puruspe 0.4.4

//! This module provides an implementation of the complex Faddeeva function.

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

use crate::{dawson, error::erfcx, INV_SQRT_PI};

/// The complex Faddeeva function, or w(z). This is Abramowitz
/// & Stegun function (7.1.3), also known as the plasma dispersion
/// function or Kramp function.
///
/// ### Definition:
/// w(z) = exp(-z^2) erfc(-iz)
///
/// ## Inputs:
/// - re: ℜ(z)
/// - im: ℑ(z)
///
/// ## Outputs:
/// A tuple of ([`f64`], [`f64`]) where the first
/// is ℜ(w(z)) and the second is ℑ(w(z)).
///
/// ### Notes on Implementation:
/// This is effectively a Rust port of `libcerf` and the
/// [Faddeeva package](http://ab-initio.mit.edu/faddeeva/)
/// written by Steven G. Johnson at MIT. (MIT License)
///
/// For sufficiently large |z|, it uses a continued-fraction expansion,
/// which is an independent reimplementation of W. Gautschi's algorithm
/// in [this paper](http://dx.doi.org/10.1137/0707012). It is roughly
/// similar to [Algorithm 680](http://dx.doi.org/10.1145/77626.77629).
///
/// For ℑ(z) near the real axis or small |z|, it uses an independent reimplementation
/// of M. Zaghloul's [Algorithm 916](http://dx.doi.org/10.1145/2049673.2049679).
pub fn faddeeva(re: f64, im: f64) -> (f64, f64) {
    use num_complex::Complex64 as c64;
    let mut res: c64;

    let z = c64::new(re, im);
    let x = re;
    let xabs = re.abs();
    let y = im;
    let yabs = im.abs();
    let z_sqared = xabs * xabs + y * y;

    /* **************************************************************** */
    /*     Case when |ℜ(z)| >> |ℑ(z)|: In this scenario we may have    */
    /* precision issues because the complex norm can be very accurate,  */
    /* but either of the components may be off by orders of magnitude.  */
    /* **************************************************************** */

    if yabs < 1e-8 * xabs {
        let wi = im_w_of_x(x);
        let e2 = if xabs > 27. {
            0.
        } else {
            f64::exp(-xabs * xabs)
        };
        if yabs == 0. {
            return (e2, wi);
        }
        return (e2 + y * 2. * (x * wi - INV_SQRT_PI), wi - 2. * x * y * e2);
    }

    /* **************************************************************** */
    /*     Case when |ℑ(z)| >> |ℜ(z)|: In this scenario we may have    */
    /* precision issues because the complex norm can be very accurate,  */
    /* but either of the components may be off by orders of magnitude.  */
    /* **************************************************************** */

    if xabs < 1e-8 * yabs {
        let wr = erfcx(y);
        if xabs == 0. {
            return (wr, 0.);
        }
        return (wr, x * (2. * (INV_SQRT_PI - y * wr)));
    }

    /* **************************************************************** */
    /*     Case when |z| -> 0: Use the MacLaurin series.                */
    /* **************************************************************** */

    if z_sqared < 0.053 {
        if z_sqared < 0.00689 {
            if z_sqared < 4e-7 {
                res = ((((
                              c64::new(5e-1, 0.) ) * z // z^4
                          + c64::new(0., -7.522_527_780_636_751e-1) ) * z // z^3
                         + c64::new(-1.0, 0.) ) * z // z^2
                        + c64::new(0., FRAC_2_SQRT_PI) ) * z // z^1
                    + 1.;

                return (res.re, res.im);
            }

            res = (((((((((((((
                                   c64::new(0., 5.344_009_079_373_427e-4) ) * z // z^13
                               + c64::new(1.388_888_888_888_889e-3, 0.) ) * z // z^12
                              + c64::new(0., -3.473_605_901_592_727_4e-3) ) * z // z^11
                             + c64::new(-8.333_333_333_333_333e-3, 0.) ) * z // z^10
                            + c64::new(0., 1.910_483_245_876e-2) ) * z // z^9
                           + c64::new(4.166_666_666_666_666_4e-2, 0.) ) * z // z^8
                          + c64::new(0., -8.597_174_606_442e-2) ) * z // z^7
                         + c64::new(-1.666_666_666_666_666_6e-1, 0.) ) * z // z^6
                        + c64::new(0., 3.009_011_112_254_700_3e-1) ) * z // z^5
                       + c64::new(5e-1, 0.) ) * z // z^4
                      + c64::new(0., -7.522_527_780_636_751e-1) ) * z // z^3
                     + c64::new(-1.0, 0.) ) * z // z^2
                    + c64::new(0., FRAC_2_SQRT_PI) ) * z // z^1
                + 1.;

            return (res.re, res.im);
        }

        res = (((((((((((((((((((
				     c64::new(0., -8.823_957_200_203_801e-7) ) * z // z^19
				 + c64::new(-2.755_731_922_398_589_3e-6, 0.) ) * z // z^18
				+ c64::new(0., 8.382_759_340_193_61e-6) ) * z // z^17
			       + c64::new(2.480_158_730_158_73e-5, 0.) ) * z // z^16
			      + c64::new(0., -7.125_345_439_164_569e-5) ) * z // z^15
			     + c64::new(-1.984_126_984_126_984e-4, 0.) ) * z // z^14
			    + c64::new(0., 5.344_009_079_373_427e-4) ) * z // z^13
			   + c64::new(1.388_888_888_888_889e-3, 0.) ) * z // z^12
			  + c64::new(0., -3.473_605_901_592_727_4e-3) ) * z // z^11
			 + c64::new(-8.333_333_333_333_333e-3, 0.) ) * z // z^10
			+ c64::new(0., 1.910_483_245_876e-2) ) * z // z^9
		       + c64::new(4.166_666_666_666_666_4e-2, 0.) ) * z // z^8
		      + c64::new(0., -8.597_174_606_442e-2) ) * z // z^7
		     + c64::new(-1.666_666_666_666_666_6e-1, 0.) ) * z // z^6
		    + c64::new(0., 3.009_011_112_254_700_3e-1) ) * z // z^5
		   + c64::new(5e-1, 0.) ) * z // z^4
		  + c64::new(0., -7.522_527_780_636_751e-1) ) * z // z^3
		 + c64::new(-1.0, 0.) ) * z // z^2
		+ c64::new(0., FRAC_2_SQRT_PI) ) * z // z^1
	    + 1.;

        return (res.re, res.im);
    }

    /* **************************************************************** */
    /*     Case when |z| -> infinity: Use the MacLaurin series.         */
    /* **************************************************************** */

    if z_sqared >= 49. {
        let xs = if y < 0. { -z.re } else { z.re }; // compute for -z if y < 0

        if z_sqared > 4.8e15 {
            // Scale to prevent overflow.
            if xabs > yabs {
                let yax = yabs / xs;
                let denom = INV_SQRT_PI / (xs + yax * yabs);
                res = c64::new(denom * yax, denom);
            } else if yabs.is_infinite() {
                res = if xabs.is_nan() || y < 0. {
                    c64::new(f64::NAN, f64::NAN)
                } else {
                    c64::new(0., 0.)
                };
                return (res.re, res.im);
            } else {
                let xya = xs / yabs;
                let denom = INV_SQRT_PI / (xya * xs + yabs);
                res = c64::new(denom, denom * xya);
            }
        } else {
            let zm2 = 1. / z_sqared; // 1/|z|^2
            let r = c64::new(yabs * zm2, xs * zm2); // i/z
            let zm4 = zm2 * zm2; // 1/|z|^4
            let r2 = c64::new(zm4 * (xs + yabs) * (xs - yabs), -2. * zm4 * xs * yabs); // 1/z^2

            if z_sqared > 540. {
                if z_sqared > 22500. {
                    res = (((1.057_855_469_152_043 * r2 // n=3
			     + 4.231_421_876_608_172_4e-1) * r2 // n=2
			    + 2.820_947_917_738_781_4e-1) * r2 // n=1
			   + 5.641_895_835_477_563e-1)
                        * r; // n=0
                } else {
                    res = (((((((((((3.787_704_007_508_795e6 * r2 // n=11
				     + 3.607_337_150_008_376e5) * r2 // n=10
				    + 3.797_197_000_008_816_4e4) * r2 // n=9
				   + 4.467_290_588_245_667e3) * r2 // n=8
				  + 5.956_387_450_994_222e2) * r2 // n=7
				 + 9.163_673_001_529_573e1) * r2 // n=6
				+ 1.666_122_363_914_467_6e1) * r2 // n=5
			       + 3.702_494_142_032_150_7) * r2 // n=4
			      + 1.057_855_469_152_043) * r2 // n=3
			     + 4.231_421_876_608_172_4e-1) * r2 // n=2
			    + 2.820_947_917_738_781_4e-1) * r2 // n=1
			   + 5.641_895_835_477_563e-1)
                        * r; // n=0
                }
            } else {
                res = (((((((((((((((((((8.824_926_094_302_537e15 * r2 // n=19
                                         + 4.770_230_321_244_615e14) * r2 // n=18
                                        + 2.725_845_897_854_065_6e13) * r2 // n=17
                                       + 1.652_027_816_881_252e12) * r2 // n=16
                                      + 1.065_824_397_987_904_4e11) * r2 // n=15
                                     + 7.350_513_089_571_754e9) * r2 // n=14
                                    + 5.444_824_510_793_892e8) * r2 // n=13
                                   + 4.355_859_608_635_114e7) * r2 // n=12
                                  + 3.787_704_007_508_795e6) * r2 // n=11
                                 + 3.607_337_150_008_376e5) * r2 // n=10
                                + 3.797_197_000_008_816_4e4) * r2 // n=9
                               + 4.467_290_588_245_667e3) * r2 // n=8
                              + 5.956_387_450_994_222e2) * r2 // n=7
                             + 9.163_673_001_529_573e1) * r2 // n=6
                            + 1.666_122_363_914_467_6e1) * r2 // n=5
                           + 3.702_494_142_032_150_7) * r2 // n=4
                          + 1.057_855_469_152_043) * r2 // n=3
                         + 4.231_421_876_608_172_4e-1) * r2 // n=2
                        + 2.820_947_917_738_781_4e-1) * r2 // n=1
                       + 5.641_895_835_477_563e-1)
                    * r; // n=0
            }
        }
        if y < 0. {
            // Use w(z) = 2.0*exp(-z*z) - w(-z),
            // but be careful of overflow in exp(-z*z) = exp(-(xs*xs-ya*ya) -2*i*xs*ya)
            res = 2.0 * c64::new((yabs - xs) * (xs + yabs), 2. * xs * y).exp() - res;
            return (res.re, res.im);
        } else {
            return (res.re, res.im);
        }
    }

    if xabs.is_nan() {
        return (xabs, xabs);
    }
    if y.is_nan() {
        return (y, y);
    }

    /* **************************************************************** */
    /*     Fall back to a Taylor series around z.                       */
    /* **************************************************************** */

    let i_tile =
        2 * ((INV_A * xabs) as isize * N_X_COVER as isize + (INV_A * yabs) as isize) as usize;
    let kp = TILES[i_tile];
    assert!(kp >= 0, "kP is always >= 0 in the libcerf code.");
    let idx = 2 * (NTAY + 1) * kp as usize;
    let t = TAYLOR_COEFFS[idx];
    let t_next = TAYLOR_COEFFS[idx + 1];
    let dz = c64::new(xabs - t, yabs - t_next);

    // Horner's method: loop stops at k=1 (not k=0), matching libcerf
    let nk = TILES[i_tile + 1];
    res = c64::new(
        TAYLOR_COEFFS[(idx as isize + 2 * nk as isize) as usize],
        TAYLOR_COEFFS[(idx as isize + 2 * nk as isize) as usize + 1],
    );
    // Equivalent to `for (int k = Nk-1; k >= 1; --k)` in libcerf.
    for k in (1..nk).rev() {
        res = res * dz
            + c64::new(
                TAYLOR_COEFFS[(idx as isize + (2 * k as isize)) as usize],
                TAYLOR_COEFFS[(idx as isize + (2 * k as isize)) as usize + 1],
            );
    }

    if y < 0. {
        if x < 0. {
            res = 2.0 * c64::new((y - x) * (x + y), -2. * x * y).exp() - res;
            return (res.re, res.im);
        }
        res = 2. * (c64::new((y - x) * (x + y), -2. * x * y)).exp() - c64::new(res.re, -(res.im));
        return (res.re, res.im);
    }
    if x < 0. {
        res = c64::new(res.re, -res.im);
    }
    (res.re, res.im)
}

/// Computes the scaled Dawson integral of x,
/// which is 2 * Dawson(x) / sqrt(pi).
/// This equals the imaginary part of the Faddeeva
/// function of x, for real x.
fn im_w_of_x(x: f64) -> f64 {
    2. * dawson(x) * INV_SQRT_PI
}

// As in erfcx_chebyshev_coeffs.rs, these are coefficients for the Taylor expansion of w().
//
// 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 INV_A: f64 = 7.000000;
const N_X_COVER: usize = 50;

const TILES: [i8; 5000] = [
    -2, -2, 3, 18, 3, 15, 3, 19, 13, 20, 13, 17, 13, 14, 13, 17, 13, 23, 29, 20, 29, 18, 29, 15,
    29, 13, 29, 15, 29, 18, 29, 23, 38, 17, 38, 15, 38, 13, 38, 13, 38, 15, 38, 18, 38, 20, 38, 23,
    48, 20, 48, 18, 48, 16, 48, 15, 48, 13, 48, 11, 48, 13, 48, 15, 48, 17, 48, 19, 48, 21, 48, 23,
    58, 19, 58, 18, 58, 16, 58, 15, 58, 13, 58, 12, 58, 10, 58, 12, 58, 13, 58, 15, 58, 16, 58, 18,
    58, 19, -2, -2, 1, 16, 0, 15, 0, 15, 8, 18, 8, 18, 13, 19, 13, 18, 13, 20, 21, 19, 21, 20, 29,
    19, 29, 17, 29, 16, 29, 18, 29, 20, 38, 21, 38, 19, 38, 17, 38, 15, 38, 15, 38, 17, 38, 19, 38,
    21, 48, 22, 48, 20, 48, 19, 48, 17, 48, 16, 48, 14, 48, 13, 48, 14, 48, 16, 48, 17, 48, 19, 48,
    21, 48, 23, 58, 19, 58, 18, 58, 17, 58, 15, 58, 14, 58, 13, 58, 12, 58, 13, 58, 14, 58, 15, 58,
    17, 58, 18, 58, 20, -2, -2, 4, 16, 0, 19, 0, 23, 8, 15, 8, 15, 15, 18, 15, 16, 15, 18, 21, 16,
    21, 16, 21, 19, 29, 20, 29, 19, 29, 20, 29, 23, 30, 19, 38, 20, 38, 19, 38, 18, 38, 18, 38, 19,
    38, 21, 38, 23, 46, 22, 46, 22, 48, 20, 48, 18, 48, 17, 48, 16, 48, 15, 48, 16, 48, 17, 48, 18,
    48, 20, 48, 22, 58, 21, 58, 20, 58, 18, 58, 17, 58, 16, 58, 15, 58, 14, 58, 14, 58, 14, 58, 15,
    58, 16, 58, 17, 58, 19, 58, 20, -2, -2, 5, 16, 2, 16, 2, 17, 8, 18, 8, 19, 15, 16, 15, 12, 15,
    16, 21, 14, 21, 14, 21, 17, 30, 19, 30, 17, 30, 15, 30, 15, 30, 17, 30, 21, 38, 21, 38, 20, 38,
    20, 38, 21, 38, 23, 46, 20, 46, 20, 46, 20, 46, 20, 48, 20, 48, 18, 48, 18, 48, 17, 48, 18, 48,
    19, 48, 20, 48, 21, 48, 23, 58, 22, 58, 20, 58, 19, 58, 18, 58, 17, 58, 16, 58, 16, 58, 15, 58,
    16, 58, 16, 58, 17, 58, 18, 58, 20, 58, 21, -2, -2, 6, 16, 2, 16, 2, 17, 11, 17, 11, 18, 15,
    18, 15, 16, 15, 19, 21, 16, 21, 16, 21, 20, 30, 18, 30, 16, 30, 13, 30, 13, 30, 16, 30, 19, 41,
    21, 38, 22, 38, 23, 41, 18, 41, 20, 46, 18, 46, 17, 46, 18, 46, 18, 46, 20, 46, 22, 48, 19, 48,
    19, 48, 20, 48, 20, 48, 21, 48, 23, 53, 22, 53, 22, 58, 21, 58, 20, 58, 19, 58, 18, 58, 18, 58,
    17, 58, 17, 58, 17, 58, 18, 58, 18, 58, 19, 58, 20, 58, 22, -2, -2, 9, 16, 7, 16, 7, 17, 11,
    14, 11, 15, 19, 18, 19, 16, 19, 18, 21, 20, 21, 23, 26, 19, 30, 19, 30, 17, 30, 15, 30, 15, 30,
    17, 30, 23, 41, 19, 41, 17, 41, 16, 41, 16, 41, 17, 46, 17, 46, 15, 46, 15, 46, 17, 46, 18, 46,
    20, 46, 23, 48, 21, 48, 21, 48, 22, 48, 23, 53, 20, 53, 20, 53, 20, 53, 21, 53, 22, 58, 20, 58,
    19, 58, 19, 58, 18, 58, 18, 58, 19, 58, 19, 58, 20, 58, 21, 58, 22, 58, 23, -2, -2, 12, 16, 7,
    16, 7, 17, 11, 18, 11, 18, 19, 16, 19, 12, 19, 16, 26, 18, 26, 16, 26, 16, 26, 18, 30, 19, 30,
    18, 30, 18, 35, 17, 35, 17, 35, 18, 41, 16, 41, 14, 41, 14, 41, 16, 46, 15, 46, 13, 46, 13, 46,
    15, 46, 17, 46, 19, 46, 23, 48, 23, 53, 20, 53, 19, 53, 19, 53, 18, 53, 18, 53, 19, 53, 20, 53,
    21, 53, 22, 58, 21, 58, 20, 58, 20, 58, 20, 58, 20, 58, 20, 58, 21, 58, 22, 58, 23, 62, 19, -2,
    -2, 14, 16, 10, 18, 10, 18, 16, 17, 16, 17, 19, 18, 19, 16, 19, 18, 26, 17, 26, 13, 26, 13, 26,
    17, 26, 23, 35, 19, 35, 16, 35, 14, 35, 15, 35, 17, 41, 15, 41, 12, 41, 12, 41, 15, 46, 14, 46,
    12, 46, 12, 46, 14, 46, 16, 46, 19, 46, 22, 53, 20, 53, 19, 53, 18, 53, 17, 53, 17, 53, 17, 53,
    17, 53, 18, 53, 19, 53, 21, 62, 19, 58, 22, 58, 21, 58, 21, 58, 21, 58, 22, 58, 22, 58, 23, 62,
    17, 62, 18, -2, -2, 17, 16, 10, 15, 10, 15, 16, 14, 16, 14, 16, 23, 23, 19, 23, 20, 26, 18, 26,
    16, 26, 16, 26, 18, 35, 20, 35, 18, 35, 15, 35, 12, 35, 12, 35, 15, 41, 16, 41, 14, 41, 14, 41,
    16, 46, 15, 46, 13, 46, 13, 46, 15, 46, 17, 46, 19, 46, 23, 53, 19, 53, 18, 53, 17, 53, 16, 53,
    15, 53, 15, 53, 16, 53, 17, 53, 18, 53, 20, 62, 18, 58, 23, 58, 23, 58, 23, 58, 23, 58, 23, 62,
    14, 62, 15, 62, 16, 62, 17, -2, -2, 18, 17, 10, 19, 10, 23, 16, 17, 16, 18, 23, 19, 23, 16, 23,
    16, 23, 19, 26, 19, 26, 19, 33, 18, 33, 20, 35, 19, 35, 16, 35, 15, 35, 15, 35, 17, 41, 17, 41,
    16, 41, 16, 41, 18, 46, 17, 46, 15, 46, 15, 46, 17, 46, 18, 46, 20, 46, 23, 53, 19, 53, 17, 53,
    16, 53, 14, 53, 13, 53, 13, 53, 14, 53, 16, 53, 17, 53, 19, 62, 17, 62, 16, 62, 15, 62, 14, 62,
    13, 62, 12, 62, 13, 62, 14, 62, 15, 62, 16, -2, -2, 18, 18, 20, 18, 20, 15, 20, 18, 25, 19, 23,
    17, 23, 14, 23, 14, 23, 17, 33, 17, 33, 15, 33, 15, 33, 18, 33, 21, 35, 18, 35, 17, 35, 17, 35,
    19, 35, 23, 41, 19, 41, 19, 41, 20, 46, 18, 46, 18, 46, 18, 46, 19, 46, 20, 46, 23, 53, 20, 53,
    18, 53, 16, 53, 15, 53, 13, 53, 11, 53, 11, 53, 13, 53, 15, 53, 17, 53, 18, 62, 17, 62, 16, 62,
    14, 62, 13, 62, 11, 62, 10, 62, 11, 62, 13, 62, 15, -2, -2, -2, -2, 22, 17, 20, 18, 20, 15, 20,
    18, 25, 17, 25, 14, 23, 16, 23, 16, 23, 20, 33, 16, 33, 13, 33, 13, 33, 16, 33, 20, 39, 20, 35,
    20, 35, 20, 35, 23, 44, 21, 41, 21, 41, 21, 46, 21, 46, 20, 46, 20, 46, 20, 46, 21, 46, 23, 53,
    21, 53, 20, 53, 18, 53, 16, 53, 15, 53, 13, 53, 11, 53, 11, 53, 13, 53, 15, 53, 17, 53, 18, 62,
    17, 62, 16, 62, 14, 62, 13, 62, 11, 62, 10, 62, 11, 62, 13, 62, 15, -2, -2, -2, -2, 22, 17, 24,
    17, 24, 14, 24, 18, 25, 17, 25, 14, 25, 17, 23, 23, 33, 20, 33, 18, 33, 15, 33, 15, 33, 18, 39,
    17, 39, 17, 39, 19, 39, 21, 44, 20, 44, 19, 44, 19, 44, 20, 44, 22, 49, 21, 49, 20, 49, 20, 49,
    19, 49, 20, 49, 21, 49, 22, 53, 19, 53, 17, 53, 16, 53, 14, 53, 13, 53, 13, 53, 14, 53, 16, 53,
    17, 53, 19, 62, 18, 62, 16, 62, 15, 62, 14, 62, 13, 62, 12, 62, 13, 62, 14, 62, 15, -2, -2, -2,
    -2, 27, 18, 24, 17, 24, 15, 24, 18, 25, 20, 25, 18, 25, 23, 31, 18, 31, 19, 33, 20, 33, 18, 33,
    18, 39, 16, 39, 15, 39, 15, 39, 17, 39, 19, 44, 18, 44, 17, 44, 17, 44, 18, 44, 20, 44, 22, 49,
    19, 49, 18, 49, 18, 49, 18, 49, 19, 49, 20, 53, 19, 53, 18, 53, 17, 53, 16, 53, 15, 53, 15, 53,
    16, 53, 17, 53, 18, 53, 20, 62, 18, 62, 17, 62, 16, 62, 15, 62, 14, 62, 14, 62, 14, 62, 15, 62,
    16, -2, -2, -2, -2, 27, 15, 28, 20, 28, 17, 28, 17, 28, 20, 31, 19, 31, 17, 31, 15, 31, 17, 31,
    20, 39, 21, 39, 18, 39, 15, 39, 12, 39, 12, 39, 15, 39, 18, 44, 16, 44, 15, 44, 15, 44, 16, 44,
    18, 44, 20, 49, 17, 49, 16, 49, 16, 49, 16, 49, 17, 49, 19, 49, 21, 53, 19, 53, 18, 53, 17, 53,
    17, 53, 17, 53, 17, 53, 18, 53, 19, 53, 21, 62, 19, 62, 18, 62, 17, 62, 16, 62, 15, 62, 15, 62,
    15, 62, 16, -2, -2, -2, -2, -2, -2, 27, 19, 28, 18, 28, 14, 28, 14, 28, 18, 31, 18, 31, 15, 31,
    12, 31, 15, 31, 18, 36, 20, 39, 19, 39, 17, 39, 15, 39, 15, 39, 17, 44, 17, 44, 15, 44, 13, 44,
    13, 44, 15, 44, 17, 49, 17, 49, 16, 49, 14, 49, 13, 49, 14, 49, 16, 49, 18, 49, 20, 49, 22, 53,
    19, 53, 19, 53, 18, 53, 18, 53, 19, 53, 20, 53, 21, 53, 23, 62, 20, 62, 19, 62, 18, 62, 17, 62,
    17, 62, 16, 62, 17, 62, 17, -2, -2, -2, -2, -2, -2, 32, 21, 28, 20, 28, 17, 28, 17, 28, 21, 31,
    20, 31, 17, 31, 15, 31, 17, 36, 18, 36, 18, 36, 19, 39, 19, 39, 17, 39, 17, 39, 19, 44, 17, 44,
    15, 44, 13, 44, 13, 44, 15, 44, 17, 49, 17, 49, 15, 49, 13, 49, 11, 49, 13, 49, 15, 49, 17, 49,
    19, 49, 21, 53, 21, 53, 20, 53, 20, 53, 20, 53, 21, 53, 22, 53, 23, 57, 20, 57, 21, 62, 20, 62,
    19, 62, 19, 62, 18, 62, 18, 62, 18, 62, 19, -2, -2, -2, -2, -2, -2, 32, 17, 32, 20, 34, 21, 34,
    19, 34, 19, 34, 21, 31, 20, 31, 19, 36, 17, 36, 15, 36, 15, 36, 17, 36, 20, 39, 20, 39, 20, 39,
    23, 44, 18, 44, 16, 44, 15, 44, 15, 44, 16, 44, 18, 49, 17, 49, 15, 49, 13, 49, 11, 49, 13, 49,
    15, 49, 17, 49, 19, 49, 21, 57, 22, 53, 22, 53, 22, 57, 19, 57, 19, 57, 19, 57, 19, 57, 19, 57,
    20, 57, 20, 57, 21, 62, 20, 62, 20, 62, 19, 62, 20, -2, -2, -2, -2, -2, -2, -2, -2, 32, 15, 32,
    19, 34, 19, 34, 16, 34, 16, 34, 19, 36, 21, 36, 19, 36, 16, 36, 13, 36, 13, 36, 16, 36, 19, 36,
    22, 47, 22, 47, 21, 44, 20, 44, 18, 44, 17, 44, 17, 44, 18, 44, 20, 49, 18, 49, 16, 49, 14, 49,
    14, 49, 14, 49, 16, 49, 18, 49, 20, 49, 22, 57, 21, 57, 20, 57, 19, 57, 18, 57, 17, 57, 17, 57,
    17, 57, 18, 57, 18, 57, 19, 57, 20, 57, 21, 62, 21, 62, 21, 62, 21, -2, -2, -2, -2, -2, -2, -2,
    -2, 32, 18, 32, 21, 34, 17, 34, 14, 34, 14, 34, 17, 34, 21, 36, 20, 36, 17, 36, 15, 36, 15, 36,
    17, 36, 20, 42, 20, 42, 22, 47, 19, 47, 18, 47, 18, 47, 19, 44, 20, 44, 21, 44, 23, 49, 19, 49,
    17, 49, 16, 49, 16, 49, 16, 49, 17, 49, 19, 49, 21, 49, 23, 57, 20, 57, 19, 57, 18, 57, 17, 57,
    16, 57, 16, 57, 16, 57, 16, 57, 17, 57, 18, 57, 19, 57, 21, 57, 22, 62, 23, 62, 23, -2, -2, -2,
    -2, -2, -2, -2, -2, 32, 22, 37, 22, 34, 19, 34, 16, 34, 16, 34, 19, 40, 22, 36, 21, 36, 19, 36,
    18, 36, 18, 42, 17, 42, 17, 42, 18, 42, 20, 47, 17, 47, 16, 47, 16, 47, 17, 47, 19, 47, 21, 49,
    22, 49, 20, 49, 19, 49, 18, 49, 18, 49, 18, 49, 19, 49, 21, 49, 23, 57, 21, 57, 19, 57, 18, 57,
    17, 57, 16, 57, 15, 57, 14, 57, 14, 57, 15, 57, 16, 57, 17, 57, 18, 57, 20, 57, 21, 57, 23, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, 37, 17, 37, 19, 34, 22, 34, 20, 34, 20, 40, 18, 40, 20, 40,
    22, 36, 22, 42, 18, 42, 16, 42, 14, 42, 14, 42, 16, 42, 19, 47, 15, 47, 14, 47, 14, 47, 15, 47,
    17, 47, 20, 47, 22, 49, 22, 49, 21, 49, 20, 49, 20, 49, 20, 49, 21, 49, 23, 57, 22, 57, 20, 57,
    19, 57, 17, 57, 16, 57, 15, 57, 14, 57, 13, 57, 13, 57, 14, 57, 15, 57, 16, 57, 18, 57, 19, 57,
    21, 57, 23, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 37, 14, 37, 18, 37, 22, 40, 18, 40, 15, 40,
    15, 40, 18, 40, 20, 42, 20, 42, 18, 42, 15, 42, 12, 42, 12, 42, 15, 42, 18, 47, 14, 47, 12, 47,
    12, 47, 14, 47, 17, 47, 19, 47, 22, 54, 21, 54, 20, 54, 20, 54, 19, 54, 19, 54, 20, 54, 21, 54,
    22, 57, 20, 57, 18, 57, 17, 57, 16, 57, 14, 57, 13, 57, 11, 57, 11, 57, 13, 57, 14, 57, 16, 57,
    17, 57, 19, 57, 20, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 37, 17, 37, 20, 37, 23, 40,
    16, 40, 13, 40, 13, 40, 16, 40, 19, 40, 23, 42, 18, 42, 16, 42, 14, 42, 14, 42, 16, 42, 19, 47,
    15, 47, 14, 47, 14, 47, 15, 47, 17, 47, 20, 47, 22, 54, 20, 54, 19, 54, 18, 54, 18, 54, 18, 54,
    18, 54, 19, 54, 20, 54, 22, 57, 18, 57, 17, 57, 16, 57, 14, 57, 13, 57, 11, 57, 11, 57, 13, 57,
    14, 57, 16, 57, 17, 57, 19, 57, 20, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 37, 20, 37,
    23, 40, 20, 40, 18, 40, 15, 40, 15, 40, 18, 40, 21, 42, 22, 42, 20, 42, 18, 42, 17, 42, 17, 42,
    18, 42, 20, 47, 17, 47, 16, 47, 16, 47, 17, 47, 19, 47, 21, 54, 20, 54, 19, 54, 18, 54, 17, 54,
    16, 54, 16, 54, 17, 54, 18, 54, 19, 54, 21, 57, 19, 57, 17, 57, 16, 57, 15, 57, 14, 57, 13, 57,
    13, 57, 14, 57, 15, 57, 16, 57, 18, 57, 19, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, 43, 22, 43, 23, 40, 22, 40, 20, 40, 18, 40, 18, 40, 20, 40, 23, 45, 21, 42, 22, 42, 21, 42,
    19, 42, 19, 42, 21, 42, 23, 47, 19, 47, 18, 47, 18, 47, 19, 47, 21, 47, 23, 54, 19, 54, 18, 54,
    16, 54, 15, 54, 14, 54, 14, 54, 15, 54, 16, 54, 18, 54, 20, 54, 21, 57, 18, 57, 17, 57, 16, 57,
    15, 57, 14, 57, 14, 57, 15, 57, 16, 57, 17, 57, 18, 57, 20, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, 43, 19, 43, 20, 43, 23, 40, 23, 40, 22, 40, 22, 45, 18, 45, 18, 45, 18, 45,
    20, 45, 22, 42, 23, 42, 23, 51, 22, 47, 23, 47, 21, 47, 21, 47, 21, 47, 22, 47, 23, 54, 20, 54,
    19, 54, 17, 54, 15, 54, 14, 54, 12, 54, 12, 54, 14, 54, 15, 54, 17, 54, 19, 54, 21, 57, 19, 57,
    18, 57, 17, 57, 16, 57, 16, 57, 16, 57, 16, 57, 17, 57, 18, 57, 19, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, 43, 16, 43, 18, 43, 21, 45, 23, 45, 20, 45, 18, 45, 16, 45,
    15, 45, 16, 45, 18, 45, 20, 45, 23, 51, 21, 51, 20, 51, 19, 51, 19, 51, 20, 51, 21, 51, 22, 54,
    22, 54, 20, 54, 18, 54, 17, 54, 15, 54, 13, 54, 11, 54, 11, 54, 13, 54, 15, 54, 17, 54, 19, 54,
    21, 57, 20, 57, 19, 57, 18, 57, 18, 57, 17, 57, 17, 57, 18, 57, 18, 57, 19, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 43, 13, 43, 17, 43, 20, 43, 23, 45, 19, 45,
    17, 45, 14, 45, 12, 45, 14, 45, 17, 45, 19, 45, 22, 51, 19, 51, 18, 51, 17, 51, 17, 51, 18, 51,
    19, 51, 21, 51, 23, 54, 20, 54, 19, 54, 17, 54, 15, 54, 14, 54, 12, 54, 12, 54, 14, 54, 15, 54,
    17, 54, 19, 54, 21, 54, 23, 57, 20, 57, 20, 57, 19, 57, 19, 57, 19, 57, 19, 57, 20, 57, 21, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 43, 16, 43, 18, 43, 21, 45,
    22, 45, 19, 45, 17, 45, 14, 45, 12, 45, 14, 45, 17, 45, 19, 45, 22, 51, 18, 51, 16, 51, 15, 51,
    15, 51, 16, 51, 18, 51, 19, 51, 21, 54, 21, 54, 19, 54, 18, 54, 16, 54, 15, 54, 14, 54, 14, 54,
    15, 54, 16, 54, 18, 54, 20, 54, 22, 61, 21, 57, 22, 57, 21, 57, 21, 57, 20, 57, 20, 57, 21, 57,
    21, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 43, 19, 43,
    21, 43, 23, 45, 23, 45, 20, 45, 18, 45, 16, 45, 15, 45, 16, 45, 18, 45, 21, 51, 18, 51, 16, 51,
    15, 51, 13, 51, 13, 51, 15, 51, 17, 51, 19, 51, 21, 51, 23, 54, 20, 54, 19, 54, 18, 54, 17, 54,
    16, 54, 16, 54, 17, 54, 18, 54, 19, 54, 21, 61, 19, 61, 19, 61, 20, 61, 21, 61, 21, 57, 22, 57,
    22, 57, 23, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, 50, 16, 50, 18, 50, 20, 50, 23, 45, 22, 45, 20, 45, 19, 45, 18, 45, 19, 45, 20, 45, 22, 51,
    18, 51, 16, 51, 14, 51, 11, 51, 11, 51, 14, 51, 16, 51, 18, 51, 20, 51, 23, 54, 22, 54, 20, 54,
    19, 54, 18, 54, 18, 54, 18, 54, 18, 54, 19, 61, 18, 61, 17, 61, 17, 61, 18, 61, 19, 61, 19, 61,
    20, 61, 21, 61, 23, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, 50, 13, 50, 16, 50, 19, 50, 22, 52, 22, 45, 23, 45, 21, 45, 21, 45, 21, 45,
    23, 51, 21, 51, 18, 51, 17, 51, 15, 51, 13, 51, 13, 51, 15, 51, 17, 51, 19, 51, 21, 51, 23, 56,
    23, 54, 22, 54, 21, 54, 20, 54, 20, 54, 20, 54, 20, 61, 17, 61, 16, 61, 16, 61, 16, 61, 17, 61,
    17, 61, 18, 61, 19, 61, 20, 61, 22, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, 50, 13, 50, 16, 50, 19, 50, 22, 52, 20, 52, 19, 52, 18, 52,
    17, 52, 18, 52, 19, 52, 21, 51, 19, 51, 18, 51, 16, 51, 15, 51, 15, 51, 16, 51, 18, 51, 20, 51,
    22, 56, 20, 56, 21, 56, 22, 54, 23, 54, 22, 54, 22, 61, 18, 61, 17, 61, 16, 61, 15, 61, 15, 61,
    15, 61, 15, 61, 16, 61, 17, 61, 18, 61, 20, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 50, 16, 50, 18, 50, 20, 50, 23, 52, 19, 52,
    17, 52, 16, 52, 15, 52, 16, 52, 17, 52, 19, 52, 21, 51, 19, 51, 18, 51, 17, 51, 17, 51, 18, 51,
    19, 56, 18, 56, 18, 56, 19, 56, 19, 56, 20, 56, 22, 61, 20, 61, 19, 61, 17, 61, 16, 61, 15, 61,
    14, 61, 13, 61, 13, 61, 14, 61, 15, 61, 16, 61, 18, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 50, 19, 50, 20, 50, 22, 52,
    20, 52, 18, 52, 16, 52, 14, 52, 12, 52, 14, 52, 16, 52, 18, 52, 20, 51, 21, 51, 20, 51, 19, 51,
    19, 51, 20, 56, 17, 56, 16, 56, 16, 56, 17, 56, 18, 56, 19, 56, 20, 56, 22, 61, 18, 61, 17, 61,
    16, 61, 14, 61, 13, 61, 12, 61, 12, 61, 13, 61, 14, 61, 16, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 50, 22, 50,
    23, 55, 22, 52, 20, 52, 17, 52, 15, 52, 13, 52, 10, 52, 13, 52, 15, 52, 17, 52, 20, 52, 22, 51,
    22, 51, 22, 56, 18, 56, 17, 56, 15, 56, 15, 56, 15, 56, 15, 56, 17, 56, 18, 56, 19, 56, 21, 61,
    18, 61, 17, 61, 15, 61, 14, 61, 12, 61, 10, 61, 10, 61, 12, 61, 14, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, 55, 17, 55, 18, 55, 20, 52, 20, 52, 18, 52, 16, 52, 14, 52, 13, 52, 14, 52, 16, 52, 18, 52,
    20, 52, 23, 56, 20, 56, 19, 56, 17, 56, 15, 56, 14, 56, 13, 56, 13, 56, 14, 56, 16, 56, 17, 56,
    19, 56, 20, 61, 18, 61, 17, 61, 16, 61, 14, 61, 13, 61, 12, 61, 12, 61, 13, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, 55, 15, 55, 16, 55, 18, 55, 20, 52, 19, 52, 17, 52, 16, 52, 15, 52, 16, 52,
    17, 52, 19, 52, 21, 56, 22, 56, 20, 56, 18, 56, 16, 56, 15, 56, 13, 56, 11, 56, 11, 56, 13, 56,
    15, 56, 16, 56, 18, 56, 20, 61, 19, 61, 18, 61, 16, 61, 15, 61, 14, 61, 13, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, 55, 13, 55, 15, 55, 17, 55, 19, 52, 21, 52, 19, 52, 18, 52,
    17, 52, 18, 52, 19, 52, 21, 52, 23, 56, 22, 56, 20, 56, 18, 56, 16, 56, 15, 56, 13, 56, 11, 56,
    11, 56, 13, 56, 15, 56, 16, 56, 18, 56, 20, 61, 20, 61, 18, 61, 17, 61, 16, 61, 15, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 55, 11, 55, 13, 55, 16, 55, 19, 55, 21, 52,
    21, 52, 20, 52, 20, 52, 20, 52, 21, 52, 23, 59, 20, 59, 21, 56, 20, 56, 19, 56, 17, 56, 16, 56,
    14, 56, 13, 56, 13, 56, 14, 56, 16, 56, 17, 56, 19, 56, 21, 56, 22, 61, 19, 61, 18, 61, 17, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 55, 13, 55, 15, 55, 17, 55,
    19, 55, 22, 59, 21, 52, 23, 59, 19, 59, 18, 59, 18, 59, 18, 59, 19, 59, 20, 59, 21, 56, 19, 56,
    18, 56, 17, 56, 16, 56, 15, 56, 15, 56, 16, 56, 17, 56, 18, 56, 20, 56, 21, 56, 23, 61, 20, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 60, 13, 60,
    14, 60, 16, 60, 18, 60, 20, 59, 20, 59, 18, 59, 17, 59, 16, 59, 16, 59, 16, 59, 17, 59, 18, 59,
    20, 56, 21, 56, 19, 56, 18, 56, 17, 56, 16, 56, 16, 56, 17, 56, 18, 56, 19, 56, 21, 56, 22, 61,
    23, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, 60, 11, 60, 13, 60, 15, 60, 17, 60, 20, 59, 18, 59, 17, 59, 16, 59, 15, 59, 14, 59, 15, 59,
    16, 59, 17, 59, 19, 59, 20, 56, 21, 56, 20, 56, 19, 56, 18, 56, 18, 56, 19, 56, 20, 56, 21, 56,
    22, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, 60, 11, 60, 13, 60, 15, 60, 17, 60, 20, 59, 18, 59, 16, 59, 15, 59, 13, 59,
    12, 59, 13, 59, 15, 59, 16, 59, 18, 59, 19, 59, 21, 56, 21, 56, 21, 56, 20, 56, 20, 56, 21, 56,
    21, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, 60, 13, 60, 15, 60, 16, 60, 18, 60, 20, 59, 17, 59, 15, 59,
    14, 59, 12, 59, 10, 59, 12, 59, 14, 59, 15, 59, 17, 59, 19, 59, 21, 56, 23, 56, 23, 56, 22, 56,
    22, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 60, 15, 60, 16, 60, 18, 60, 20, 59, 19, 59,
    17, 59, 15, 59, 14, 59, 12, 59, 10, 59, 12, 59, 14, 59, 15, 59, 17, 59, 19, 59, 21, 59, 23, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 60, 17, 60, 18, 60, 20, 60,
    21, 59, 19, 59, 18, 59, 16, 59, 15, 59, 13, 59, 12, 59, 13, 59, 15, 59, 16, 59, 18, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 60, 20, 60,
    21, 60, 22, 59, 22, 59, 20, 59, 19, 59, 17, 59, 16, 59, 15, 59, 14, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
    -2, -2, -2, -2, -2,
];

const NTAY: usize = 23;
static TAYLOR_COEFFS: [f64; 2 * 24 * 63] = [
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