✨ mulSqrt

[Vectorized]

Description

Short version:

/// @dev Returns `sqrt(x * y)`. Also called the geometric mean.
function mulSqrt(uint256 x, uint256 y) internal pure returns (uint256 z) {
    if (x == y) return x;
    uint256 p = rawMul(x, y);
    if (y == rawDiv(p, x)) return sqrt(p);
    for (z = saturatingMul(rawAdd(sqrt(x), 1), rawAdd(sqrt(y), 1));; z = avg(z, p)) {
        if ((p = fullMulDivUnchecked(x, y, z)) >= z) break;
    }
}

[PASS] testMulSqrt() (gas: 287258)
[PASS] test__codesize() (gas: 62144)

Inlined version:

/// @dev Returns `sqrt(x * y)`. Also called the geometric mean.
function mulSqrt(uint256 x, uint256 y) internal pure returns (uint256 z) {
    if (x == y) return x; // Identity.
    uint256 p0 = rawMul(x, y); // Lower 256 bits of `x * y`.
    if (y == rawDiv(p0, x)) return sqrt(p0);
    z = saturatingMul(rawAdd(sqrt(x), 1), rawAdd(sqrt(y), 1)); // Initial over-estimate.
    /// @solidity memory-safe-assembly
    assembly {
        let mm := mulmod(x, y, not(0))
        let p1 := sub(mm, add(p0, lt(mm, p0))) // Upper 256 bits of `x * y`.
        // Babylonian with inlined `fullMulDiv`.
        for {} 1 {} {
            let t := and(z, sub(0, z))
            let r := mulmod(x, y, z)
            let d := div(z, t)
            let inv := xor(2, mul(3, d))
            inv := mul(inv, sub(2, mul(d, inv)))
            inv := mul(inv, sub(2, mul(d, inv)))
            inv := mul(inv, sub(2, mul(d, inv)))
            inv := mul(inv, sub(2, mul(d, inv)))
            inv := mul(inv, sub(2, mul(d, inv)))
            let q :=
                mul(
                    or(mul(sub(p1, gt(r, p0)), add(div(sub(0, t), t), 1)), div(sub(p0, r), t)),
                    mul(sub(2, mul(d, inv)), inv)
                )
            if iszero(lt(q, z)) { break }
            z := add(and(z, q), shr(1, xor(z, q))) // Non-overflowing average.
        }
    }
}

[PASS] testMulSqrt() (gas: 262692)
[PASS] test__codesize() (gas: 62243)

Checklist

Ensure you completed all of the steps below before submitting your pull request:

• Ran forge fmt?
• Ran forge test?

Pull requests with an incomplete checklist will be thrown out.

[github-actions]

Gas Snapshot Comparison Report

Generated at commit : d74d21e, Compared to commit : ba711c9

Contract Name	Test Name	Main Gas	PR Gas	Diff
FixedPointMathLibTest	testAbs()	679	701	22
	testAvgEdgeCase()	470	448	-22
	testAvgSigned()	901	923	22
	testCbrt()	10328	10283	-45
	testCbrtWadConverged()	2725	2716	-9
	testCbrtWadDebug()	7339	7383	44
	testCbrtWadMonotonicallyIncreasing()	4183	4227	44
	testDist()	656	678	22
	testDistEdgeCases()	526	570	44
	testDivWad()	692	736	44
	testDivWadUp()	2986	2964	-22
	testDivWadUpEdgeCases()	451	450	-1
	testDivWadUpZeroDenominatorReverts()	3982	3960	-22
	testDivWadZeroDenominatorReverts()	4025	3963	-62
	testFactorial()	99191	99190	-1
	testFullMulDiv()	1128	1106	-22
	testFullMulDivN()	874	896	22
	testFullMulDivUnchecked()	1613	1635	22
	testFullMulDivUpRevertsIfRoundedUpResultOverflowsCase1()	4512	4468	-44
	testFullMulDivUpRevertsIfRoundedUpResultOverflowsCase2()	4556	4512	-44
	testGcd()	4207	4229	22
	testInvMod()	12915	12892	-23
	testLambertW0WadKnownValues()	1685617	1685605	-12
	testLambertW0WadMonoDebug()	770690	770668	-22
	testLambertW0WadMonotonicallyIncreasing()	18416531	18416509	-22
	testLambertW0WadRevertsForOutOfDomain()	58122	58144	22
	testLambertW0WadWithinBounds()	122062	122084	22
	testLerpUint()	6398	6442	44
	testLnWad()	2054	2076	22
	testLnWadBig()	2066	2088	22
	testLnWadNegativeReverts()	6191	6257	66
	testLnWadOverflowReverts()	4028	4051	23
	testLnWadSmall()	2667	2708	41
	testLog10()	76190	76212	22
	testLog10Up()	4413	4435	22
	testLog256()	22853	22809	-44
	testLog256Up()	1294	1293	-1
	testLog2Up()	297369	297346	-23
	testMulDiv()	1890	1912	22
	testMulDivEdgeCases()	741	763	22
	testMulDivUp()	2169	2126	-43
	testMulDivUpEdgeCases()	862	839	-23
	testMulDivUpZeroDenominator()	4077	4056	-21
	testMulDivZeroDenominatorReverts()	4055	4033	-22
	testMulWad()	690	712	22
	testMulWadEdgeCases()	754	709	-45
	testMulWadUp()	757	779	22
	testMulWadUpEdgeCases()	799	776	-23
	testPackUnpackSci()	130063	130040	-23
	testRPow()	3276	3298	22
	testRPowOverflowReverts()	8168	8256	88
	testSDivWad()	840	862	22
	testSMulWad()	1053	1033	-20
	testSMulWadEdgeCases()	1265	1287	22
	testSaturatingAdd()	1774	1729	-45
	testSaturatingMul()	1838	1837	-1
	testSci()	1838614	1838658	44
	testSqrt()	43842	45393	1551
	testSqrtHashedSingle()	54610	54632	22
	testSqrtWad()	7981	7937	-44
	test__codesize()	49928	62144	12216
	testMulSqrt()	-	287258	-