From d2a4aca790aeef5b56324a8a9431b9e2da059761 Mon Sep 17 00:00:00 2001
From: Duncan Townsend <git@duncancmt.com>
Date: Tue, 12 May 2026 16:53:05 +0200
Subject: [PATCH 1/2] =?UTF-8?q?=E2=9A=A1=EF=B8=8F=20Golf=20`cbrt`=20by=20u?=
 =?UTF-8?q?sing=20a=20small=20table=20for=20an=20even=20better=20initial?=
 =?UTF-8?q?=20estimate=20and=20removing=20another=20iteration?=
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Content-Type: text/plain; charset=UTF-8
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---
 src/utils/clz/FixedPointMathLib.sol | 20 ++++++++++----------
 1 file changed, 10 insertions(+), 10 deletions(-)

diff --git a/src/utils/clz/FixedPointMathLib.sol b/src/utils/clz/FixedPointMathLib.sol
index c349e0b59..1d4ba89bb 100644
--- a/src/utils/clz/FixedPointMathLib.sol
+++ b/src/utils/clz/FixedPointMathLib.sol
@@ -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)

From 6b69bf36cb16400068476dce9f0709aebcedcc77 Mon Sep 17 00:00:00 2001
From: Duncan Townsend <git@duncancmt.com>
Date: Tue, 12 May 2026 17:03:48 +0200
Subject: [PATCH 2/2] Formatting

---
 src/utils/clz/FixedPointMathLib.sol | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/utils/clz/FixedPointMathLib.sol b/src/utils/clz/FixedPointMathLib.sol
index 1d4ba89bb..c3be70416 100644
--- a/src/utils/clz/FixedPointMathLib.sol
+++ b/src/utils/clz/FixedPointMathLib.sol
@@ -808,7 +808,7 @@ library FixedPointMathLib {
             // 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
+            // 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)