import { twoSum } from "big-float-ts";
import { EFTHorner } from "./eft-horner.js";
import { HornerSum } from "./horner-sum.js";
import { HornerAbsSum } from "./horner-abs-sum.js";
import { γs } from "../../error-analysis/gamma.js";


const u = Number.EPSILON;


/**
 * Returns the result of evaluating a univariate polynomial using once compensated
 * Horner's method, including a dynamic check for faithfull rounding and a 
 * certified running error bound.
 * 
 * * once compensated means the error in the evaluation is reduced by roughly
 * `1 / Number.EPSILON` which is again roughly `2^53` - it is the same as using
 * double-double precision in a normal Horner evaluation
 * 
 * * see [Algorithms for Accurate, Validated and Fast Polynomial Evaluation, *Stef Graillat, Philippe Langlois and Nicolas Louvet*](https://projecteuclid.org/download/pdf_1/euclid.jjiam/1265033778)
 * * see also [*Philippe Langlois, Nicolas Louvet.* Faithful Polynomial Evaluation with Compensated Horner Algorithm. ARITH18: 18th IEEE International Symposium on Computer Arithmetic, Jun 2007, Montpellier, France. pp.141–149. ffhal-00107222f](https://hal.archives-ouvertes.fr/hal-00107222/document)
 * * see also [Horner's Method](https://en.wikipedia.org/wiki/Horner%27s_method)
 * 
 * @param p a polynomial with coefficients given densely as an array of double
 * floating point numbers from highest to lowest power, e.g. `[5,-3,0]` 
 * represents the polynomial `5x^2 - 3x`
 * @param x the value at which to evaluate the polynomial
 * 
 * @doc
 */
function compHornerIsFaithful(
        p: number[], x: number): { isFaithful: boolean; errBound: number; r̄: number } {

    const n = p.length-1;
    
    const { r̂, pπ, pσ } = EFTHorner(p,x);
    const ĉ = HornerSum(pπ, pσ, x);
    const [e, r̄] = twoSum(r̂, ĉ);

    const b̂ = HornerAbsSum(pπ, pσ, Math.abs(x));
    const α̂  = (γs(2*n - 1) * b̂) / ((1 - 2*(n+1) * u));
    const β̂  = (α̂ + Math.abs(e)) / (1 - 2*u);

    return { 
        isFaithful: α̂ < (u/2) * Math.abs(r̄),
        errBound: β̂,
        r̄
    };
}


export { compHornerIsFaithful }