SqrtPriceMath.sol

// SPDX-License-Identifier: GPL-2.0-or-later
pragma solidity ^0.8.20;

import {SafeCast} from "./SafeCast.sol";

import {FullMath} from "./FullMath.sol";
import {UnsafeMath} from "./UnsafeMath.sol";
import {FixedPoint96} from "./FixedPoint96.sol";

/// @title Functions based on Q64.96 sqrt price and liquidity
/// @notice Contains the math that uses square root of price as a Q64.96 and liquidity to compute deltas
library SqrtPriceMath {
    using SafeCast for uint256;

    error InvalidPriceOrLiquidity();
    error InvalidPrice();
    error NotEnoughLiquidity();
    error PriceOverflow();

    /// @notice Gets the next sqrt price given a delta of currency0
    /// @dev Always rounds up, because in the exact output case (increasing price) we need to move the price at least
    /// far enough to get the desired output amount, and in the exact input case (decreasing price) we need to move the
    /// price less in order to not send too much output.
    /// The most precise formula for this is liquidity * sqrtPX96 / (liquidity +- amount * sqrtPX96),
    /// if this is impossible because of overflow, we calculate liquidity / (liquidity / sqrtPX96 +- amount).
    /// @param sqrtPX96 The starting price, i.e. before accounting for the currency0 delta
    /// @param liquidity The amount of usable liquidity
    /// @param amount How much of currency0 to add or remove from virtual reserves
    /// @param add Whether to add or remove the amount of currency0
    /// @return The price after adding or removing amount, depending on add
    function getNextSqrtPriceFromAmount0RoundingUp(uint160 sqrtPX96, uint128 liquidity, uint256 amount, bool add)
        internal
        pure
        returns (uint160)
    {
        // we short circuit amount == 0 because the result is otherwise not guaranteed to equal the input price
        if (amount == 0) return sqrtPX96;
        uint256 numerator1 = uint256(liquidity) << FixedPoint96.RESOLUTION;

        if (add) {
            unchecked {
                uint256 product = amount * sqrtPX96;
                if (product / amount == sqrtPX96) {
                    uint256 denominator = numerator1 + product;
                    if (denominator >= numerator1) {
                        // always fits in 160 bits
                        return uint160(FullMath.mulDivRoundingUp(numerator1, sqrtPX96, denominator));
                    }
                }
            }
            // denominator is checked for overflow
            return uint160(UnsafeMath.divRoundingUp(numerator1, (numerator1 / sqrtPX96) + amount));
        } else {
            unchecked {
                uint256 product = amount * sqrtPX96;
                // if the product overflows, we know the denominator underflows
                // in addition, we must check that the denominator does not underflow
                // equivalent: if (product / amount != sqrtPX96 || numerator1 <= product) revert PriceOverflow();
                assembly ("memory-safe") {
                    if iszero(and(eq(div(product, amount), sqrtPX96), gt(numerator1, product))) {
                        mstore(0, 0xf5c787f1) // selector for PriceOverflow()
                        revert(0x1c, 0x04)
                    }
                }
                uint256 denominator = numerator1 - product;
                return FullMath.mulDivRoundingUp(numerator1, sqrtPX96, denominator).toUint160();
            }
        }
    }

    /// @notice Gets the next sqrt price given a delta of currency1
    /// @dev Always rounds down, because in the exact output case (decreasing price) we need to move the price at least
    /// far enough to get the desired output amount, and in the exact input case (increasing price) we need to move the
    /// price less in order to not send too much output.
    /// The formula we compute is within <1 wei of the lossless version: sqrtPX96 +- amount / liquidity
    /// @param sqrtPX96 The starting price, i.e., before accounting for the currency1 delta
    /// @param liquidity The amount of usable liquidity
    /// @param amount How much of currency1 to add, or remove, from virtual reserves
    /// @param add Whether to add, or remove, the amount of currency1
    /// @return The price after adding or removing `amount`
    function getNextSqrtPriceFromAmount1RoundingDown(uint160 sqrtPX96, uint128 liquidity, uint256 amount, bool add)
        internal
        pure
        returns (uint160)
    {
        // if we're adding (subtracting), rounding down requires rounding the quotient down (up)
        // in both cases, avoid a mulDiv for most inputs
        if (add) {
            uint256 quotient = (
                amount <= type(uint160).max
                    ? (amount << FixedPoint96.RESOLUTION) / liquidity
                    : FullMath.mulDiv(amount, FixedPoint96.Q96, liquidity)
            );

            return (uint256(sqrtPX96) + quotient).toUint160();
        } else {
            uint256 quotient = (
                amount <= type(uint160).max
                    ? UnsafeMath.divRoundingUp(amount << FixedPoint96.RESOLUTION, liquidity)
                    : FullMath.mulDivRoundingUp(amount, FixedPoint96.Q96, liquidity)
            );

            // equivalent: if (sqrtPX96 <= quotient) revert NotEnoughLiquidity();
            assembly ("memory-safe") {
                if iszero(gt(sqrtPX96, quotient)) {
                    mstore(0, 0x4323a555) // selector for NotEnoughLiquidity()
                    revert(0x1c, 0x04)
                }
            }
            // always fits 160 bits
            unchecked {
                return uint160(sqrtPX96 - quotient);
            }
        }
    }

    /// @notice Gets the next sqrt price given an input amount of currency0 or currency1
    /// @dev Throws if price or liquidity are 0, or if the next price is out of bounds
    /// @param sqrtPX96 The starting price, i.e., before accounting for the input amount
    /// @param liquidity The amount of usable liquidity
    /// @param amountIn How much of currency0, or currency1, is being swapped in
    /// @param zeroForOne Whether the amount in is currency0 or currency1
    /// @return sqrtQX96 The price after adding the input amount to currency0 or currency1
    function getNextSqrtPriceFromInput(uint160 sqrtPX96, uint128 liquidity, uint256 amountIn, bool zeroForOne)
        internal
        pure
        returns (uint160 sqrtQX96)
    {
        // equivalent: if (sqrtPX96 == 0 || liquidity == 0) revert InvalidPriceOrLiquidity();
        assembly ("memory-safe") {
            if or(iszero(sqrtPX96), iszero(liquidity)) {
                mstore(0, 0x4f2461b8) // selector for InvalidPriceOrLiquidity()
                revert(0x1c, 0x04)
            }
        }

        // round to make sure that we don't pass the target price
        return zeroForOne
            ? getNextSqrtPriceFromAmount0RoundingUp(sqrtPX96, liquidity, amountIn, true)
            : getNextSqrtPriceFromAmount1RoundingDown(sqrtPX96, liquidity, amountIn, true);
    }

    /// @notice Gets the next sqrt price given an output amount of currency0 or currency1
    /// @dev Throws if price or liquidity are 0 or the next price is out of bounds
    /// @param sqrtPX96 The starting price before accounting for the output amount
    /// @param liquidity The amount of usable liquidity
    /// @param amountOut How much of currency0, or currency1, is being swapped out
    /// @param zeroForOne Whether the amount out is currency1 or currency0
    /// @return sqrtQX96 The price after removing the output amount of currency0 or currency1
    function getNextSqrtPriceFromOutput(uint160 sqrtPX96, uint128 liquidity, uint256 amountOut, bool zeroForOne)
        internal
        pure
        returns (uint160 sqrtQX96)
    {
        // equivalent: if (sqrtPX96 == 0 || liquidity == 0) revert InvalidPriceOrLiquidity();
        assembly ("memory-safe") {
            if or(iszero(sqrtPX96), iszero(liquidity)) {
                mstore(0, 0x4f2461b8) // selector for InvalidPriceOrLiquidity()
                revert(0x1c, 0x04)
            }
        }

        // round to make sure that we pass the target price
        return zeroForOne
            ? getNextSqrtPriceFromAmount1RoundingDown(sqrtPX96, liquidity, amountOut, false)
            : getNextSqrtPriceFromAmount0RoundingUp(sqrtPX96, liquidity, amountOut, false);
    }

    /// @notice Gets the amount0 delta between two prices
    /// @dev Calculates liquidity / sqrt(lower) - liquidity / sqrt(upper),
    /// i.e. liquidity * (sqrt(upper) - sqrt(lower)) / (sqrt(upper) * sqrt(lower))
    /// @param sqrtPriceAX96 A sqrt price
    /// @param sqrtPriceBX96 Another sqrt price
    /// @param liquidity The amount of usable liquidity
    /// @param roundUp Whether to round the amount up or down
    /// @return amount0 Amount of currency0 required to cover a position of size liquidity between the two passed prices
    function getAmount0Delta(uint160 sqrtPriceAX96, uint160 sqrtPriceBX96, uint128 liquidity, bool roundUp)
        internal
        pure
        returns (uint256 amount0)
    {
        unchecked {
            if (sqrtPriceAX96 > sqrtPriceBX96) (sqrtPriceAX96, sqrtPriceBX96) = (sqrtPriceBX96, sqrtPriceAX96);

            // equivalent: if (sqrtPriceAX96 == 0) revert InvalidPrice();
            assembly ("memory-safe") {
                if iszero(sqrtPriceAX96) {
                    mstore(0, 0x00bfc921) // selector for InvalidPrice()
                    revert(0x1c, 0x04)
                }
            }

            uint256 numerator1 = uint256(liquidity) << FixedPoint96.RESOLUTION;
            uint256 numerator2 = sqrtPriceBX96 - sqrtPriceAX96;

            return roundUp
                ? UnsafeMath.divRoundingUp(FullMath.mulDivRoundingUp(numerator1, numerator2, sqrtPriceBX96), sqrtPriceAX96)
                : FullMath.mulDiv(numerator1, numerator2, sqrtPriceBX96) / sqrtPriceAX96;
        }
    }

    /// @notice Equivalent to: `a >= b ? a - b : b - a`
    function absDiff(uint160 a, uint160 b) internal pure returns (uint256 res) {
        assembly {
            let diff := sub(a, b)
            // mask = 0 if a >= b else -1 (all 1s)
            let mask := sar(255, diff)
            // if a >= b, res = a - b = 0 ^ (a - b)
            // if a < b, res = b - a = ~~(b - a) = ~(-(b - a) - 1) = ~(a - b - 1) = (-1) ^ (a - b - 1)
            // either way, res = mask ^ (a - b + mask)
            res := xor(mask, add(mask, diff))
        }
    }

    /// @notice Gets the amount1 delta between two prices
    /// @dev Calculates liquidity * (sqrt(upper) - sqrt(lower))
    /// @param sqrtPriceAX96 A sqrt price
    /// @param sqrtPriceBX96 Another sqrt price
    /// @param liquidity The amount of usable liquidity
    /// @param roundUp Whether to round the amount up, or down
    /// @return amount1 Amount of currency1 required to cover a position of size liquidity between the two passed prices
    function getAmount1Delta(uint160 sqrtPriceAX96, uint160 sqrtPriceBX96, uint128 liquidity, bool roundUp)
        internal
        pure
        returns (uint256 amount1)
    {
        uint256 numerator = absDiff(sqrtPriceAX96, sqrtPriceBX96);
        uint256 denominator = FixedPoint96.Q96;
        uint256 _liquidity;
        assembly {
            // avoid implicit upcasting
            _liquidity := liquidity
        }
        /**
         * Equivalent to:
         *   amount1 = roundUp
         *       ? FullMath.mulDivRoundingUp(liquidity, sqrtPriceBX96 - sqrtPriceAX96, FixedPoint96.Q96)
         *       : FullMath.mulDiv(liquidity, sqrtPriceBX96 - sqrtPriceAX96, FixedPoint96.Q96);
         * Cannot overflow because `type(uint128).max * type(uint160).max >> 96 < (1 << 192)`.
         */
        amount1 = FullMath.mulDiv(_liquidity, numerator, denominator);
        assembly {
            amount1 := add(amount1, and(gt(mulmod(_liquidity, numerator, denominator), 0), roundUp))
        }
    }

    /// @notice Helper that gets signed currency0 delta
    /// @param sqrtPriceAX96 A sqrt price
    /// @param sqrtPriceBX96 Another sqrt price
    /// @param liquidity The change in liquidity for which to compute the amount0 delta
    /// @return amount0 Amount of currency0 corresponding to the passed liquidityDelta between the two prices
    function getAmount0Delta(uint160 sqrtPriceAX96, uint160 sqrtPriceBX96, int128 liquidity)
        internal
        pure
        returns (int256 amount0)
    {
        unchecked {
            return liquidity < 0
                ? getAmount0Delta(sqrtPriceAX96, sqrtPriceBX96, uint128(-liquidity), false).toInt256()
                : -getAmount0Delta(sqrtPriceAX96, sqrtPriceBX96, uint128(liquidity), true).toInt256();
        }
    }

    /// @notice Helper that gets signed currency1 delta
    /// @param sqrtPriceAX96 A sqrt price
    /// @param sqrtPriceBX96 Another sqrt price
    /// @param liquidity The change in liquidity for which to compute the amount1 delta
    /// @return amount1 Amount of currency1 corresponding to the passed liquidityDelta between the two prices
    function getAmount1Delta(uint160 sqrtPriceAX96, uint160 sqrtPriceBX96, int128 liquidity)
        internal
        pure
        returns (int256 amount1)
    {
        unchecked {
            return liquidity < 0
                ? getAmount1Delta(sqrtPriceAX96, sqrtPriceBX96, uint128(-liquidity), false).toInt256()
                : -getAmount1Delta(sqrtPriceAX96, sqrtPriceBX96, uint128(liquidity), true).toInt256();
        }
    }
}