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⚡️ Golf cbrt by using a small table for an even better initial estimate and removing another iteration #1523

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atarpara merged 2 commits into from
May 13, 2026
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⚡️ Golf cbrt by using a small table for an even better initial estimate and removing another iteration #1523

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⚡️ Golf `cbrt` by using a small table for an even better initial esti…
duncancmt May 12, 2026
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20 changes: 10 additions & 10 deletions src/utils/clz/FixedPointMathLib.sol
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Expand Up @@ -804,16 +804,16 @@ library FixedPointMathLib {
function cbrt(uint256 x) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
// 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)
// Initial guess z ≈ c · 2𐞥 where b = ⌊log₂(x)⌋, q = ⌊b / 3⌋. The
// 8-bit fixed-point multipliers `c`: 144/128, 181/128, and 229/128
// are selected by `b mod 3` to balance each octave's worst-case
// final error. This gives >98 bits of precision after only 5
// Newton-Raphson iterations. The `or(..., 1)` keeps z ≥ 1 when the
// shifted estimate is 0.
let b := sub(255, clz(x))
z := or(shr(7, shl(div(b, 3), byte(add(mod(b, 3), 29), 0x90b5e5))), 1)

// 5 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)
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