How the new CLZ opcode (EIP-7939) makes Solidity Black-Scholes pricing ~10% cheaper - by cascading through sqrt and ln

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The CLZ ("count leading zeros") opcode landed in EVM Osaka via EIP-7939, exposed in Solidity 0.8.31 as the Yul builtin `clz`. It costs 3 gas, returns the number of leading zero bits in a 256-bit value, and turns `floor(log₂(x))` into a near-free operation. The bit-length identity is the building block: bits = 256 − clz(x) // bit length of x floor(log₂(x)) = bits − 1 // for x ≥ 1 Two applications I used in DeFiMath: **1. Newton-Raphson initial guess.** `y₀ = 2^⌈bits/k⌉` lands within a factor of the k-th root of 2 of the true k-th root, so Newton converges in 6 iterations to bit-exact precision. Whole `sqrt` becomes 245 gas, `cbrt` 368 gas. // CLZ-derived initial guess: y = 2^⌈bits/2⌉, within √2 of √x y := shl(shr(1, sub(256, clz(x))), 1) // 6 Newton iterations y := shr(1, add(y, div(x, y))) // ... ×5 more **2. Range reduction for `ln`.** Find `k = floor(log₂(x))` with CLZ, divide x by `2^k`, land in `[1, 2)`. Mercator series then converges in ~10 terms. Total: 375 gas. **Compounding effect.** `sqrt` and `ln` are inside the Black-Scholes formula. Swapping the pre-CLZ versions of those two primitives dropped `callOptionPrice` from ~3,100 to 2,876 gas — about 10% cheaper with zero change to the option-pricing math. Same effect ripples through IV solving, futures, historical volatility, Sharpe ratio — anywhere a log or root appears. Full writeup with the actual assembly, the bit-length identity walked through, and a gas comparison table vs PRBMath, ABDK, and Solady: https://defimath.com/blog/clz-opcode-solidity Caveats: Solidity 0.8.31+ and EVM target `osaka` required. Older targets compile-error on the `clz` call (not a runtime surprise — fails fast). (Disclosure: I'm building DeFiMath. Posting because the CLZ trick is generalizable — any library doing log/sqrt-style math can pick up the same savings.)   submitted by   /u/nebojsakonsta [link]   [comments]