/** `2 * 2^-53` -> 2x the standard round-of unit `=== Number.EPSILON` */
const eps = Number.EPSILON;

/** `2^-53` -> the standard round-of unit `=== eps/2` */
const u = eps / 2; 

/** `2^-106` -> the standard round-of unit for double-double precision `=== (eps/2)**2` */
const uu = u*u;


/**
 * the standard floating point error function evaluated at `1`
 * 
 * * `=== 1.1102230246251568e-16`
 * * very close to being `=== Number.EPSILON / 2`
 */ 
const γ1 = γ(1);
/**
 * the standard double-double floating point error function evaluated at `3`;
 * the `3` factor is due to double-double arithmetic accuracy constraints
 * 
 * * `=== 3.697785493223493e-32`
 * * very close to being `=== 3*(Number.EPSILON / 2)**2`
 */ 
const γγ3 = γγ(3);



/**
 * The canonical floating point error function, γ.
 * 
 * * very close to being `=== n * (Number.EPSILON / 2)`
 * * see e.g. [Algorithms for Accurate, Validated and Fast Polynomial Evaluation](https://hal.archives-ouvertes.fr/hal-00285603/document)
 * @param n the parameter - typically a small positive integer, e.g. for 
 * polynomial evaluation this `=== 2*d + 1`, where `d` is the degree of the 
 * polynomial
 * 
 * @doc
 */
function γ(n: number): number {
    const nu = n*u;

    return nu/(1 - nu);
}

/**
 * The canonical, once compensated (implying double-double precision), 
 * floating point error function.
 * 
 * * very close to being `=== n * (Number.EPSILON / 2)**2`
 * * see e.g. [Algorithms for Accurate, Validated and Fast Polynomial Evaluation](https://hal.archives-ouvertes.fr/hal-00285603/document)
 * @param n the parameter - typically a small positive integer, e.g. for 
 * polynomial evaluation this === 2*d + 1, where d is the degree of the 
 * polynomial
 * 
 * @doc
 */
function γγ(n: number): number {
    const nuu = n*uu;
    
    return nuu/(1-nuu);
}


// cache some standard error bound units
const _γs: number[] = new Array(20).fill(0).map((_,i) => γ(i));


/** @internal */
function γs(n: number): number {
    return _γs[n] || γ(n);
}


export { γ, γγ, γs, γ1, γγ3, u, uu, eps }