💹Price representation

How Ekubo represents pool prices: sqrt ratios, ticks, and the compact floating-point SqrtRatio type used by Ekubo V3 on EVM chains

Every Ekubo pool tracks its current price as a square root ratio: the square root of the price of token1 in terms of token0.

$$sqrt\_ratio = \sqrt{token1/token0}$$

Working with the square root of the price (rather than the price itself) makes the concentrated liquidity math cheaper and more precise — see the Math 1-pager for why. Prices are supported over the enormous range [2^-128, 2^128], so the sqrt ratio always lies in [2^-64, 2^64].

Ticks are the same on every deployment: tick i corresponds to the sqrt ratio sqrt(1.000001^i), i.e. each tick is 1/100th of a basis point of price. The valid tick range is roughly ±88.72 million (±88,722,835 on EVM, ±88,722,883 on Starknet — the EVM range is very slightly narrower).

How the sqrt ratio is encoded differs between the Starknet and EVM deployments.

Starknet: 64.128 fixed point

On Starknet, sqrt_ratio is a u256 interpreted as a binary fixed-point number with 128 fractional bits. A sqrt ratio of 1.0 (price of 1) is 2^128. Since the sqrt ratio never exceeds 2^64, the value always fits in 192 bits, which lets Core pack the pool price and tick into a single felt252 storage slot.

• MIN_SQRT_RATIO = 18446748437148339061 (≈ 2^-64 · 2^128)

• MAX_SQRT_RATIO = 6277100250585753475930931601400621808602321654880405518632 (≈ 2^64 · 2^128)

To convert to a price: divide by 2^128, then square. See Reading pool price for a worked example.

EVM (V3): the floating-point SqrtRatio type

Ekubo V3 on EVM chains introduces a purpose-built 96-bit dynamic fixed-point type — a compact floating-point-style encoding invented for Ekubo:

type SqrtRatio is uint96;

Why a float?

A full-precision 64.128 sqrt ratio needs 192 bits, which doesn't leave room for anything else in a 256-bit storage word. The AMM doesn't actually need 192 bits of precision at every magnitude — it needs relative precision. By encoding the sqrt ratio in 96 bits, the entire pool state packs into a single storage slot:

poolState (bytes32) = sqrtRatio (96 bits) | tick (32 bits) | liquidity (128 bits)

This makes updating pool state during a swap dramatically cheaper. The arithmetic on the type is designed so that any rounding error is biased in favor of the pool.

Bit layout

The 96 bits consist of a 2-bit exponent selector (bits 95–94) and a 94-bit mantissa (bits 93–0). The exponent selects how the mantissa is interpreted as a binary fixed-point number:

Exponent (bits 95–94)	Mantissa interpretation	Value range covered
00	0.126 (mantissa / 2^126)	sqrt ratios below 2^-32
01	0.94 (mantissa / 2^94)	2^-32 up to 1
10	32.62 (mantissa / 2^62)	1 up to 2^32
11	64.30 (mantissa / 2^30)	2^32 up to 2^64

There is no implicit leading bit and no bias — each exponent value simply shifts the radix point by 32 bits. A sqrt ratio of exactly 1.0 is encoded with exponent 10 and mantissa 2^62:

SqrtRatio constant ONE = SqrtRatio.wrap(uint96((1 << 95) + (1 << 62)));

Key properties:

• Monotonic: a larger raw uint96 always means a larger sqrt ratio, so comparisons are plain integer comparisons — no decoding required.

• Normalized: a valid value always has at least 63 significant mantissa bits (isValid requires the mantissa to be ≥ 2^62, so bit 62 is set, and the raw value to be within the min/max bounds). Effective precision therefore ranges from 63 to 94 significant bits depending on where in the range the value falls — far more than enough for 1/100th-basis-point ticks.

• Bounded: MIN_SQRT_RATIO (raw 4611797791050542631) through MAX_SQRT_RATIO (raw 79227682466138141934206691491), covering approximately the same [2^-64, 2^64] sqrt ratio range as Starknet — the EVM bounds are very slightly tighter, matching its slightly narrower tick range.

Converting to and from 64.128

The canonical "expanded" form on EVM is the same 64.128 fixed point used on Starknet. Conversion is a single shift, chosen by the exponent:

// SqrtRatio -> 64.128 fixed point (uint256)
function toFixed(SqrtRatio sqrtRatio) pure returns (uint256);
// shift left by 2, 34, 66 or 98 bits for exponents 00, 01, 10, 11

// 64.128 fixed point -> SqrtRatio (rounding toward or away from zero)
function toSqrtRatio(uint256 sqrtRatioFixed, bool roundUp) pure returns (SqrtRatio);

Both directions round-trip losslessly for any valid SqrtRatio. To compute a human-readable price from the 64.128 value: divide by 2^128, square, and adjust for the two tokens' decimals (worked example in Reading pool price).

Where you'll encounter it

• Swap parameters: sqrtRatioLimit in swap params and router RouteNodes is a SqrtRatio (a value of 0 means "no limit").

• Pool state and events: PoolInitialized emits the initial SqrtRatio; extension hooks (afterSwap, afterUpdatePosition, ...) receive the packed pool state containing it.

• Reading prices: CoreDataFetcher.poolPrice() returns the price already expanded to 64.128 via toFixed(), so most integrators never need to decode the packed form manually. Its poolState() returns the packed SqrtRatio undecoded.

• SDKs: @ekubo/sdk (TypeScript) and the Rust SDK implement the exact tick and sqrt-ratio conversions for both Starknet and EVM.

Fee encoding differs too

While ticks are chain-agnostic, the pool fee encoding differs between deployments:

Deployment	Fee type	Denominator	Example: 0.3%
Starknet	u128, 0.128 fixed point	2^128	1020847100762815411640772995208708096
EVM (V3)	uint64, 0.64 fixed point	2^64	55340232221128654

In both cases the fee is a binary fraction of the full width — fee / 2^128 (Starknet) or fee / 2^64 (EVM) — applied to the swap input amount, rounded up in favor of the pool.