double-double/src/double-mixed-double-double/double-div-double.ts

Pure double-double precision functions *with strict error bounds*.

/** @internal */
const f = 134217729;  // 2**27 + 1;


/**
 * Returns the result of dividing a double-precision floating point 
 * number by a double with the result given as a double-double.
 * This is a slight modification of ddDivDd.
 * 
 * * **!! NOT an error-free transformation !!**
 * * relative error bound: 3u^2, i.e. fl(a/b) = (a/b)(1+ϵ), where ϵ <= 3u^2, 
 * u = 0.5 * Number.EPSILON 
 * 
 * * ALGORITHM 15 of https://hal.archives-ouvertes.fr/hal-01351529v3/document
 * (slightly modified)
 * @param x dividend
 * @param y divisor
 */
function doubleDivDouble(x: number, y: number): number[] {
    const th = x/y;

    //const [πl,πh] = twoProduct(th,y);
    const πh = th*y;
    const c = f * th; const ah = c - (c - th); const al = th - ah;
    const d = f * y; const bh = d - (d - y); const bl = y - bh;
    const πl = (al*bl) - ((πh - (ah*bh)) - (al*bh) - (ah*bl));

    const δh = x - πh;  // exact operation
    const δt = δh - πl;  // exact operation

    const tl = δt/y;

    //return fastTwoSum(th,tl);
    const xx = th + tl;
    return [tl - (xx - th), xx];
}


export { doubleDivDouble }