double-double/src/double-double/binary/dd-mult-dd.ts

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

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


/**
 * Returns the product of two double-double-precision floating point numbers.
 * 
 * * relative error bound: 7u^2, i.e. fl(a+b) = (a+b)(1+ϵ), 
 * where ϵ <= 7u^2, u = 0.5 * Number.EPSILON 
 * the error bound is not sharp - the worst case that could be found by the 
 * authors were 5u^2
 * 
 * * ALGORITHM 10 of https://hal.archives-ouvertes.fr/hal-01351529v3/document
 * @param x a double-double precision floating point number
 * @param y another double-double precision floating point number
 */
function ddMultDd(x: number[], y: number[]): number[] {
    const xh = x[1];
    const yh = y[1];

    //const [cl1,ch] = twoProduct(xh,yh);
    const ch = xh*yh;
    const c = f * xh; const ah = c - (c - xh); const al = xh - ah;
    const d = f * yh; const bh = d - (d - yh); const bl = yh - bh;
    const cl1 = (al*bl) - ((ch - (ah*bh)) - (al*bh) - (ah*bl));

    //return fastTwoSum(ch,cl1 + (xh*yl + xl*yh));
    const b = cl1 + (xh*y[0] + x[0]*yh);
    const xx = ch + b;

    return [b - (xx - ch), xx];
}


export { ddMultDd }