UniswapV3Broker.sol

pragma solidity ^0.8.0;

import "@uniswap/lib/contracts/libraries/TransferHelper.sol";

import "@uniswap/v3-core/contracts/interfaces/IUniswapV3Pool.sol";

import "@uniswap/v3-periphery/contracts/interfaces/ISwapRouter.sol";

import "@uniswap/v3-core/contracts/libraries/LowGasSafeMath.sol";
import "@uniswap/v3-core/contracts/libraries/LiquidityMath.sol";
import "@uniswap/v3-core/contracts/libraries/BitMath.sol";
import "@uniswap/v3-core/contracts/libraries/UnsafeMath.sol";
import "@uniswap/v3-core/contracts/libraries/SafeCast.sol";

/**
 * @title UniswapV3Broker
 * @notice Trading contract used to arb uniswapV3 pairs to a desired "true" price. Intended use is to arb UMA perpetual
 * synthetics that trade off peg. This implementation can ber used in conjunction with a DSProxy contract to atomically
 * swap and move a uniswap market.
 */

contract UniswapV3Broker {
    using SafeCast for uint256;
    using LowGasSafeMath for uint256;
    using LowGasSafeMath for int256;

    struct SwapState {
        uint160 sqrtPriceX96;
        int24 tick;
        uint128 liquidity;
        uint256 requiredInputAmount;
    }

    struct StepComputations {
        uint160 sqrtPriceStartX96;
        int24 tickNext;
        bool initialized;
        uint160 sqrtPriceNextX96;
    }

    /**
     * @notice Swaps an amount of either pool tokens such that the trade results in the uniswap pair's price equaling a
     * desired price.
     * @dev The desired price is represented as sqrtRatioTargetX96. This is the Price^(1/2) * 96^2.
     * @dev The caller must approve this contract to spend whichever token is intended to be swapped.
     * @param tradingAsEOA bool to indicate if the UniswapBroker is being called by a DSProxy or an EOA.
     * @param uniswapPool address of the pool to uniswap v3 trade against.
     * @param uniswapRouter address of the uniswap v3 router to route the trade.
     * @param sqrtRatioTargetX96 target, encoded price.
     * @param recipient address that the output tokens should be sent to.
     * @param deadline to limit when the trade can execute. If the tx is mined after this timestamp then revert.
     */
    function swapToPrice(
        bool tradingAsEOA,
        address uniswapPool,
        address uniswapRouter,
        uint160 sqrtRatioTargetX96,
        address recipient,
        uint256 deadline
    ) external returns (uint256) {
        // Create an instance of the pool and load in the current token price and the active tick.
        IUniswapV3Pool pool = IUniswapV3Pool(uniswapPool);
        (uint160 sqrtPriceX96, int24 tick, , , , , ) = pool.slot0();

        // Work out the direction we need to trade. If the current price is more than the target price then we are
        // trading token0 for token1. Else, we are trading token1 for token0.
        bool zeroForOne = sqrtPriceX96 >= sqrtRatioTargetX96;

        // Build a state object to store this information which can be re-used during.
        SwapState memory state =
            SwapState({ sqrtPriceX96: sqrtPriceX96, tick: tick, liquidity: pool.liquidity(), requiredInputAmount: 0 });

        // Iterate in a while loop that breaks when we hit the target price.
        while (true) {
            // Compute the next initialized tick. We only need to traverse initialized ticks as uninitialized ticks
            // have the same liquidity as the previous tick.
            StepComputations memory step;
            step.sqrtPriceStartX96 = state.sqrtPriceX96;
            (step.tickNext, step.initialized) = TickBitmap.nextInitializedTickWithinOneWord(
                pool,
                state.tick,
                pool.tickSpacing(),
                zeroForOne
            );

            // Double check we are not over or underflow the ticks.
            if (step.tickNext < TickMath.MIN_TICK) step.tickNext = TickMath.MIN_TICK;
            else if (step.tickNext > TickMath.MAX_TICK) step.tickNext = TickMath.MAX_TICK;

            // Find the price at the next tick. Between the current state.sqrtPriceX96 and the nextTickPriceX96 we
            // can find how much of the sold token is needed to sufficiently move the market over the interval.

            uint160 nextTickPriceX96 = TickMath.getSqrtRatioAtTick(step.tickNext);
            uint256 inputAmountForStep;

            // If zeroForOne is true, then we are moving the price UP. In this case we need to ensure that if the next
            // tick price is more than the target price, we set the set the next step price to the target price. This
            // ensures that the price does not undershoot when the next tick is the last tick. Else, traverse the whole tick.
            if (zeroForOne) {
                step.sqrtPriceNextX96 = nextTickPriceX96 > sqrtRatioTargetX96 ? sqrtRatioTargetX96 : nextTickPriceX96;
                inputAmountForStep = SqrtPriceMath.getAmount0Delta( // As we are trading token0 for token1, calculate the token0 input.
                    step.sqrtPriceStartX96,
                    step.sqrtPriceNextX96,
                    state.liquidity,
                    false
                );
                // Else, if zeroForOne is false, then we are moving the price DOWN. In this case we need to ensure that we
                // don't overshoot the price on the next step.
            } else {
                step.sqrtPriceNextX96 = nextTickPriceX96 > sqrtRatioTargetX96 ? nextTickPriceX96 : sqrtRatioTargetX96;
                inputAmountForStep = SqrtPriceMath.getAmount1Delta( // As we are trading token1 for token0, calculate the token1 input.
                    step.sqrtPriceStartX96,
                    step.sqrtPriceNextX96,
                    state.liquidity,
                    false
                );
            }

            // Add amount for this step to the total required input.
            state.requiredInputAmount = state.requiredInputAmount.add(inputAmountForStep);

            // If we have hit(or exceeded) our target price in the associate direction, then stop.
            if (zeroForOne && state.sqrtPriceX96 <= sqrtRatioTargetX96) break;
            if (!zeroForOne && state.sqrtPriceX96 >= sqrtRatioTargetX96) break;

            // If the next step is is initialized then we will need to update the liquidity for the current step.
            if (step.initialized) {
                // Fetch the net liquidity. this could be positive or negative depending on if a LP is turning on or off at this price.
                (, int128 liquidityNet, , , , , , ) = pool.ticks(step.tickNext);
                state.liquidity = LiquidityMath.addDelta(state.liquidity, zeroForOne ? liquidityNet : -liquidityNet);
            }

            // Finally, set the state price to the next price for the next iteration.
            state.sqrtPriceX96 = step.sqrtPriceNextX96;
            state.tick = step.tickNext;
        }

        // Based on the direction we are moving, set the input and output tokens.
        (address tokenIn, address tokenOut) =
            zeroForOne ? (pool.token0(), pool.token1()) : (pool.token1(), pool.token0());

        // If trading from an EOA pull tokens into this contract. If trading from a DSProxy this is redundant.
        if (tradingAsEOA)
            TransferHelper.safeTransferFrom(tokenIn, msg.sender, address(this), state.requiredInputAmount);

        // Approve the router and execute the swap.
        TransferHelper.safeApprove(tokenIn, address(uniswapRouter), state.requiredInputAmount);
        ISwapRouter(uniswapRouter).exactInputSingle(
            ISwapRouter.ExactInputSingleParams({
                tokenIn: tokenIn,
                tokenOut: tokenOut,
                fee: pool.fee(),
                recipient: recipient,
                deadline: deadline,
                amountIn: state.requiredInputAmount,
                amountOutMinimum: 0,
                sqrtPriceLimitX96: sqrtRatioTargetX96
            })
        );

        return state.requiredInputAmount;
    }
}

// The code below are taken almost verbatim from https://github.com/Uniswap/uniswap-v3-core/blob/main/contracts/libraries/TickBitmap.sol.
// They was modified slightly to enable the them to be called on an external pool by passing in a pool address and
// to accommodate solidity 0.8.

library TickBitmap {
    function nextInitializedTickWithinOneWord(
        IUniswapV3Pool pool,
        int24 tick,
        int24 tickSpacing,
        bool lte
    ) internal view returns (int24 next, bool initialized) {
        int24 compressed = tick / tickSpacing;
        if (tick < 0 && tick % tickSpacing != 0) compressed--; // round towards negative infinity

        if (lte) {
            (int16 wordPos, uint8 bitPos) = position(compressed);
            // all the 1s at or to the right of the current bitPos
            uint256 mask = (1 << bitPos) - 1 + (1 << bitPos);
            uint256 masked = pool.tickBitmap(wordPos) & mask;

            // if there are no initialized ticks to the right of or at the current tick, return rightmost in the word
            initialized = masked != 0;
            // overflow/underflow is possible, but prevented externally by limiting both tickSpacing and tick
            next = initialized
                ? (compressed - int24(int24(uint24(bitPos)) - int24(uint24(BitMath.mostSignificantBit(masked))))) *
                    tickSpacing
                : (compressed - int24(uint24(bitPos))) * tickSpacing;
        } else {
            // start from the word of the next tick, since the current tick state doesn't matter
            (int16 wordPos, uint8 bitPos) = position(compressed + 1);
            // all the 1s at or to the left of the bitPos
            uint256 mask = ~((1 << bitPos) - 1);
            uint256 masked = pool.tickBitmap(wordPos) & mask;

            // if there are no initialized ticks to the left of the current tick, return leftmost in the word
            initialized = masked != 0;
            // overflow/underflow is possible, but prevented externally by limiting both tickSpacing and tick
            next = initialized
                ? (compressed + 1 + int24(BitMath.leastSignificantBit(masked) - uint24(bitPos))) * tickSpacing
                : (compressed + 1 + int24(type(uint8).max - uint24(bitPos))) * tickSpacing;
        }
    }

    function position(int24 tick) private pure returns (int16 wordPos, uint8 bitPos) {
        wordPos = int16(tick >> 8);
        bitPos = uint8(int8(tick % 256));
    }

    function flipTick(
        mapping(int16 => uint256) storage self,
        int24 tick,
        int24 tickSpacing
    ) internal {
        require(tick % tickSpacing == 0); // ensure that the tick is spaced
        (int16 wordPos, uint8 bitPos) = position(tick / tickSpacing);
        uint256 mask = 1 << bitPos;
        self[wordPos] ^= mask;
    }
}

// Taken from https://github.com/Uniswap/uniswap-v3-core/blob/main/contracts/libraries/TickMath.sol and update
// to work with solidity 0.8.
library TickMath {
    /// @dev The minimum tick that may be passed to #getSqrtRatioAtTick computed from log base 1.0001 of 2**-128
    int24 internal constant MIN_TICK = -887272;
    /// @dev The maximum tick that may be passed to #getSqrtRatioAtTick computed from log base 1.0001 of 2**128
    int24 internal constant MAX_TICK = -MIN_TICK;

    /// @dev The minimum value that can be returned from #getSqrtRatioAtTick. Equivalent to getSqrtRatioAtTick(MIN_TICK)
    uint160 internal constant MIN_SQRT_RATIO = 4295128739;
    /// @dev The maximum value that can be returned from #getSqrtRatioAtTick. Equivalent to getSqrtRatioAtTick(MAX_TICK)
    uint160 internal constant MAX_SQRT_RATIO = 1461446703485210103287273052203988822378723970342;

    /// @notice Calculates sqrt(1.0001^tick) * 2^96
    /// @dev Throws if |tick| > max tick
    /// @param tick The input tick for the above formula
    /// @return sqrtPriceX96 A Fixed point Q64.96 number representing the sqrt of the ratio of the two assets (token1/token0)
    /// at the given tick
    function getSqrtRatioAtTick(int24 tick) internal pure returns (uint160 sqrtPriceX96) {
        uint256 absTick = tick < 0 ? uint256(-int256(tick)) : uint256(int256(tick));
        require(absTick <= uint256(int256(MAX_TICK)), "T");

        uint256 ratio = absTick & 0x1 != 0 ? 0xfffcb933bd6fad37aa2d162d1a594001 : 0x100000000000000000000000000000000;
        if (absTick & 0x2 != 0) ratio = (ratio * 0xfff97272373d413259a46990580e213a) >> 128;
        if (absTick & 0x4 != 0) ratio = (ratio * 0xfff2e50f5f656932ef12357cf3c7fdcc) >> 128;
        if (absTick & 0x8 != 0) ratio = (ratio * 0xffe5caca7e10e4e61c3624eaa0941cd0) >> 128;
        if (absTick & 0x10 != 0) ratio = (ratio * 0xffcb9843d60f6159c9db58835c926644) >> 128;
        if (absTick & 0x20 != 0) ratio = (ratio * 0xff973b41fa98c081472e6896dfb254c0) >> 128;
        if (absTick & 0x40 != 0) ratio = (ratio * 0xff2ea16466c96a3843ec78b326b52861) >> 128;
        if (absTick & 0x80 != 0) ratio = (ratio * 0xfe5dee046a99a2a811c461f1969c3053) >> 128;
        if (absTick & 0x100 != 0) ratio = (ratio * 0xfcbe86c7900a88aedcffc83b479aa3a4) >> 128;
        if (absTick & 0x200 != 0) ratio = (ratio * 0xf987a7253ac413176f2b074cf7815e54) >> 128;
        if (absTick & 0x400 != 0) ratio = (ratio * 0xf3392b0822b70005940c7a398e4b70f3) >> 128;
        if (absTick & 0x800 != 0) ratio = (ratio * 0xe7159475a2c29b7443b29c7fa6e889d9) >> 128;
        if (absTick & 0x1000 != 0) ratio = (ratio * 0xd097f3bdfd2022b8845ad8f792aa5825) >> 128;
        if (absTick & 0x2000 != 0) ratio = (ratio * 0xa9f746462d870fdf8a65dc1f90e061e5) >> 128;
        if (absTick & 0x4000 != 0) ratio = (ratio * 0x70d869a156d2a1b890bb3df62baf32f7) >> 128;
        if (absTick & 0x8000 != 0) ratio = (ratio * 0x31be135f97d08fd981231505542fcfa6) >> 128;
        if (absTick & 0x10000 != 0) ratio = (ratio * 0x9aa508b5b7a84e1c677de54f3e99bc9) >> 128;
        if (absTick & 0x20000 != 0) ratio = (ratio * 0x5d6af8dedb81196699c329225ee604) >> 128;
        if (absTick & 0x40000 != 0) ratio = (ratio * 0x2216e584f5fa1ea926041bedfe98) >> 128;
        if (absTick & 0x80000 != 0) ratio = (ratio * 0x48a170391f7dc42444e8fa2) >> 128;

        if (tick > 0) ratio = type(uint256).max / ratio;

        // this divides by 1<<32 rounding up to go from a Q128.128 to a Q128.96.
        // we then downcast because we know the result always fits within 160 bits due to our tick input constraint
        // we round up in the division so getTickAtSqrtRatio of the output price is always consistent
        sqrtPriceX96 = uint160((ratio >> 32) + (ratio % (1 << 32) == 0 ? 0 : 1));
    }

    function getTickAtSqrtRatio(uint160 sqrtPriceX96) internal pure returns (int24 tick) {
        // second inequality must be < because the price can never reach the price at the max tick
        require(sqrtPriceX96 >= MIN_SQRT_RATIO && sqrtPriceX96 < MAX_SQRT_RATIO, "R");
        uint256 ratio = uint256(sqrtPriceX96) << 32;

        uint256 r = ratio;
        uint256 msb = 0;

        assembly {
            let f := shl(7, gt(r, 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF))
            msb := or(msb, f)
            r := shr(f, r)
        }
        assembly {
            let f := shl(6, gt(r, 0xFFFFFFFFFFFFFFFF))
            msb := or(msb, f)
            r := shr(f, r)
        }
        assembly {
            let f := shl(5, gt(r, 0xFFFFFFFF))
            msb := or(msb, f)
            r := shr(f, r)
        }
        assembly {
            let f := shl(4, gt(r, 0xFFFF))
            msb := or(msb, f)
            r := shr(f, r)
        }
        assembly {
            let f := shl(3, gt(r, 0xFF))
            msb := or(msb, f)
            r := shr(f, r)
        }
        assembly {
            let f := shl(2, gt(r, 0xF))
            msb := or(msb, f)
            r := shr(f, r)
        }
        assembly {
            let f := shl(1, gt(r, 0x3))
            msb := or(msb, f)
            r := shr(f, r)
        }
        assembly {
            let f := gt(r, 0x1)
            msb := or(msb, f)
        }

        if (msb >= 128) r = ratio >> (msb - 127);
        else r = ratio << (127 - msb);

        int256 log_2 = (int256(msb) - 128) << 64;

        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(63, f))
            r := shr(f, r)
        }
        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(62, f))
            r := shr(f, r)
        }
        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(61, f))
            r := shr(f, r)
        }
        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(60, f))
            r := shr(f, r)
        }
        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(59, f))
            r := shr(f, r)
        }
        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(58, f))
            r := shr(f, r)
        }
        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(57, f))
            r := shr(f, r)
        }
        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(56, f))
            r := shr(f, r)
        }
        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(55, f))
            r := shr(f, r)
        }
        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(54, f))
            r := shr(f, r)
        }
        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(53, f))
            r := shr(f, r)
        }
        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(52, f))
            r := shr(f, r)
        }
        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(51, f))
            r := shr(f, r)
        }
        assembly {
            r := shr(127, mul(r, r))
            let f := shr(128, r)
            log_2 := or(log_2, shl(50, f))
        }

        int256 log_sqrt10001 = log_2 * 255738958999603826347141; // 128.128 number

        int24 tickLow = int24((log_sqrt10001 - 3402992956809132418596140100660247210) >> 128);
        int24 tickHi = int24((log_sqrt10001 + 291339464771989622907027621153398088495) >> 128);

        tick = tickLow == tickHi ? tickLow : getSqrtRatioAtTick(tickHi) <= sqrtPriceX96 ? tickHi : tickLow;
    }
}

library SqrtPriceMath {
    using SafeCast for uint256;

    function getAmount0Delta(
        uint160 sqrtRatioAX96,
        uint160 sqrtRatioBX96,
        uint128 liquidity,
        bool roundUp
    ) internal pure returns (uint256 amount0) {
        if (sqrtRatioAX96 > sqrtRatioBX96) (sqrtRatioAX96, sqrtRatioBX96) = (sqrtRatioBX96, sqrtRatioAX96);

        uint256 numerator1 = uint256(liquidity) << FixedPoint96.RESOLUTION;
        uint256 numerator2 = sqrtRatioBX96 - sqrtRatioAX96;

        require(sqrtRatioAX96 > 0);

        return
            roundUp
                ? UnsafeMath.divRoundingUp(
                    FullMath.mulDivRoundingUp(numerator1, numerator2, sqrtRatioBX96),
                    sqrtRatioAX96
                )
                : FullMath.mulDiv(numerator1, numerator2, sqrtRatioBX96) / sqrtRatioAX96;
    }

    /// @notice Gets the amount1 delta between two prices
    /// @dev Calculates liquidity * (sqrt(upper) - sqrt(lower))
    /// @param sqrtRatioAX96 A sqrt price
    /// @param sqrtRatioBX96 Another sqrt price
    /// @param liquidity The amount of usable liquidity
    /// @param roundUp Whether to round the amount up, or down
    /// @return amount1 Amount of token1 required to cover a position of size liquidity between the two passed prices
    function getAmount1Delta(
        uint160 sqrtRatioAX96,
        uint160 sqrtRatioBX96,
        uint128 liquidity,
        bool roundUp
    ) internal pure returns (uint256 amount1) {
        if (sqrtRatioAX96 > sqrtRatioBX96) (sqrtRatioAX96, sqrtRatioBX96) = (sqrtRatioBX96, sqrtRatioAX96);

        return
            roundUp
                ? FullMath.mulDivRoundingUp(liquidity, sqrtRatioBX96 - sqrtRatioAX96, FixedPoint96.Q96)
                : FullMath.mulDiv(liquidity, sqrtRatioBX96 - sqrtRatioAX96, FixedPoint96.Q96);
    }

    /// @notice Helper that gets signed token0 delta
    /// @param sqrtRatioAX96 A sqrt price
    /// @param sqrtRatioBX96 Another sqrt price
    /// @param liquidity The change in liquidity for which to compute the amount0 delta
    /// @return amount0 Amount of token0 corresponding to the passed liquidityDelta between the two prices
    function getAmount0Delta(
        uint160 sqrtRatioAX96,
        uint160 sqrtRatioBX96,
        int128 liquidity
    ) internal pure returns (int256 amount0) {
        return
            liquidity < 0
                ? -getAmount0Delta(sqrtRatioAX96, sqrtRatioBX96, uint128(-liquidity), false).toInt256()
                : getAmount0Delta(sqrtRatioAX96, sqrtRatioBX96, uint128(liquidity), true).toInt256();
    }

    /// @notice Helper that gets signed token1 delta
    /// @param sqrtRatioAX96 A sqrt price
    /// @param sqrtRatioBX96 Another sqrt price
    /// @param liquidity The change in liquidity for which to compute the amount1 delta
    /// @return amount1 Amount of token1 corresponding to the passed liquidityDelta between the two prices
    function getAmount1Delta(
        uint160 sqrtRatioAX96,
        uint160 sqrtRatioBX96,
        int128 liquidity
    ) internal pure returns (int256 amount1) {
        return
            liquidity < 0
                ? -getAmount1Delta(sqrtRatioAX96, sqrtRatioBX96, uint128(-liquidity), false).toInt256()
                : getAmount1Delta(sqrtRatioAX96, sqrtRatioBX96, uint128(liquidity), true).toInt256();
    }
}

library FullMath {
    /// @notice Calculates floor(a×b÷denominator) with full precision. Throws if result overflows a uint256 or denominator == 0
    /// @param a The multiplicand
    /// @param b The multiplier
    /// @param denominator The divisor
    /// @return result The 256-bit result
    /// @dev Credit to Remco Bloemen under MIT license https://xn--2-umb.com/21/muldiv
    function mulDiv(
        uint256 a,
        uint256 b,
        uint256 denominator
    ) internal pure returns (uint256 result) {
        // 512-bit multiply [prod1 prod0] = a * b
        // Compute the product mod 2**256 and mod 2**256 - 1
        // then use the Chinese Remainder Theorem to reconstruct
        // the 512 bit result. The result is stored in two 256
        // variables such that product = prod1 * 2**256 + prod0
        uint256 prod0; // Least significant 256 bits of the product
        uint256 prod1; // Most significant 256 bits of the product
        assembly {
            let mm := mulmod(a, b, not(0))
            prod0 := mul(a, b)
            prod1 := sub(sub(mm, prod0), lt(mm, prod0))
        }

        // Handle non-overflow cases, 256 by 256 division
        if (prod1 == 0) {
            require(denominator > 0);
            assembly {
                result := div(prod0, denominator)
            }
            return result;
        }

        // Make sure the result is less than 2**256.
        // Also prevents denominator == 0
        require(denominator > prod1);

        ///////////////////////////////////////////////
        // 512 by 256 division.
        ///////////////////////////////////////////////

        // Make division exact by subtracting the remainder from [prod1 prod0]
        // Compute remainder using mulmod
        uint256 remainder;
        assembly {
            remainder := mulmod(a, b, denominator)
        }
        // Subtract 256 bit number from 512 bit number
        assembly {
            prod1 := sub(prod1, gt(remainder, prod0))
            prod0 := sub(prod0, remainder)
        }

        // Factor powers of two out of denominator
        // Compute largest power of two divisor of denominator.
        // Always >= 1.
        // NOTE: this is modified from the original Full math implementation to work with solidity 8
        uint256 twos = (type(uint256).max - denominator + 1) & denominator;
        // Divide denominator by power of two
        assembly {
            denominator := div(denominator, twos)
        }

        // Divide [prod1 prod0] by the factors of two
        assembly {
            prod0 := div(prod0, twos)
        }
        // Shift in bits from prod1 into prod0. For this we need
        // to flip `twos` such that it is 2**256 / twos.
        // If twos is zero, then it becomes one
        assembly {
            twos := add(div(sub(0, twos), twos), 1)
        }
        unchecked {
            prod0 |= prod1 * twos;

            // Invert denominator mod 2**256
            // Now that denominator is an odd number, it has an inverse
            // modulo 2**256 such that denominator * inv = 1 mod 2**256.
            // Compute the inverse by starting with a seed that is correct
            // correct for four bits. That is, denominator * inv = 1 mod 2**4
            uint256 inv = (3 * denominator) ^ 2;
            // Now use Newton-Raphson iteration to improve the precision.
            // Thanks to Hensel's lifting lemma, this also works in modular
            // arithmetic, doubling the correct bits in each step.
            // NOTE: this is modified from the original Full math implementation to work with solidity 8 with the unchecked syntax.
            inv *= 2 - denominator * inv; // inverse mod 2**8
            inv *= 2 - denominator * inv; // inverse mod 2**16
            inv *= 2 - denominator * inv; // inverse mod 2**32
            inv *= 2 - denominator * inv; // inverse mod 2**64
            inv *= 2 - denominator * inv; // inverse mod 2**128
            inv *= 2 - denominator * inv; // inverse mod 2**256
            // Because the division is now exact we can divide by multiplying
            // with the modular inverse of denominator. This will give us the
            // correct result modulo 2**256. Since the precoditions guarantee
            // that the outcome is less than 2**256, this is the final result.
            // We don't need to compute the high bits of the result and prod1
            // is no longer required.
            result = prod0 * inv;
        }
        return result;
    }

    /// @notice Calculates ceil(a×b÷denominator) with full precision. Throws if result overflows a uint256 or denominator == 0
    /// @param a The multiplicand
    /// @param b The multiplier
    /// @param denominator The divisor
    /// @return result The 256-bit result
    function mulDivRoundingUp(
        uint256 a,
        uint256 b,
        uint256 denominator
    ) internal pure returns (uint256 result) {
        result = mulDiv(a, b, denominator);
        if (mulmod(a, b, denominator) > 0) {
            require(result < type(uint256).max);
            result++;
        }
    }
}

library FixedPoint96 {
    uint8 internal constant RESOLUTION = 96;
    uint256 internal constant Q96 = 0x1000000000000000000000000;
}