/** @internal */
const eps = Number.EPSILON;


/**
 * Returns the result of the square root of a double floating point number  
 * together with an absolute error bound where x_ is an absolute error 
 * bound on the input value.
 * * see also "A Reduced Product of Absolute and Relative Error Bounds for Floating-point Analysis"
 * by Maxime Jacquemin, Sylvie Putot, and Franck Vedrine
 * @param x numerator
 * @param x_ absolute value error bound in numerator
 */
function sqrtWithErr(
        x: number, 
        x_: number) {

    // Note: it is assumed x + x_ >= 0, else the error in x_ was wrong in the
    // first place (since we can't have a negative input to the square root)

    // estimate the result of the square root
    if (x - x_ <= 0) {
        const est = x > 0 ? Math.sqrt(x) : 0;
        return { 
            est,
            err: Math.max(Math.sqrt(x + x_) - est, est)
        }
    }

    const est = Math.sqrt(x);
    const minSqrt = Math.sqrt(x - x_);
    const maxSqrt = Math.sqrt(x + x_);
    const err = Math.max(
        Math.abs(minSqrt - est), 
        Math.abs(maxSqrt - est)
    );
    
    //err += eps*abs(est + err);
    //err = eps*abs(est + err);

    // approx relative input error
    //const rel = x_/abs(x);

    // propogated error bound
    //const err = est*(Math.sqrt(1 + rel) - 1) + u*abs(est);
    
    return { est, err };

}


export { sqrtWithErr }