sqrt

Math

Computes square root of x in 18-decimal fixed-point.

Avg. gas

197

Max abs. error

1.0e-18

when sqrt(x) < 1

Max rel. error

2.0e-18

when sqrt(x) ≥ 1

Signature

solidity

function sqrt(uint256 x) internal pure returns (uint256 y)

Parameters

Name	Type	Description
x	uint256	Input in 18-decimal fixed-point format (1e18 = 1.0). Any value in [0, uint256.max] accepted.

Returns

Name	Type	Description
y	uint256	Square root √x in 18-decimal fixed-point format.

Bounds

Bound	Value
Input domain	Full `uint256` domain — the function has no named bounds and accepts any input in `[0, uint256.max]`. The internal branch cutoff at `type(uint128).max` is an implementation detail, not a limit on callers.

Behavior

Returns 0 when x == 0 — handled by the algorithm's natural underflow via EVM's div(0, 0) = 0 semantic, no explicit guard.
Never reverts. Handles the full [0, uint256.max] range via a two-branch split at type(uint128).max.
Uses the CLZ opcode (Osaka) inside the range reduction; see Counting leading zeros in Solidity using CLZ opcode.
Pure assembly hot path; no external calls or storage.

How it works

Pre-scale, seed, refine — all in assembly:

// 1. Pre-scale x to 1e36 base so div(x, y) lands in 1e18 — cheap div, no muldiv
x := mul(x, 1e18)

// 2. CLZ seed:  y = 2^floor(msb/2),  msb = 256 − clz(x)   — within √2 of √x
y := shl(shr(1, sub(254, clz(x))), 2)

// 3. Five Newton steps — shr halves; quadratic convergence to bit-perfect FP18
y := shr(1, add(y, div(x, y)))             // ×5,  ~20 gas each

The seed lands within a factor of √2 of the root (~41% worst case); five Newton steps drive that to ~80 bits — bit-exact when √x < 1, under 2e-18 relative error above. A second branch post-scales instead of pre-scaling for x > type(uint128).max, where x · 1e18 would overflow. Full derivation in the walkthrough: How I wrote a fixed-point Solidity sqrt that runs in 197 gas.

Errors

Error	Trigger
None	Never reverts. Accepts any `uint256` input. `x == 0` returns `0`; large `x` takes the post-scale branch and stays FP18-accurate.

Example

solidity

import "defimath-lib/contracts/math/Math.sol";

uint256 x = 2e18;             // x = 2.0
uint256 y = Math.sqrt(x); // y ≈ 1.41421356e18

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