nodef/extra-math

// function powAbs(x: number, n: number): number {
//   return Math.sign(x) * Math.pow(Math.abs(x), n);
// }

/**
 * Compute the gamma function of a number (Γ).
 * @param x a number
 * @returns Γ(x); for +ve integer Γ(x) = x!
 */
// function gamma(x: number): number {
//   var x  = x - 1;
//   var e1 = Math.sqrt(Math.abs(2*Math.PI*x));
//   var e2 = powAbs(x/Math.E, x);
//   var e3 = powAbs(x*Math.sinh(1/x), x/2);
//   var e4 = Math.exp((7/324)*(1/((x**3) * (35*x**2 + 33))));
//   return e1*e2*e3*e4;
// }
// - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5840229/


/**
 * Compute the natural logarithm of the absolute value of the gamma function of a number (log-Γ).
 * @param x a number
 * @returns log(|Γ(x)|)
 */
// function lgamma(x: number): number {
//   return Math.log(Math.abs(gamma(x)));
// }
// - https://en.cppreference.com/w/cpp/numeric/math/lgamma


const ERF_A1 =  0.254829592;
const ERF_A2 = -0.284496736;
const ERF_A3 =  1.421413741;
const ERF_A4 = -1.453152027;
const ERF_A5 =  1.061405429;
const ERF_P  =  0.3275911;

/**
 * Find error function value of number (approximation).
 * @param x a number
 * @returns erf(x)
 */
export function erf(x: number): number {
  var sgn = x<0? -1:1;
  var x = Math.abs(x);
  var t = 1/(1+ ERF_P*x);
  var y = 1-(((((ERF_A5*t+ERF_A4)*t)+ERF_A3)*t+ERF_A2)*t+ERF_A1)*t*Math.exp(-x*x);
  return sgn*y;
}
// - https://github.com/AndreasMadsen/mathfn/blob/master/functions/erf.js
// - http://www.johndcook.com/cpp_erf.html
// - https://en.wikipedia.org/wiki/Error_function


const ERFC_TABLE = [
  -2.8e-17, 1.21e-16, -9.4e-17, -1.523e-15, 7.106e-15,
   3.81e-16, -1.12708e-13, 3.13092e-13, 8.94487e-13,
  -6.886027e-12, 2.394038e-12, 9.6467911e-11,
  -2.27365122e-10, -9.91364156e-10, 5.059343495e-9,
   6.529054439e-9, -8.5238095915e-8, 1.5626441722e-8,
   1.303655835580e-6, -1.624290004647e-6,
  -2.0278578112534e-5, 4.2523324806907e-5,
   3.66839497852761e-4, -9.46595344482036e-4,
  -9.561514786808631e-3, 1.9476473204185836e-2,
   6.4196979235649026e-1, -1.3026537197817094
];

/**
 * Find the complementary error function value of number (approximation).
 * @param x a number
 * @returns erfc(x)
 */
export function erfc(x: number): number {
  if(x<0) return 2-erfc(-x);
  var c = 0, d = 0, y = 2/(2+x), z = 4*y-2;
  for(var i=0, I=ERFC_TABLE.length-1; i<I; i++) {
    var t = d;
    d = z*d-c+ERFC_TABLE[i];
    c = t;
  }
  return y*Math.exp(-x*x+0.5*(ERFC_TABLE[I]+z*d)-c);
}
// - https://github.com/AndreasMadsen/mathfn/blob/master/functions/erf.js
// - http://pydoc.net/Python/timbre/1.0.0/timbre.stats/
// - https://github.com/jstat/jstat/blob/master/src/special.js
// - https://en.wikipedia.org/wiki/Error_function