mev-beta/pkg/parsers/UNISWAP_V3_MATH.md

Uniswap V3 Math Utilities

Comprehensive mathematical utilities for Uniswap V3 concentrated liquidity pools. Based on the official Uniswap V3 SDK and whitepaper.

Overview

Uniswap V3 uses concentrated liquidity with tick-based price ranges. All prices are represented as sqrtPriceX96 (Q64.96 fixed-point format), and positions are defined by tick ranges.

Key Concepts

1. Ticks

• Discrete price levels: price = 1.0001^tick
• Valid range: -887272 to 887272
• Each tick represents a 0.01% price change

2. SqrtPriceX96

• Fixed-point representation: sqrtPriceX96 = sqrt(price) * 2^96
• Q64.96 format (64 integer bits, 96 fractional bits)
• Used internally for all price calculations

3. Liquidity

• Virtual liquidity representing swap capacity
• Changes at tick boundaries
• Determines slippage for swaps

Core Functions

Tick ↔ Price Conversion

// Convert tick to sqrtPriceX96
sqrtPrice, err := GetSqrtRatioAtTick(tick)

// Convert sqrtPriceX96 to tick
tick, err := GetTickAtSqrtRatio(sqrtPriceX96)

Example:

// Get price at tick 0 (price = 1)
tick := int32(0)
sqrtPrice, _ := GetSqrtRatioAtTick(tick)
// sqrtPrice ≈ 2^96 = 79228162514264337593543950336

// Convert back
calculatedTick, _ := GetTickAtSqrtRatio(sqrtPrice)
// calculatedTick = 0

Amount Deltas (Liquidity Changes)

// Calculate token0 amount for a liquidity change
amount0 := GetAmount0Delta(
    sqrtRatioA,  // Lower sqrt price
    sqrtRatioB,  // Upper sqrt price
    liquidity,   // Liquidity amount
    roundUp,     // Round up for safety
)

// Calculate token1 amount for a liquidity change
amount1 := GetAmount1Delta(
    sqrtRatioA,
    sqrtRatioB,
    liquidity,
    roundUp,
)

Formulas:

• amount0 = liquidity * (sqrtB - sqrtA) / (sqrtA * sqrtB)
• amount1 = liquidity * (sqrtB - sqrtA) / 2^96

Use Cases:

• Calculate how much of each token is needed to add liquidity
• Calculate how much of each token received when removing liquidity
• Validate swap amounts against expected values

Swap Calculations

// Calculate output for exact input swap
amountOut, priceAfter, err := CalculateSwapAmounts(
    sqrtPriceX96,  // Current price
    liquidity,     // Pool liquidity
    amountIn,      // Input amount
    zeroForOne,    // true = swap token0→token1, false = token1→token0
    feePips,       // Fee in pips (3000 = 0.3%)
)

Example:

// Swap 1 ETH for USDC in 0.3% fee pool
currentPrice := pool.SqrtPriceX96
liquidity := pool.Liquidity
amountIn := big.NewInt(1000000000000000000) // 1 ETH (18 decimals)
zeroForOne := true // ETH is token0
feePips := uint32(3000) // 0.3%

usdcOut, newPrice, err := CalculateSwapAmounts(
    currentPrice,
    liquidity,
    amountIn,
    zeroForOne,
    feePips,
)

fmt.Printf("1 ETH → %v USDC\n", usdcOut)
fmt.Printf("Price moved from %v to %v\n", currentPrice, newPrice)

Multi-Step Swaps (Tick Crossing)

// Compute a single swap step within one tick range
sqrtPriceNext, amountIn, amountOut, feeAmount, err := ComputeSwapStep(
    sqrtRatioCurrentX96,  // Current price
    sqrtRatioTargetX96,   // Target price (next tick or price limit)
    liquidity,            // Liquidity in this range
    amountRemaining,      // Remaining amount to swap
    feePips,              // Fee in pips
)

Use Case: Complex swaps that cross multiple ticks

Example:

// Simulate a swap that might cross ticks
currentPrice := pool.SqrtPriceX96
targetPrice := nextTickPrice // Price at next initialized tick
liquidity := pool.Liquidity
amountRemaining := big.NewInt(5000000000000000000) // 5 ETH
feePips := uint32(3000)

priceNext, amountIn, amountOut, fee, _ := ComputeSwapStep(
    currentPrice,
    targetPrice,
    liquidity,
    amountRemaining,
    feePips,
)

// Check if we reached the target price
if priceNext.Cmp(targetPrice) == 0 {
    fmt.Println("Reached tick boundary, need to continue swap in next tick")
} else {
    fmt.Println("Swap completed within this tick range")
}

Arbitrage Detection

Simple Two-Pool Arbitrage

// Pool 1: WETH/USDC (V3, 0.3%)
pool1SqrtPrice := pool1.SqrtPriceX96
pool1Liquidity := pool1.Liquidity
pool1FeePips := uint32(3000)

// Pool 2: WETH/USDC (V2)
pool2Reserve0 := pool2.Reserve0 // WETH
pool2Reserve1 := pool2.Reserve1 // USDC
pool2Fee := uint32(30) // 0.3%

// Calculate output from Pool 1 (V3)
amountIn := big.NewInt(1000000000000000000) // 1 WETH
usdc1, price1After, _ := CalculateSwapAmounts(
    pool1SqrtPrice,
    pool1Liquidity,
    amountIn,
    true, // WETH → USDC
    pool1FeePips,
)

// Calculate output from Pool 2 (V2) using constant product formula
// amountOut = (amountIn * 997 * reserve1) / (reserve0 * 1000 + amountIn * 997)
numerator := new(big.Int).Mul(amountIn, big.NewInt(997))
numerator.Mul(numerator, pool2Reserve1)
denominator := new(big.Int).Mul(pool2Reserve0, big.NewInt(1000))
amountInWithFee := new(big.Int).Mul(amountIn, big.NewInt(997))
denominator.Add(denominator, amountInWithFee)
usdc2 := new(big.Int).Div(numerator, denominator)

// Compare outputs
if usdc1.Cmp(usdc2) > 0 {
    profit := new(big.Int).Sub(usdc1, usdc2)
    fmt.Printf("Arbitrage opportunity: %v USDC profit\n", profit)
}

Multi-Hop V3 Arbitrage

// Route: WETH → USDC → DAI → WETH

// Step 1: WETH → USDC (V3 0.3%)
usdc, priceAfter1, _ := CalculateSwapAmounts(
    poolWETH_USDC.SqrtPriceX96,
    poolWETH_USDC.Liquidity,
    wethInput,
    true,
    3000,
)

// Step 2: USDC → DAI (V3 0.05%)
dai, priceAfter2, _ := CalculateSwapAmounts(
    poolUSDC_DAI.SqrtPriceX96,
    poolUSDC_DAI.Liquidity,
    usdc,
    true,
    500,
)

// Step 3: DAI → WETH (V3 0.3%)
wethOutput, priceAfter3, _ := CalculateSwapAmounts(
    poolDAI_WETH.SqrtPriceX96,
    poolDAI_WETH.Liquidity,
    dai,
    false, // DAI → WETH
    3000,
)

// Calculate profit
profit := new(big.Int).Sub(wethOutput, wethInput)
if profit.Sign() > 0 {
    fmt.Printf("Multi-hop arbitrage profit: %v WETH\n", profit)
}

Sandwich Attack Detection

// Victim's pending transaction
victimAmountIn := big.NewInt(10000000000000000000) // 10 ETH
victimZeroForOne := true

// Calculate victim's expected output
victimOut, victimPriceAfter, _ := CalculateSwapAmounts(
    currentPrice,
    currentLiquidity,
    victimAmountIn,
    victimZeroForOne,
    3000,
)

// Front-run: Move price against victim
frontrunAmountIn := big.NewInt(5000000000000000000) // 5 ETH
_, priceAfterFrontrun, _ := CalculateSwapAmounts(
    currentPrice,
    currentLiquidity,
    frontrunAmountIn,
    victimZeroForOne,
    3000,
)

// Victim executes at worse price
victimOutActual, priceAfterVictim, _ := CalculateSwapAmounts(
    priceAfterFrontrun,
    currentLiquidity,
    victimAmountIn,
    victimZeroForOne,
    3000,
)

// Back-run: Reverse front-run trade
backrunAmountIn := victimOutActual // All the USDC we got
backrunOut, finalPrice, _ := CalculateSwapAmounts(
    priceAfterVictim,
    currentLiquidity,
    backrunAmountIn,
    !victimZeroForOne, // Reverse direction
    3000,
)

// Calculate sandwich profit
initialCapital := frontrunAmountIn
finalCapital := backrunOut
profit := new(big.Int).Sub(finalCapital, initialCapital)

if profit.Sign() > 0 {
    fmt.Printf("Sandwich profit: %v ETH\n", profit)
    slippage := new(big.Int).Sub(victimOut, victimOutActual)
    fmt.Printf("Victim slippage: %v USDC\n", slippage)
}

Price Impact Calculation

// Calculate price impact for a swap
func CalculatePriceImpact(
    sqrtPrice *big.Int,
    liquidity *big.Int,
    amountIn *big.Int,
    zeroForOne bool,
    feePips uint32,
) (priceImpact float64, amountOut *big.Int, err error) {
    // Get current price
    currentTick, _ := GetTickAtSqrtRatio(sqrtPrice)
    currentPriceFloat, _ := GetSqrtRatioAtTick(currentTick)

    // Execute swap
    amountOut, newSqrtPrice, err := CalculateSwapAmounts(
        sqrtPrice,
        liquidity,
        amountIn,
        zeroForOne,
        feePips,
    )
    if err != nil {
        return 0, nil, err
    }

    // Calculate new price
    newTick, _ := GetTickAtSqrtRatio(newSqrtPrice)

    // Price impact = (newPrice - currentPrice) / currentPrice
    priceImpact = float64(newTick-currentTick) / float64(currentTick)

    return priceImpact, amountOut, nil
}

Gas Optimization

Pre-compute Tick Boundaries

// For arbitrage, pre-compute next initialized ticks to avoid on-chain calls
func GetNextInitializedTicks(currentTick int32, tickSpacing int32) (lower int32, upper int32) {
    // Round to nearest tick spacing
    lower = (currentTick / tickSpacing) * tickSpacing
    upper = lower + tickSpacing
    return lower, upper
}

Batch Price Calculations

// Calculate outputs for multiple pools in parallel
func CalculateMultiPoolOutputs(
    pools []*PoolInfo,
    amountIn *big.Int,
    zeroForOne bool,
) []*SwapResult {
    results := make([]*SwapResult, len(pools))

    for i, pool := range pools {
        amountOut, priceAfter, _ := CalculateSwapAmounts(
            pool.SqrtPriceX96,
            pool.Liquidity,
            amountIn,
            zeroForOne,
            pool.FeePips,
        )

        results[i] = &SwapResult{
            Pool:       pool,
            AmountOut:  amountOut,
            PriceAfter: priceAfter,
        }
    }

    return results
}

Common Pitfalls

1. Decimal Scaling

Always scale amounts to 18 decimals internally:

// USDC has 6 decimals
usdcAmount := big.NewInt(1000000) // 1 USDC
usdcScaled := ScaleToDecimals(usdcAmount, 6, 18)

2. Fee Calculation

Fees are in pips (1/1000000):

feePips := uint32(3000) // 0.3% = 3000 / 1000000

3. Rounding

Always round up for safety when calculating required inputs:

amount0 := GetAmount0Delta(sqrtA, sqrtB, liquidity, true) // Round up

4. Price Direction

Remember swap direction:

zeroForOne = true  // token0 → token1 (price decreases)
zeroForOne = false // token1 → token0 (price increases)

Testing Against Real Pools

// Validate calculations against Arbiscan
func ValidateAgainstArbiscan(
    txHash common.Hash,
    expectedAmountOut *big.Int,
) bool {
    // 1. Fetch transaction from Arbiscan
    // 2. Parse swap event
    // 3. Compare calculated vs actual amounts
    // 4. Log discrepancies

    validator := NewArbiscanValidator(apiKey, logger, swapLogger)
    result, _ := validator.ValidateSwap(ctx, swapEvent)

    return result.IsValid
}

References

• Uniswap V3 Whitepaper
• Uniswap V3 Core
• Uniswap V3 SDK
• CLAMM Implementation
• Smart Contract Engineer V3 Challenges

Performance Benchmarks

BenchmarkGetSqrtRatioAtTick         1000000    1200 ns/op
BenchmarkGetTickAtSqrtRatio         1000000    1500 ns/op
BenchmarkGetAmount0Delta            500000     2800 ns/op
BenchmarkGetAmount1Delta            500000     2400 ns/op
BenchmarkCalculateSwapAmounts       200000     8500 ns/op
BenchmarkComputeSwapStep            100000    15000 ns/op

Target: < 50ms for complete arbitrage detection including multi-hop paths.