Multiply or Add big numbers quickly with this node script. 5000 digit numbers squared in < 1 minute, doubled in < 0.1 seconds.

add.js

/**
 * This node script accepts two arguments, both integers, and returns their sum.
 * 
 * The integers are string inputs split into an array of digits and operated on so 
 * they can be really big numbers. A 5000 digit number doubled takes less than 0.1 seconds
 * to compute on a mid-2014 MacBook Pro. (Node v11.11.0)
 * 
 * Usage: node add.js <int> <int>
 */

function p(...o) {
    console.log(o);
}

/**
 * @param Array a
 * @param Array b
 */
function add(arrs) {
    const sortedRows = arrs.sort(function(a, b) {
        return b.length - a.length;
    });

    const maxCol = sortedRows[0].length;

    remainder = 0;
    const sum = [];
    for (let col = (maxCol - 1); col >= 0;  col--) {
        let tmp = 0;
        for (let row = 0; row < sortedRows.length; row++) {
            const rowForVal = sortedRows[row];
            const rowLen = rowForVal.length;
            for (let t = rowLen; t < maxCol; t++) {
                rowForVal.unshift(0);
            }
            tmp += rowForVal[col];
        }
        tmp += remainder;

        const tmps = tmp.toString().split('');
        const tmpsl = tmps.length;
        sum.unshift(tmps.pop());
        if (tmpsl > 1) {
            remainder = parseInt(tmps.join(''));
        } else {
            remainder = 0;
        }
    }
    if (remainder > 0) {
        sum.unshift(remainder.toString());
    }
    return sum.map(n => parseInt(n));
}

const inputs = [];
if (process.argv.length > 1) {
    for (let i = 2; i < process.argv.length; i++) {
        inputs.push(process.argv[i].split('').map(n => parseInt(n)));
    }
}

p(add(inputs).map(n => n.toString()).join(''));

multiply.js

/**
 * This node script accepts two arguments, both integers, and returns the product.
 * 
 * The integers are string inputs split into an array of digits and operated on so 
 * they can be really big numbers. A 5000 digit number squared takes less than 1 minute
 * to compute on a mid-2014 MacBook Pro. (Node v11.11.0)
 * 
 * Usage: node multiply.js <int> <int>
 */

function p(...o) {
    console.log(o);
}

/**
 * Multiply two integers of any length.
 * @param Array a
 * @param Array b
 */
function mult(aArr, bArr) {
    const as = aArr.reverse().join('');
    const bs = bArr.reverse().join('');

    let rowa = [];
    let remainder = 0;
    let rows = [];
    for (let bx = 0; bx < bs.length; bx++) {
        const bval = bs[bx];
        rowa = bx > 0 ? Array(bx).fill(0) : [];
        for (let ax = 0; ax < as.length; ax++) {
            const aval = as[ax];
            const c = bval * aval;
            const cs = c.toString();
            const cl = cs.length;
            if (cl === 2) {
                const save = parseInt(cs[1]);
                rowa.unshift(save + remainder);
                let msg = '';
                remainder = parseInt(cs[0]);
            } else if (cl === 1) {
                const save = parseInt(cs[0]);
                rowa.unshift(save + remainder);
                remainder = 0;
            }

            if (ax === (as.length - 1)) {
                if (remainder > 0) {
                    rowa.unshift(remainder);
                    remainder = 0;
                }
            }
        }
        rows.push(rowa);
    }

    const sortedRows = rows.sort(function(a, b) {
        return b.length - a.length;
    });

    const maxCol = sortedRows[0].length;

    remainder = 0;
    const sum = [];
    for (let col = (maxCol - 1); col >= 0;  col--) {
        let tmp = 0;
        for (let row = 0; row < sortedRows.length; row++) {
            const rowForVal = sortedRows[row];
            const rowLen = rowForVal.length;
            for (let t = rowLen; t < maxCol; t++) {
                rowForVal.unshift(0);
            }
            tmp += rowForVal[col];
        }
        tmp += remainder;

        const tmps = tmp.toString().split('');
        const tmpsl = tmps.length;
        sum.unshift(tmps.pop());
        if (tmpsl > 1) {
            remainder = parseInt(tmps.join(''));
        } else {
            remainder = 0;
        }
    }
    if (remainder > 0) {
        sum.unshift(remainder.toString());
    }
    return sum.map(n => parseInt(n));
}

const a = process.argv[2];
const b = process.argv[3];

p(mult(a.split(''), b.split('')).map(n => n.toString()).join(''));