double-double/src/double-mixed-double-double/dd-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-double-precision floating point 
 * number by a double.
 * 
 * * relative error bound: 3u^2, i.e. fl(a/b) = (a/b)(1+ϵ), where ϵ <= 3u^2, 
 * u = 0.5 * Number.EPSILON 
 * * the bound is very sharp
 * 
 * * ALGORITHM 15 of https://hal.archives-ouvertes.fr/hal-01351529v3/document
 * @param x a double-double precision floating point number
 * @param y the double-precision divisor
 */
function ddDivDouble(x: number[], y: number): number[] {
    const xl = x[0];
    const xh = x[1];

    const th = xh/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 = xh - πh;  // exact operation
    const δt = δh - πl;  // exact operation
    const δ = δt + xl;
    const tl = δ/y;

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


export { ddDivDouble }