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⚡️ Golf sqrt and cbrt by getting a better initial estimate and removing an iteration #1515

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⚡️ Golf sqrt and cbrt by getting a better initial estimate and removing an iteration #1515

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e53dc06
⚡️Golf `sqrt` and `cbrt` by getting a better initial estimate and rem…
duncancmt Feb 27, 2026
e29ead0
⚡️Golf `cbrt` a little bit more by cutting corners during initialization
duncancmt Feb 28, 2026
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26 changes: 15 additions & 11 deletions src/utils/clz/FixedPointMathLib.sol
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Expand Up @@ -774,14 +774,16 @@ library FixedPointMathLib {
// n = floor(log2(x))
// For x ≈ 2^n, we know sqrt(x) ≈ 2^(n/2)
// We use (n+1)/2 instead of n/2 to round up slightly
// This gives a better initial approximation
// This gives a better initial approximation. This seed gives
// ε₁ = 0.0607 after one Babylonian step for all inputs. With
// ε_{n+1} ≈ ε²/2, 6 steps yield 2⁻¹⁶⁰ relative error (>128 correct
// bits).
//
// Formula: z = 2^((n+1)/2) = 2^(floor((n+1)/2))
// Implemented as: z = 1 << ((n+1) >> 1)
z := shl(shr(1, sub(256, clz(x))), 1)

/// (x/z + z) / 2
z := shr(1, add(z, div(x, z)))
// 6 Babylonian steps; z = (x/z + z) / 2
z := shr(1, add(z, div(x, z)))
z := shr(1, add(z, div(x, z)))
z := shr(1, add(z, div(x, z)))
Expand All @@ -799,23 +801,25 @@ library FixedPointMathLib {
/// @dev Returns the cube root of `x`, rounded down.
/// Credit to bout3fiddy and pcaversaccio under AGPLv3 license:
/// https://github.com/pcaversaccio/snekmate/blob/main/src/snekmate/utils/math.vy
/// Formally verified by xuwinnie:
/// https://github.com/vectorized/solady/blob/main/audits/xuwinnie-solady-cbrt-proof.pdf
function cbrt(uint256 x) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
// Initial guess z = 2^ceil((log2(x) + 2) / 3).
// Since log2(x) = 255 - clz(x), the expression shl((257 - clz(x)) / 3, 1)
// computes this over-estimate. Guaranteed ≥ cbrt(x) and safe for Newton-Raphson's.
z := shl(div(sub(257, clz(x)), 3), 1)
// Newton-Raphson's.
z := div(add(add(div(x, mul(z, z)), z), z), 3)
// Initial guess z ≈ ∛(3/4) · 2𐞥 where q = ⌊(257 − clz(x)) / 3⌋. The
// multiplier 233/256 ≈ 0.909 ≈ ∛(3/4) balances the worst-case
// over/underestimate across each octave triplet (ε_over ≈ 0.445,
// ε_under ≈ −0.278), giving >85 bits of precision after 6 N-R
// iterations. The `add(..., 1)` term ensures z ≥ 1 when x > 0 (the
// `shr` can produce 0 for small `q`)
z := add(shr(8, shl(div(sub(257, clz(x)), 3), 233)), 1)

// 6 Newton-Raphson iterations.
z := div(add(add(div(x, mul(z, z)), z), z), 3)
z := div(add(add(div(x, mul(z, z)), z), z), 3)
z := div(add(add(div(x, mul(z, z)), z), z), 3)
z := div(add(add(div(x, mul(z, z)), z), z), 3)
z := div(add(add(div(x, mul(z, z)), z), z), 3)
z := div(add(add(div(x, mul(z, z)), z), z), 3)

// Round down.
z := sub(z, lt(div(x, mul(z, z)), z))
}
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