Integrate f(x) from [-1.0, 1.0]
fn integrate_core<F>(f: F, target_absolute_error: f64) -> Complex64
where
    F: Fn(f64) -> Complex64,
{
    let mut error_estimate = ::std::f64::MAX;
    let mut num_function_evaluations = 1;
    let mut current_delta = ::std::f64::MAX;

    let mut integral = 2.0 * ::std::f64::consts::FRAC_PI_2 * f(0.0);

    for &weight in &WEIGHTS {
        let new_contribution = weight
            .iter()
            .map(|&(w, x)| w * (f(x) + f(-x)))
            .fold(Complex64::new(0.0, 0.0), |sum, x| sum + x);
        num_function_evaluations += 2 * weight.len();

        // difference in consecutive integral estimates
        let previous_delta_ln = current_delta.ln();
        current_delta = (0.5 * integral - new_contribution).norm();
        integral = 0.5 * integral + new_contribution;

        // Once convergence kicks in, error is approximately squared at each step.
        // Determine whether we're in the convergent region by looking at the trend in the error.
        if num_function_evaluations <= 13 {
            // level <= 1
            continue; // previousDelta meaningless, so cannot check convergence.
        }

        // Exact comparison with zero is harmless here.  Could possibly be replaced with
        // a small positive upper limit on the size of currentDelta, but determining
        // that upper limit would be difficult.  At worse, the loop is executed more
        // times than necessary.  But no infinite loop can result since there is
        // an upper bound on the loop variable.
        if current_delta == 0.0 {
            error_estimate = 0.0;
            break;
        }
        // previousDelta != 0 or would have been kicked out previously
        let r = current_delta.ln() / previous_delta_ln;

        if r > 1.9 && r < 2.1 {
            // If convergence theory applied perfectly, r would be 2 in the convergence region.
            // r close to 2 is good enough. We expect the difference between this integral estimate
            // and the next one to be roughly delta^2.
            error_estimate = current_delta * current_delta;
        } else {
            // Not in the convergence region.  Assume only that error is decreasing.
            error_estimate = current_delta;
        }

        if error_estimate < target_absolute_error {
            break;
        }
    }

    integral
}

#[cfg(test)]
mod tests {
    use std::f64::consts::PI;

    use super::*;

    #[test]
    fn trivial_function_works() {
        let unit_circle =
            |t: f64| -> Complex64 { Complex64::new((PI * 2.0 * t).cos(), (PI * 2.0 * t).sin()) };
        let dzdt = |t: f64| {
            Complex64::new(
                -2.0 * PI * (2.0 * PI * t).sin(),
                2.0 * PI * (2.0 * PI * t).cos(),
            )
        };
        let f = |t: f64| dzdt(t) / unit_circle(t);
        let o = integrate(f, 0.0, 1.0, 1e-14);
        println!("{}", o);
        assert!((o - Complex64::new(0.0, 2.0 * PI)).norm() < 1e-14);
    }
}

pub const WEIGHTS: [&'static [(f64, f64)]; 7] = [
    &[
        // First layer weights
        (0.230022394514788685, 0.95136796407274694573),
        (0.00026620051375271690866, 0.99997747719246159286),
        (1.3581784274539090834e-12, 0.99999999999995705839),
    ],
    &[
        // 2nd layer weights and abcissas: transformed 1/2, 3/2, 5/2
        (0.5 * 0.96597657941230114801, 0.67427149224843582608),
        (0.5 * 0.018343166989927842087, 0.99751485645722438683),
        (0.5 * 2.1431204556943039358e-7, 0.99999998887566488198),
    ],
    &[
        // 3rd layer weights and abcissas: transformed 1/4, 3/4, ...
        (0.25 * 1.3896147592472563229, 0.37720973816403417379),
        (0.25 * 0.53107827542805397476, 0.85956905868989663517),
        (0.25 * 0.076385743570832304188, 0.98704056050737689169),
        (0.25 * 0.0029025177479013135936, 0.99968826402835320905),
        (0.25 * 0.000011983701363170720047, 0.99999920473711471266),
        (0.25 * 1.1631165814255782766e-9, 0.99999999995285644818),
    ],
    &[
        // 4th layer weights and abcissas: transformed 1/8, 3/8, ...
        (0.125 * 1.5232837186347052132, 0.19435700332493543161),
        (0.125 * 1.1934630258491569639, 0.53914670538796776905),
        (0.125 * 0.73743784836154784136, 0.78060743898320029925),
        (0.125 * 0.36046141846934367417, 0.91487926326457461091),
        (0.125 * 0.13742210773316772341, 0.97396686819567744856),
        (0.125 * 0.039175005493600779072, 0.99405550663140214329),
        (0.125 * 0.0077426010260642407123, 0.99906519645578584642),
        (0.125 * 0.00094994680428346871691, 0.99990938469514399984),
        (0.125 * 0.000062482559240744082891, 0.99999531604122052843),
        (0.125 * 1.8263320593710659699e-6, 0.99999989278161241838),
        (0.125 * 1.8687282268736410132e-8, 0.99999999914270509218),
        (0.125 * 4.9378538776631926964e-11, 0.99999999999823216531),
    ],
    &[
        //  5th layer weights and abcissa: transformed 1/16, 3/16, ...
        (0.0625 * 1.5587733555333301451, 0.097923885287832333262),
        (0.0625 * 1.466014426716965781, 0.28787993274271591456),
        (0.0625 * 1.297475750424977998, 0.46125354393958570440),
        (0.0625 * 1.0816349854900704074, 0.61027365750063894488),
        (0.0625 * 0.85017285645662006895, 0.73101803479256151149),
        (0.0625 * 0.63040513516474369106, 0.82331700550640237006),
        (0.0625 * 0.44083323627385823707, 0.88989140278426019808),
        (0.0625 * 0.290240679312454185, 0.93516085752198468323),
        (0.0625 * 0.17932441211072829296, 0.96411216422354729193),
        (0.0625 * 0.10343215422333290062, 0.98145482667733517003),
        (0.0625 * 0.055289683742240583845, 0.99112699244169880223),
        (0.0625 * 0.027133510013712003219, 0.99610866543750854254),
        (0.0625 * 0.012083543599157953493, 0.99845420876769773751),
        (0.0625 * 0.0048162981439284630173, 0.99945143443527460584),
        (0.0625 * 0.0016908739981426396472, 0.99982882207287494166),
        (0.0625 * 0.00051339382406790336017, 0.99995387100562796075),
        (0.0625 * 0.00013205234125609974879, 0.99998948201481850361),
        (0.0625 * 0.000028110164327940134749, 0.99999801714059543208),
        (0.0625 * 4.8237182032615502124e-6, 0.99999969889415261122),
        (0.0625 * 6.4777566035929719908e-7, 0.99999996423908091534),
        (0.0625 * 6.5835185127183396672e-8, 0.99999999678719909830),
        (0.0625 * 4.8760060974240625869e-9, 0.99999999978973286224),
        (0.0625 * 2.5216347918530148572e-10, 0.99999999999039393352),
        (0.0625 * 8.6759314149796046502e-12, 0.99999999999970809734),
    ],
    &[
        // 6th layer weights and abcissas: transformed 1/32, 3/32, ...
        (0.03125 * 1.5677814313072218572, 0.049055967305077886315),
        (0.03125 * 1.5438811161769592204, 0.14641798429058794053),
        (0.03125 * 1.4972262225410362896, 0.24156631953888365838),
        (0.03125 * 1.4300083548722996676, 0.33314226457763809244),
        (0.03125 * 1.3452788847662516615, 0.41995211127844715849),
        (0.03125 * 1.2467012074518577048, 0.50101338937930910152),
        (0.03125 * 1.1382722433763053734, 0.57558449063515165995),
        (0.03125 * 1.0240449331118114483, 0.64317675898520470128),
        (0.03125 * 0.90787937915489531693, 0.70355000514714201566),
        (0.03125 * 0.79324270082051671787, 0.75669390863372994941),
        (0.03125 * 0.68306851634426375464, 0.80279874134324126576),
        (0.03125 * 0.57967810308778764708, 0.84221924635075686382),
        (0.03125 * 0.48475809121475539287, 0.87543539763040867837),
        (0.03125 * 0.39938474152571713515, 0.90301328151357387064),
        (0.03125 * 0.32408253961152890402, 0.92556863406861266645),
        (0.03125 * 0.258904639514053516, 0.94373478605275715685),
        (0.03125 * 0.20352399885860174519, 0.95813602271021369012),
        (0.03125 * 0.15732620348436615027, 0.96936673289691733517),
        (0.03125 * 0.11949741128869592428, 0.97797623518666497298),
        (0.03125 * 0.089103139240941462841, 0.98445883116743083087),
        (0.03125 * 0.065155533432536205042, 0.98924843109013389601),
        (0.03125 * 0.046668208054846613644, 0.99271699719682728538),
        (0.03125 * 0.032698732726609031113, 0.99517602615532735426),
        (0.03125 * 0.022379471063648476483, 0.99688031812819187372),
        (0.03125 * 0.014937835096050129696, 0.99803333631543375402),
        (0.03125 * 0.0097072237393916892692, 0.99879353429880589929),
        (0.03125 * 0.0061300376320830301252, 0.99928111192179195541),
        (0.03125 * 0.0037542509774318343023, 0.99958475035151758732),
        (0.03125 * 0.0022250827064786427022, 0.99976797159956083506),
        (0.03125 * 0.0012733279447082382027, 0.99987486504878034648),
        (0.03125 * 0.0007018595156842422708, 0.99993501992508242369),
        (0.03125 * 0.00037166693621677760301, 0.99996759306794345976),
        (0.03125 * 0.00018856442976700318572, 0.99998451990227082442),
        (0.03125 * 0.000091390817490710122732, 0.99999293787666288565),
        (0.03125 * 0.000042183183841757600604, 0.99999693244919035751),
        (0.03125 * 0.000018481813599879217116, 0.99999873547186590954),
        (0.03125 * 7.6595758525203162562e-6, 0.99999950700571943689),
        (0.03125 * 2.9916615878138787094e-6, 0.99999981889371276701),
        (0.03125 * 1.0968835125901264732e-6, 0.99999993755407837378),
        (0.03125 * 3.7595411862360630091e-7, 0.99999997987450320175),
        (0.03125 * 1.1992442782902770219e-7, 0.99999999396413420165),
        (0.03125 * 3.5434777171421953043e-8, 0.99999999832336194826),
        (0.03125 * 9.6498888961089633609e-9, 0.99999999957078777261),
        (0.03125 * 2.4091773256475940779e-9, 0.99999999989927772326),
        (0.03125 * 5.482835779709497755e-10, 0.99999999997845533741),
        (0.03125 * 1.1306055347494680536e-10, 0.99999999999582460688),
        (0.03125 * 2.0989335404511469109e-11, 0.99999999999927152627),
        (0.03125 * 3.4841937670261059685e-12, 0.99999999999988636130),
    ],
    &[
        // 7th layer weights and abcissas: transformed 1/64, 3/64, ...
        (0.015625 * 1.5700420292795931467, 0.024539763574649160379),
        (0.015625 * 1.5640214037732320999, 0.073525122985671294475),
        (0.015625 * 1.5520531698454121192, 0.12222912220155764235),
        (0.015625 * 1.5342817381543034316, 0.17046797238201051811),
        (0.015625 * 1.5109197230741697127, 0.21806347346971200463),
        (0.015625 * 1.48224329788553807, 0.26484507658344795046),
        (0.015625 * 1.4485862549613225916, 0.31065178055284596083),
        (0.015625 * 1.4103329714462590129, 0.35533382516507453330),
        (0.015625 * 1.3679105116808964881, 0.39875415046723775644),
        (0.015625 * 1.3217801174437728579, 0.44078959903390086627),
        (0.015625 * 1.2724283455378627082, 0.48133184611690504422),
        (0.015625 * 1.2203581095793582207, 0.52028805069123015958),
        (0.015625 * 1.1660798699324345766, 0.55758122826077823080),
        (0.015625 * 1.1101031939653403796, 0.59315035359195315880),
        (0.015625 * 1.0529288799552666556, 0.62695020805104287950),
        (0.015625 * 0.99504180404613271514, 0.65895099174335012438),
        (0.015625 * 0.93690461274566793366, 0.68913772506166767176),
        (0.015625 * 0.87895234555278212039, 0.71750946748732412721),
        (0.015625 * 0.82158803526696470334, 0.74407838354734739913),
        (0.015625 * 0.7651792989089561367, 0.76886868676824658459),
        (0.015625 * 0.71005590120546898385, 0.79191549237614211447),
        (0.015625 * 0.65650824613162753076, 0.81326360850297385168),
        (0.015625 * 0.60478673057840362158, 0.83296629391941087564),
        (0.015625 * 0.55510187800363350959, 0.85108400798784873261),
        (0.015625 * 0.5076251588319080997, 0.86768317577564598669),
        (0.015625 * 0.4624903980553677613, 0.88283498824466895513),
        (0.015625 * 0.41979566844501548066, 0.89661425428007602579),
        (0.015625 * 0.37960556938665160999, 0.90909831816302043511),
        (0.015625 * 0.3419537959230168323, 0.92036605303195280235),
        (0.015625 * 0.30684590941791694932, 0.93049693799715340631),
        (0.015625 * 0.27426222968906810637, 0.93957022393327475539),
        (0.015625 * 0.24416077786983990868, 0.94766419061515309734),
        (0.015625 * 0.21648020911729617038, 0.95485549580502268541),
        (0.015625 * 0.19114268413342749532, 0.96121861515111640753),
        (0.015625 * 0.16805663794826916233, 0.96682537031235585284),
        (0.015625 * 0.14711941325785693248, 0.97174454156548730892),
        (0.015625 * 0.12821973363120098675, 0.97604156025657673933),
        (0.015625 * 0.11123999898874453035, 0.97977827580061576265),
        (0.015625 * 0.096058391865189467849, 0.98301279148110110558),
        (0.015625 * 0.082550788110701737654, 0.98579936302528343597),
        (0.015625 * 0.070592469906866999352, 0.98818835380074264243),
        (0.015625 * 0.060059642358636300319, 0.99022624046752774694),
        (0.015625 * 0.05083075757257047107, 0.99195566300267761562),
        (0.015625 * 0.042787652157725676034, 0.99341551316926403900),
        (0.015625 * 0.035816505604196436523, 0.99464105571251119672),
        (0.015625 * 0.029808628117310126969, 0.99566407681695316965),
        (0.015625 * 0.024661087314753282511, 0.99651305464025377317),
        (0.015625 * 0.020277183817500123926, 0.99721334704346870224),
        (0.015625 * 0.016566786254247575375, 0.99778739195890653083),
        (0.015625 * 0.013446536605285730674, 0.99825491617199629344),
        (0.015625 * 0.010839937168255907211, 0.99863314864067747762),
        (0.015625 * 0.0086773307495391815854, 0.99893703483351217373),
        (0.015625 * 0.0068957859690660035329, 0.99917944893488591716),
        (0.015625 * 0.0054388997976239984331, 0.99937140114093768690),
        (0.015625 * 0.0042565295990178580165, 0.99952223765121720422),
        (0.015625 * 0.0033044669940348302363, 0.99963983134560036519),
        (0.015625 * 0.0025440657675291729678, 0.99973076151980848263),
        (0.015625 * 0.0019418357759843675814, 0.99980048143113838630),
        (0.015625 * 0.0014690143599429791058, 0.99985347277311141171),
        (0.015625 * 0.0011011261134519383862, 0.99989338654759256426),
        (0.015625 * 0.00081754101332469493115, 0.99992317012928932869),
        (0.015625 * 0.00060103987991147422573, 0.99994518061445869309),
        (0.015625 * 0.00043739495615911687786, 0.99996128480785666613),
        (0.015625 * 0.00031497209186021200274, 0.99997294642523223656),
        (0.015625 * 0.00022435965205008549104, 0.99998130127012072679),
        (0.015625 * 0.00015802788400701191949, 0.99998722128200062811),
        (0.015625 * 0.00011002112846666697224, 0.99999136844834487344),
        (
            0.015625 * 0.000075683996586201477788,
            0.99999423962761663478,
        ),
        (
            0.015625 * 0.000051421497447658802092,
            0.99999620334716617675,
        ),
        (0.015625 * 0.0000344921247593431977, 0.99999752962380516793),
        (
            0.015625 * 0.000022832118109036146591,
            0.99999841381096473542,
        ),
        (
            0.015625 * 0.000014908514031870608449,
            0.99999899541068996962,
        ),
        (0.015625 * 9.5981941283784710776e-6, 0.99999937270733536947),
        (0.015625 * 6.0899100320949039256e-6, 0.99999961398855024275),
        (0.015625 * 3.8061983264644899045e-6, 0.99999976602333243312),
        (0.015625 * 2.3421667208528096843e-6, 0.99999986037121459941),
        (0.015625 * 1.4183067155493917523e-6, 0.99999991800479471056),
        (0.015625 * 8.4473756384859863469e-7, 0.99999995264266446185),
        (0.015625 * 4.9458288702754198508e-7, 0.99999997311323594362),
        (0.015625 * 2.8449923659159806339e-7, 0.99999998500307631173),
        (0.015625 * 1.6069394579076224911e-7, 0.99999999178645609907),
        (0.015625 * 8.9071395140242387124e-8, 0.99999999558563361584),
        (0.015625 * 4.8420950198072369669e-8, 0.99999999767323673790),
        (0.015625 * 2.579956822953589238e-8, 0.99999999879798350040),
        (0.015625 * 1.3464645522302038796e-8, 0.99999999939177687583),
        (0.015625 * 6.8784610955899001111e-9, 0.99999999969875436925),
        (0.015625 * 3.4371856744650090511e-9, 0.99999999985405611550),
        (0.015625 * 1.6788897682161906807e-9, 0.99999999993088839501),
        (0.015625 * 8.0099784479729665356e-10, 0.99999999996803321674),
        (0.015625 * 3.7299501843052790038e-10, 0.99999999998556879008),
        (0.015625 * 1.6939457789411646876e-10, 0.99999999999364632387),
        (0.015625 * 7.4967397573818224522e-11, 0.99999999999727404948),
        (0.015625 * 3.230446433325236576e-11, 0.99999999999886126543),
        (0.015625 * 1.3542512912336274432e-11, 0.99999999999953722654),
        (0.015625 * 5.5182369468174885821e-12, 0.99999999999981720098),
        (0.015625 * 2.1835922099233609052e-12, 0.99999999999992987953),
    ],
]; // end weights