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Contract Name:
GDACurve
Compiler Version
v0.8.20+commit.a1b79de6
Optimization Enabled:
Yes with 1000000 runs
Other Settings:
default evmVersion
Contract Source Code (Solidity Standard Json-Input format)
// SPDX-License-Identifier: AGPL-3.0 pragma solidity ^0.8.13; import {ICurve} from "./ICurve.sol"; import {CurveErrorCodes} from "./CurveErrorCodes.sol"; import {UD60x18, ud, unwrap, uUNIT, UNIT, convert} from "@prb/math/UD60x18.sol"; /** * @author 0xmons, boredGenius, 0xCygaar * @notice Bonding curve logic for a gradual dutch auction based on https://www.paradigm.xyz/2022/04/gda. * @dev Trade pools will result in unexpected behavior due to the time factor always increasing. Buying an NFT * and selling it back into the pool will result in a non-zero difference. Therefore it is recommended to only * use this curve for single-sided pools. */ contract GDACurve is ICurve, CurveErrorCodes { uint256 internal constant _SCALE_FACTOR = 1e9; uint256 internal constant _TIME_SCALAR = 2 * uUNIT; // Used in place of Euler's number uint256 internal constant _MAX_TIME_EXPONENT = 10; /** * @notice Minimum price to prevent numerical issues */ uint256 public constant MIN_PRICE = 1 gwei; /** * @dev See {ICurve-validateDelta} */ function validateDelta(uint128 delta) external pure override returns (bool) { (UD60x18 alpha,,) = _parseDelta(delta); return alpha.gt(UNIT); } /** * @dev See {ICurve-validateSpotPrice} */ function validateSpotPrice(uint128 newSpotPrice) external pure override returns (bool) { return newSpotPrice >= MIN_PRICE; } /** * @dev See {ICurve-getBuyInfo} */ function getBuyInfo( uint128 spotPrice, uint128 delta, uint256 numItems, uint256 feeMultiplier, uint256 protocolFeeMultiplier ) external view override returns ( Error error, uint128 newSpotPrice, uint128 newDelta, uint256 inputValue, uint256 tradeFee, uint256 protocolFee ) { // NOTE: we assume alpha is > 1, as checked by validateDelta() // We only calculate changes for buying 1 or more NFTs if (numItems == 0) { return (Error.INVALID_NUMITEMS, 0, 0, 0, 0, 0); } UD60x18 spotPrice_ = ud(spotPrice); UD60x18 decayFactor; { (, uint256 lambda, uint256 prevTime) = _parseDelta(delta); UD60x18 exponent = ud((block.timestamp - prevTime) * lambda); if (convert(exponent) > _MAX_TIME_EXPONENT) { // Cap the max decay factor to 2^20 exponent = convert(_MAX_TIME_EXPONENT); } decayFactor = ud(_TIME_SCALAR).pow(exponent); } (UD60x18 alpha,,) = _parseDelta(delta); UD60x18 alphaPowN = alpha.powu(numItems); // The new spot price is multiplied by alpha^n and divided by the time decay so future // calculations do not need to track number of items sold or the initial time/price. This new spot price // implicitly stores the the initial price, total items sold so far, and time elapsed since the start. { UD60x18 newSpotPrice_ = spotPrice_.mul(alphaPowN); newSpotPrice_ = newSpotPrice_.div(decayFactor); if (newSpotPrice_.gt(ud(type(uint128).max))) { return (Error.SPOT_PRICE_OVERFLOW, 0, 0, 0, 0, 0); } if (newSpotPrice_.lt(ud(MIN_PRICE))) { return (Error.SPOT_PRICE_UNDERFLOW, 0, 0, 0, 0, 0); } newSpotPrice = uint128(unwrap(newSpotPrice_)); } // If the user buys n items, then the total cost is equal to: // buySpotPrice + (alpha * buySpotPrice) + (alpha^2 * buySpotPrice) + ... (alpha^(numItems - 1) * buySpotPrice). // This is equal to buySpotPrice * (alpha^n - 1) / (alpha - 1). // We then divide the value by scalar^(lambda * timeElapsed) to factor in the exponential decay. { UD60x18 inputValue_ = spotPrice_.mul(alphaPowN.sub(UNIT)).div(alpha.sub(UNIT)).div(decayFactor); // Account for the protocol fee, a flat percentage of the buy amount protocolFee = unwrap(inputValue_.mul(ud(protocolFeeMultiplier))); // Account for the trade fee, only for Trade pools tradeFee = unwrap(inputValue_.mul(ud(feeMultiplier))); // Add the protocol and trade fees to the required input amount and unwrap to uint256 inputValue = unwrap(inputValue_.add(ud(protocolFee)).add(ud(tradeFee))); } // Update delta with the current timestamp newDelta = _getNewDelta(delta); // If we got all the way here, no math error happened error = Error.OK; } /** * @dev See {ICurve-getSellInfo} */ function getSellInfo( uint128 spotPrice, uint128 delta, uint256 numItems, uint256 feeMultiplier, uint256 protocolFeeMultiplier ) external view override returns ( Error error, uint128 newSpotPrice, uint128 newDelta, uint256 outputValue, uint256 tradeFee, uint256 protocolFee ) { // We only calculate changes for buying 1 or more NFTs if (numItems == 0) { return (Error.INVALID_NUMITEMS, 0, 0, 0, 0, 0); } UD60x18 spotPrice_ = ud(spotPrice); UD60x18 boostFactor; { (, uint256 lambda, uint256 prevTime) = _parseDelta(delta); UD60x18 exponent = ud((block.timestamp - prevTime) * lambda); if (convert(exponent) > _MAX_TIME_EXPONENT) { // Cap the max boost factor to 2^20 exponent = convert(_MAX_TIME_EXPONENT); } boostFactor = ud(_TIME_SCALAR).pow(exponent); } (UD60x18 alpha,,) = _parseDelta(delta); UD60x18 alphaPowN = alpha.powu(numItems); // The new spot price is multiplied by the time boost and divided by alpha^n so future // calculations do not need to track number of items sold or the initial time/price. This new spot price // implicitly stores the the initial price, total items sold so far, and time elapsed since the start. { UD60x18 newSpotPrice_ = spotPrice_.mul(boostFactor); newSpotPrice_ = newSpotPrice_.div(alphaPowN); if (newSpotPrice_.gt(ud(type(uint128).max))) { return (Error.SPOT_PRICE_OVERFLOW, 0, 0, 0, 0, 0); } if (newSpotPrice_.lt(ud(MIN_PRICE))) { return (Error.SPOT_PRICE_UNDERFLOW, 0, 0, 0, 0, 0); } newSpotPrice = uint128(unwrap(newSpotPrice_)); } // The expected output for an auction at index n is defined by the formula: p(t) = k * scalar^(lambda * t) / alpha^n // where k is the initial price, lambda is the boost constant, t is time elapsed, alpha is the scale factor, and // n is the number of items sold. The amount to receive for selling into a pool can thus be written as: // k * scalar^(lambda * t) / alpha^(m + q - 1) * (alpha^q - 1) / (alpha - 1) where m is the number of items purchased thus far // and q is the number of items to sell. // Our spot price implicity embeds the number of items already purchased and the previous time boost, so we just need to // do some simple adjustments to get the current scalar^(lambda * t) and alpha^(m + q - 1) values. UD60x18 outputValue_ = spotPrice_.mul(boostFactor).div(alphaPowN.div(alpha)).mul(alphaPowN.sub(UNIT)).div(alpha.sub(UNIT)); // Account for the protocol fee, a flat percentage of the sell amount protocolFee = unwrap(outputValue_.mul(ud(protocolFeeMultiplier))); // Account for the trade fee, only for Trade pools tradeFee = unwrap(outputValue_.mul(ud(feeMultiplier))); // Remove the protocol and trade fees from the output amount and unwrap to uint256 outputValue = unwrap(outputValue_.sub(ud(protocolFee)).sub(ud(tradeFee))); // Update delta with the current timestamp newDelta = _getNewDelta(delta); // If we got all the way here, no math error happened error = Error.OK; } function _parseDelta(uint128 delta) internal pure returns (UD60x18 alpha, uint256 lambda, uint256 prevTime) { // The highest 40 bits are alpha with 9 decimals of precision. // However, because our alpha value needs to be 18 decimals of precision, we multiply by a scaling factor alpha = ud(uint40(delta >> 88) * _SCALE_FACTOR); // The middle 40 bits are lambda with 9 decimals of precision // lambda determines the exponential decay (when buying) or exponential boost (when selling) over time // See https://www.paradigm.xyz/2022/04/gda // lambda also needs to be 18 decimals of precision so we multiply by a scaling factor lambda = uint40(delta >> 48) * _SCALE_FACTOR; // The lowest 48 bits are the start timestamp // This works because solidity cuts off higher bits when converting from a larger type to a smaller type // See https://docs.soliditylang.org/en/latest/types.html#explicit-conversions prevTime = uint256(uint48(delta)); } function _getNewDelta(uint128 delta) internal view returns (uint128) { // Clear lower 48 bits delta = (delta >> 48) << 48; // Set lower 48 bits to be the current timestamp return delta | uint48(block.timestamp); } }
// SPDX-License-Identifier: AGPL-3.0 pragma solidity ^0.8.0; import {CurveErrorCodes} from "./CurveErrorCodes.sol"; interface ICurve { /** * @notice Validates if a delta value is valid for the curve. The criteria for * validity can be different for each type of curve, for instance ExponentialCurve * requires delta to be greater than 1. * @param delta The delta value to be validated * @return valid True if delta is valid, false otherwise */ function validateDelta(uint128 delta) external pure returns (bool valid); /** * @notice Validates if a new spot price is valid for the curve. Spot price is generally assumed to be the immediate sell price of 1 NFT to the pool, in units of the pool's paired token. * @param newSpotPrice The new spot price to be set * @return valid True if the new spot price is valid, false otherwise */ function validateSpotPrice(uint128 newSpotPrice) external view returns (bool valid); /** * @notice Given the current state of the pair and the trade, computes how much the user * should pay to purchase an NFT from the pair, the new spot price, and other values. * @param spotPrice The current selling spot price of the pair, in tokens * @param delta The delta parameter of the pair, what it means depends on the curve * @param numItems The number of NFTs the user is buying from the pair * @param feeMultiplier Determines how much fee the LP takes from this trade, 18 decimals * @param protocolFeeMultiplier Determines how much fee the protocol takes from this trade, 18 decimals * @return error Any math calculation errors, only Error.OK means the returned values are valid * @return newSpotPrice The updated selling spot price, in tokens * @return newDelta The updated delta, used to parameterize the bonding curve * @return inputValue The amount that the user should pay, in tokens * @return tradeFee The amount that is sent to the trade fee recipient * @return protocolFee The amount of fee to send to the protocol, in tokens */ function getBuyInfo( uint128 spotPrice, uint128 delta, uint256 numItems, uint256 feeMultiplier, uint256 protocolFeeMultiplier ) external view returns ( CurveErrorCodes.Error error, uint128 newSpotPrice, uint128 newDelta, uint256 inputValue, uint256 tradeFee, uint256 protocolFee ); /** * @notice Given the current state of the pair and the trade, computes how much the user * should receive when selling NFTs to the pair, the new spot price, and other values. * @param spotPrice The current selling spot price of the pair, in tokens * @param delta The delta parameter of the pair, what it means depends on the curve * @param numItems The number of NFTs the user is selling to the pair * @param feeMultiplier Determines how much fee the LP takes from this trade, 18 decimals * @param protocolFeeMultiplier Determines how much fee the protocol takes from this trade, 18 decimals * @return error Any math calculation errors, only Error.OK means the returned values are valid * @return newSpotPrice The updated selling spot price, in tokens * @return newDelta The updated delta, used to parameterize the bonding curve * @return outputValue The amount that the user should receive, in tokens * @return tradeFee The amount that is sent to the trade fee recipient * @return protocolFee The amount of fee to send to the protocol, in tokens */ function getSellInfo( uint128 spotPrice, uint128 delta, uint256 numItems, uint256 feeMultiplier, uint256 protocolFeeMultiplier ) external view returns ( CurveErrorCodes.Error error, uint128 newSpotPrice, uint128 newDelta, uint256 outputValue, uint256 tradeFee, uint256 protocolFee ); }
// SPDX-License-Identifier: AGPL-3.0 pragma solidity ^0.8.0; contract CurveErrorCodes { enum Error { OK, // No error INVALID_NUMITEMS, // The numItem value is 0 SPOT_PRICE_OVERFLOW, // The updated spot price doesn't fit into 128 bits DELTA_OVERFLOW, // The updated delta doesn't fit into 128 bits SPOT_PRICE_UNDERFLOW // The updated spot price goes too low } }
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import "./ud60x18/Casting.sol"; import "./ud60x18/Constants.sol"; import "./ud60x18/Conversions.sol"; import "./ud60x18/Errors.sol"; import "./ud60x18/Helpers.sol"; import "./ud60x18/Math.sol"; import "./ud60x18/ValueType.sol";
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { MAX_UINT128, MAX_UINT40 } from "../Common.sol"; import { uMAX_SD1x18 } from "../sd1x18/Constants.sol"; import { SD1x18 } from "../sd1x18/ValueType.sol"; import { uMAX_SD59x18 } from "../sd59x18/Constants.sol"; import { SD59x18 } from "../sd59x18/ValueType.sol"; import { uMAX_UD2x18 } from "../ud2x18/Constants.sol"; import { UD2x18 } from "../ud2x18/ValueType.sol"; import { PRBMath_UD60x18_IntoSD1x18_Overflow, PRBMath_UD60x18_IntoUD2x18_Overflow, PRBMath_UD60x18_IntoSD59x18_Overflow, PRBMath_UD60x18_IntoUint128_Overflow, PRBMath_UD60x18_IntoUint40_Overflow } from "./Errors.sol"; import { UD60x18 } from "./ValueType.sol"; /// @notice Casts an UD60x18 number into SD1x18. /// @dev Requirements: /// - x must be less than or equal to `uMAX_SD1x18`. function intoSD1x18(UD60x18 x) pure returns (SD1x18 result) { uint256 xUint = UD60x18.unwrap(x); if (xUint > uint256(int256(uMAX_SD1x18))) { revert PRBMath_UD60x18_IntoSD1x18_Overflow(x); } result = SD1x18.wrap(int64(uint64(xUint))); } /// @notice Casts an UD60x18 number into UD2x18. /// @dev Requirements: /// - x must be less than or equal to `uMAX_UD2x18`. function intoUD2x18(UD60x18 x) pure returns (UD2x18 result) { uint256 xUint = UD60x18.unwrap(x); if (xUint > uMAX_UD2x18) { revert PRBMath_UD60x18_IntoUD2x18_Overflow(x); } result = UD2x18.wrap(uint64(xUint)); } /// @notice Casts an UD60x18 number into SD59x18. /// @dev Requirements: /// - x must be less than or equal to `uMAX_SD59x18`. function intoSD59x18(UD60x18 x) pure returns (SD59x18 result) { uint256 xUint = UD60x18.unwrap(x); if (xUint > uint256(uMAX_SD59x18)) { revert PRBMath_UD60x18_IntoSD59x18_Overflow(x); } result = SD59x18.wrap(int256(xUint)); } /// @notice Casts an UD60x18 number into uint128. /// @dev This is basically a functional alias for the `unwrap` function. function intoUint256(UD60x18 x) pure returns (uint256 result) { result = UD60x18.unwrap(x); } /// @notice Casts an UD60x18 number into uint128. /// @dev Requirements: /// - x must be less than or equal to `MAX_UINT128`. function intoUint128(UD60x18 x) pure returns (uint128 result) { uint256 xUint = UD60x18.unwrap(x); if (xUint > MAX_UINT128) { revert PRBMath_UD60x18_IntoUint128_Overflow(x); } result = uint128(xUint); } /// @notice Casts an UD60x18 number into uint40. /// @dev Requirements: /// - x must be less than or equal to `MAX_UINT40`. function intoUint40(UD60x18 x) pure returns (uint40 result) { uint256 xUint = UD60x18.unwrap(x); if (xUint > MAX_UINT40) { revert PRBMath_UD60x18_IntoUint40_Overflow(x); } result = uint40(xUint); } /// @notice Alias for the `wrap` function. function ud(uint256 x) pure returns (UD60x18 result) { result = UD60x18.wrap(x); } /// @notice Alias for the `wrap` function. function ud60x18(uint256 x) pure returns (UD60x18 result) { result = UD60x18.wrap(x); } /// @notice Unwraps an UD60x18 number into uint256. function unwrap(UD60x18 x) pure returns (uint256 result) { result = UD60x18.unwrap(x); } /// @notice Wraps an uint256 number into the UD60x18 value type. function wrap(uint256 x) pure returns (UD60x18 result) { result = UD60x18.wrap(x); }
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { UD60x18 } from "./ValueType.sol"; /// @dev Euler's number as an UD60x18 number. UD60x18 constant E = UD60x18.wrap(2_718281828459045235); /// @dev Half the UNIT number. uint256 constant uHALF_UNIT = 0.5e18; UD60x18 constant HALF_UNIT = UD60x18.wrap(uHALF_UNIT); /// @dev log2(10) as an UD60x18 number. uint256 constant uLOG2_10 = 3_321928094887362347; UD60x18 constant LOG2_10 = UD60x18.wrap(uLOG2_10); /// @dev log2(e) as an UD60x18 number. uint256 constant uLOG2_E = 1_442695040888963407; UD60x18 constant LOG2_E = UD60x18.wrap(uLOG2_E); /// @dev The maximum value an UD60x18 number can have. uint256 constant uMAX_UD60x18 = 115792089237316195423570985008687907853269984665640564039457_584007913129639935; UD60x18 constant MAX_UD60x18 = UD60x18.wrap(uMAX_UD60x18); /// @dev The maximum whole value an UD60x18 number can have. uint256 constant uMAX_WHOLE_UD60x18 = 115792089237316195423570985008687907853269984665640564039457_000000000000000000; UD60x18 constant MAX_WHOLE_UD60x18 = UD60x18.wrap(uMAX_WHOLE_UD60x18); /// @dev PI as an UD60x18 number. UD60x18 constant PI = UD60x18.wrap(3_141592653589793238); /// @dev The unit amount that implies how many trailing decimals can be represented. uint256 constant uUNIT = 1e18; UD60x18 constant UNIT = UD60x18.wrap(uUNIT); /// @dev Zero as an UD60x18 number. UD60x18 constant ZERO = UD60x18.wrap(0);
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { uMAX_UD60x18, uUNIT } from "./Constants.sol"; import { PRBMath_UD60x18_Convert_Overflow } from "./Errors.sol"; import { UD60x18 } from "./ValueType.sol"; /// @notice Converts an UD60x18 number to a simple integer by dividing it by `UNIT`. Rounds towards zero in the process. /// @dev Rounds down in the process. /// @param x The UD60x18 number to convert. /// @return result The same number in basic integer form. function convert(UD60x18 x) pure returns (uint256 result) { result = UD60x18.unwrap(x) / uUNIT; } /// @notice Converts a simple integer to UD60x18 by multiplying it by `UNIT`. /// /// @dev Requirements: /// - x must be less than or equal to `MAX_UD60x18` divided by `UNIT`. /// /// @param x The basic integer to convert. /// @param result The same number converted to UD60x18. function convert(uint256 x) pure returns (UD60x18 result) { if (x > uMAX_UD60x18 / uUNIT) { revert PRBMath_UD60x18_Convert_Overflow(x); } unchecked { result = UD60x18.wrap(x * uUNIT); } } /// @notice Alias for the `convert` function defined above. /// @dev Here for backward compatibility. Will be removed in V4. function fromUD60x18(UD60x18 x) pure returns (uint256 result) { result = convert(x); } /// @notice Alias for the `convert` function defined above. /// @dev Here for backward compatibility. Will be removed in V4. function toUD60x18(uint256 x) pure returns (UD60x18 result) { result = convert(x); }
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { UD60x18 } from "./ValueType.sol"; /// @notice Emitted when ceiling a number overflows UD60x18. error PRBMath_UD60x18_Ceil_Overflow(UD60x18 x); /// @notice Emitted when converting a basic integer to the fixed-point format overflows UD60x18. error PRBMath_UD60x18_Convert_Overflow(uint256 x); /// @notice Emitted when taking the natural exponent of a base greater than 133.084258667509499441. error PRBMath_UD60x18_Exp_InputTooBig(UD60x18 x); /// @notice Emitted when taking the binary exponent of a base greater than 192. error PRBMath_UD60x18_Exp2_InputTooBig(UD60x18 x); /// @notice Emitted when taking the geometric mean of two numbers and multiplying them overflows UD60x18. error PRBMath_UD60x18_Gm_Overflow(UD60x18 x, UD60x18 y); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in SD1x18. error PRBMath_UD60x18_IntoSD1x18_Overflow(UD60x18 x); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in SD59x18. error PRBMath_UD60x18_IntoSD59x18_Overflow(UD60x18 x); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in UD2x18. error PRBMath_UD60x18_IntoUD2x18_Overflow(UD60x18 x); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in uint128. error PRBMath_UD60x18_IntoUint128_Overflow(UD60x18 x); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in uint40. error PRBMath_UD60x18_IntoUint40_Overflow(UD60x18 x); /// @notice Emitted when taking the logarithm of a number less than 1. error PRBMath_UD60x18_Log_InputTooSmall(UD60x18 x); /// @notice Emitted when calculating the square root overflows UD60x18. error PRBMath_UD60x18_Sqrt_Overflow(UD60x18 x);
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { unwrap, wrap } from "./Casting.sol"; import { UD60x18 } from "./ValueType.sol"; /// @notice Implements the checked addition operation (+) in the UD60x18 type. function add(UD60x18 x, UD60x18 y) pure returns (UD60x18 result) { result = wrap(unwrap(x) + unwrap(y)); } /// @notice Implements the AND (&) bitwise operation in the UD60x18 type. function and(UD60x18 x, uint256 bits) pure returns (UD60x18 result) { result = wrap(unwrap(x) & bits); } /// @notice Implements the equal operation (==) in the UD60x18 type. function eq(UD60x18 x, UD60x18 y) pure returns (bool result) { result = unwrap(x) == unwrap(y); } /// @notice Implements the greater than operation (>) in the UD60x18 type. function gt(UD60x18 x, UD60x18 y) pure returns (bool result) { result = unwrap(x) > unwrap(y); } /// @notice Implements the greater than or equal to operation (>=) in the UD60x18 type. function gte(UD60x18 x, UD60x18 y) pure returns (bool result) { result = unwrap(x) >= unwrap(y); } /// @notice Implements a zero comparison check function in the UD60x18 type. function isZero(UD60x18 x) pure returns (bool result) { // This wouldn't work if x could be negative. result = unwrap(x) == 0; } /// @notice Implements the left shift operation (<<) in the UD60x18 type. function lshift(UD60x18 x, uint256 bits) pure returns (UD60x18 result) { result = wrap(unwrap(x) << bits); } /// @notice Implements the lower than operation (<) in the UD60x18 type. function lt(UD60x18 x, UD60x18 y) pure returns (bool result) { result = unwrap(x) < unwrap(y); } /// @notice Implements the lower than or equal to operation (<=) in the UD60x18 type. function lte(UD60x18 x, UD60x18 y) pure returns (bool result) { result = unwrap(x) <= unwrap(y); } /// @notice Implements the checked modulo operation (%) in the UD60x18 type. function mod(UD60x18 x, UD60x18 y) pure returns (UD60x18 result) { result = wrap(unwrap(x) % unwrap(y)); } /// @notice Implements the not equal operation (!=) in the UD60x18 type function neq(UD60x18 x, UD60x18 y) pure returns (bool result) { result = unwrap(x) != unwrap(y); } /// @notice Implements the OR (|) bitwise operation in the UD60x18 type. function or(UD60x18 x, UD60x18 y) pure returns (UD60x18 result) { result = wrap(unwrap(x) | unwrap(y)); } /// @notice Implements the right shift operation (>>) in the UD60x18 type. function rshift(UD60x18 x, uint256 bits) pure returns (UD60x18 result) { result = wrap(unwrap(x) >> bits); } /// @notice Implements the checked subtraction operation (-) in the UD60x18 type. function sub(UD60x18 x, UD60x18 y) pure returns (UD60x18 result) { result = wrap(unwrap(x) - unwrap(y)); } /// @notice Implements the unchecked addition operation (+) in the UD60x18 type. function uncheckedAdd(UD60x18 x, UD60x18 y) pure returns (UD60x18 result) { unchecked { result = wrap(unwrap(x) + unwrap(y)); } } /// @notice Implements the unchecked subtraction operation (-) in the UD60x18 type. function uncheckedSub(UD60x18 x, UD60x18 y) pure returns (UD60x18 result) { unchecked { result = wrap(unwrap(x) - unwrap(y)); } } /// @notice Implements the XOR (^) bitwise operation in the UD60x18 type. function xor(UD60x18 x, UD60x18 y) pure returns (UD60x18 result) { result = wrap(unwrap(x) ^ unwrap(y)); }
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { msb, mulDiv, mulDiv18, prbExp2, prbSqrt } from "../Common.sol"; import { unwrap, wrap } from "./Casting.sol"; import { uHALF_UNIT, uLOG2_10, uLOG2_E, uMAX_UD60x18, uMAX_WHOLE_UD60x18, UNIT, uUNIT, ZERO } from "./Constants.sol"; import { PRBMath_UD60x18_Ceil_Overflow, PRBMath_UD60x18_Exp_InputTooBig, PRBMath_UD60x18_Exp2_InputTooBig, PRBMath_UD60x18_Gm_Overflow, PRBMath_UD60x18_Log_InputTooSmall, PRBMath_UD60x18_Sqrt_Overflow } from "./Errors.sol"; import { UD60x18 } from "./ValueType.sol"; /*////////////////////////////////////////////////////////////////////////// MATHEMATICAL FUNCTIONS //////////////////////////////////////////////////////////////////////////*/ /// @notice Calculates the arithmetic average of x and y, rounding down. /// /// @dev Based on the formula: /// /// $$ /// avg(x, y) = (x & y) + ((xUint ^ yUint) / 2) /// $$ // /// In English, what this formula does is: /// /// 1. AND x and y. /// 2. Calculate half of XOR x and y. /// 3. Add the two results together. /// /// This technique is known as SWAR, which stands for "SIMD within a register". You can read more about it here: /// https://devblogs.microsoft.com/oldnewthing/20220207-00/?p=106223 /// /// @param x The first operand as an UD60x18 number. /// @param y The second operand as an UD60x18 number. /// @return result The arithmetic average as an UD60x18 number. function avg(UD60x18 x, UD60x18 y) pure returns (UD60x18 result) { uint256 xUint = unwrap(x); uint256 yUint = unwrap(y); unchecked { result = wrap((xUint & yUint) + ((xUint ^ yUint) >> 1)); } } /// @notice Yields the smallest whole UD60x18 number greater than or equal to x. /// /// @dev This is optimized for fractional value inputs, because for every whole value there are "1e18 - 1" fractional /// counterparts. See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions. /// /// Requirements: /// - x must be less than or equal to `MAX_WHOLE_UD60x18`. /// /// @param x The UD60x18 number to ceil. /// @param result The least number greater than or equal to x, as an UD60x18 number. function ceil(UD60x18 x) pure returns (UD60x18 result) { uint256 xUint = unwrap(x); if (xUint > uMAX_WHOLE_UD60x18) { revert PRBMath_UD60x18_Ceil_Overflow(x); } assembly ("memory-safe") { // Equivalent to "x % UNIT" but faster. let remainder := mod(x, uUNIT) // Equivalent to "UNIT - remainder" but faster. let delta := sub(uUNIT, remainder) // Equivalent to "x + delta * (remainder > 0 ? 1 : 0)" but faster. result := add(x, mul(delta, gt(remainder, 0))) } } /// @notice Divides two UD60x18 numbers, returning a new UD60x18 number. Rounds towards zero. /// /// @dev Uses `mulDiv` to enable overflow-safe multiplication and division. /// /// Requirements: /// - The denominator cannot be zero. /// /// @param x The numerator as an UD60x18 number. /// @param y The denominator as an UD60x18 number. /// @param result The quotient as an UD60x18 number. function div(UD60x18 x, UD60x18 y) pure returns (UD60x18 result) { result = wrap(mulDiv(unwrap(x), uUNIT, unwrap(y))); } /// @notice Calculates the natural exponent of x. /// /// @dev Based on the formula: /// /// $$ /// e^x = 2^{x * log_2{e}} /// $$ /// /// Requirements: /// - All from `log2`. /// - x must be less than 133.084258667509499441. /// /// @param x The exponent as an UD60x18 number. /// @return result The result as an UD60x18 number. function exp(UD60x18 x) pure returns (UD60x18 result) { uint256 xUint = unwrap(x); // Without this check, the value passed to `exp2` would be greater than 192. if (xUint >= 133_084258667509499441) { revert PRBMath_UD60x18_Exp_InputTooBig(x); } unchecked { // We do the fixed-point multiplication inline rather than via the `mul` function to save gas. uint256 doubleUnitProduct = xUint * uLOG2_E; result = exp2(wrap(doubleUnitProduct / uUNIT)); } } /// @notice Calculates the binary exponent of x using the binary fraction method. /// /// @dev See https://ethereum.stackexchange.com/q/79903/24693. /// /// Requirements: /// - x must be 192 or less. /// - The result must fit within `MAX_UD60x18`. /// /// @param x The exponent as an UD60x18 number. /// @return result The result as an UD60x18 number. function exp2(UD60x18 x) pure returns (UD60x18 result) { uint256 xUint = unwrap(x); // Numbers greater than or equal to 2^192 don't fit within the 192.64-bit format. if (xUint >= 192e18) { revert PRBMath_UD60x18_Exp2_InputTooBig(x); } // Convert x to the 192.64-bit fixed-point format. uint256 x_192x64 = (xUint << 64) / uUNIT; // Pass x to the `prbExp2` function, which uses the 192.64-bit fixed-point number representation. result = wrap(prbExp2(x_192x64)); } /// @notice Yields the greatest whole UD60x18 number less than or equal to x. /// @dev Optimized for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts. /// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions. /// @param x The UD60x18 number to floor. /// @param result The greatest integer less than or equal to x, as an UD60x18 number. function floor(UD60x18 x) pure returns (UD60x18 result) { assembly ("memory-safe") { // Equivalent to "x % UNIT" but faster. let remainder := mod(x, uUNIT) // Equivalent to "x - remainder * (remainder > 0 ? 1 : 0)" but faster. result := sub(x, mul(remainder, gt(remainder, 0))) } } /// @notice Yields the excess beyond the floor of x. /// @dev Based on the odd function definition https://en.wikipedia.org/wiki/Fractional_part. /// @param x The UD60x18 number to get the fractional part of. /// @param result The fractional part of x as an UD60x18 number. function frac(UD60x18 x) pure returns (UD60x18 result) { assembly ("memory-safe") { result := mod(x, uUNIT) } } /// @notice Calculates the geometric mean of x and y, i.e. $$sqrt(x * y)$$, rounding down. /// /// @dev Requirements: /// - x * y must fit within `MAX_UD60x18`, lest it overflows. /// /// @param x The first operand as an UD60x18 number. /// @param y The second operand as an UD60x18 number. /// @return result The result as an UD60x18 number. function gm(UD60x18 x, UD60x18 y) pure returns (UD60x18 result) { uint256 xUint = unwrap(x); uint256 yUint = unwrap(y); if (xUint == 0 || yUint == 0) { return ZERO; } unchecked { // Checking for overflow this way is faster than letting Solidity do it. uint256 xyUint = xUint * yUint; if (xyUint / xUint != yUint) { revert PRBMath_UD60x18_Gm_Overflow(x, y); } // We don't need to multiply the result by `UNIT` here because the x*y product had picked up a factor of `UNIT` // during multiplication. See the comments in the `prbSqrt` function. result = wrap(prbSqrt(xyUint)); } } /// @notice Calculates 1 / x, rounding toward zero. /// /// @dev Requirements: /// - x cannot be zero. /// /// @param x The UD60x18 number for which to calculate the inverse. /// @return result The inverse as an UD60x18 number. function inv(UD60x18 x) pure returns (UD60x18 result) { unchecked { // 1e36 is UNIT * UNIT. result = wrap(1e36 / unwrap(x)); } } /// @notice Calculates the natural logarithm of x. /// /// @dev Based on the formula: /// /// $$ /// ln{x} = log_2{x} / log_2{e}$$. /// $$ /// /// Requirements: /// - All from `log2`. /// /// Caveats: /// - All from `log2`. /// - This doesn't return exactly 1 for 2.718281828459045235, for that more fine-grained precision is needed. /// /// @param x The UD60x18 number for which to calculate the natural logarithm. /// @return result The natural logarithm as an UD60x18 number. function ln(UD60x18 x) pure returns (UD60x18 result) { unchecked { // We do the fixed-point multiplication inline to save gas. This is overflow-safe because the maximum value // that `log2` can return is 196.205294292027477728. result = wrap((unwrap(log2(x)) * uUNIT) / uLOG2_E); } } /// @notice Calculates the common logarithm of x. /// /// @dev First checks if x is an exact power of ten and it stops if yes. If it's not, calculates the common /// logarithm based on the formula: /// /// $$ /// log_{10}{x} = log_2{x} / log_2{10} /// $$ /// /// Requirements: /// - All from `log2`. /// /// Caveats: /// - All from `log2`. /// /// @param x The UD60x18 number for which to calculate the common logarithm. /// @return result The common logarithm as an UD60x18 number. function log10(UD60x18 x) pure returns (UD60x18 result) { uint256 xUint = unwrap(x); if (xUint < uUNIT) { revert PRBMath_UD60x18_Log_InputTooSmall(x); } // Note that the `mul` in this assembly block is the assembly multiplication operation, not the UD60x18 `mul`. // prettier-ignore assembly ("memory-safe") { switch x case 1 { result := mul(uUNIT, sub(0, 18)) } case 10 { result := mul(uUNIT, sub(1, 18)) } case 100 { result := mul(uUNIT, sub(2, 18)) } case 1000 { result := mul(uUNIT, sub(3, 18)) } case 10000 { result := mul(uUNIT, sub(4, 18)) } case 100000 { result := mul(uUNIT, sub(5, 18)) } case 1000000 { result := mul(uUNIT, sub(6, 18)) } case 10000000 { result := mul(uUNIT, sub(7, 18)) } case 100000000 { result := mul(uUNIT, sub(8, 18)) } case 1000000000 { result := mul(uUNIT, sub(9, 18)) } case 10000000000 { result := mul(uUNIT, sub(10, 18)) } case 100000000000 { result := mul(uUNIT, sub(11, 18)) } case 1000000000000 { result := mul(uUNIT, sub(12, 18)) } case 10000000000000 { result := mul(uUNIT, sub(13, 18)) } case 100000000000000 { result := mul(uUNIT, sub(14, 18)) } case 1000000000000000 { result := mul(uUNIT, sub(15, 18)) } case 10000000000000000 { result := mul(uUNIT, sub(16, 18)) } case 100000000000000000 { result := mul(uUNIT, sub(17, 18)) } case 1000000000000000000 { result := 0 } case 10000000000000000000 { result := uUNIT } case 100000000000000000000 { result := mul(uUNIT, 2) } case 1000000000000000000000 { result := mul(uUNIT, 3) } case 10000000000000000000000 { result := mul(uUNIT, 4) } case 100000000000000000000000 { result := mul(uUNIT, 5) } case 1000000000000000000000000 { result := mul(uUNIT, 6) } case 10000000000000000000000000 { result := mul(uUNIT, 7) } case 100000000000000000000000000 { result := mul(uUNIT, 8) } case 1000000000000000000000000000 { result := mul(uUNIT, 9) } case 10000000000000000000000000000 { result := mul(uUNIT, 10) } case 100000000000000000000000000000 { result := mul(uUNIT, 11) } case 1000000000000000000000000000000 { result := mul(uUNIT, 12) } case 10000000000000000000000000000000 { result := mul(uUNIT, 13) } case 100000000000000000000000000000000 { result := mul(uUNIT, 14) } case 1000000000000000000000000000000000 { result := mul(uUNIT, 15) } case 10000000000000000000000000000000000 { result := mul(uUNIT, 16) } case 100000000000000000000000000000000000 { result := mul(uUNIT, 17) } case 1000000000000000000000000000000000000 { result := mul(uUNIT, 18) } case 10000000000000000000000000000000000000 { result := mul(uUNIT, 19) } case 100000000000000000000000000000000000000 { result := mul(uUNIT, 20) } case 1000000000000000000000000000000000000000 { result := mul(uUNIT, 21) } case 10000000000000000000000000000000000000000 { result := mul(uUNIT, 22) } case 100000000000000000000000000000000000000000 { result := mul(uUNIT, 23) } case 1000000000000000000000000000000000000000000 { result := mul(uUNIT, 24) } case 10000000000000000000000000000000000000000000 { result := mul(uUNIT, 25) } case 100000000000000000000000000000000000000000000 { result := mul(uUNIT, 26) } case 1000000000000000000000000000000000000000000000 { result := mul(uUNIT, 27) } case 10000000000000000000000000000000000000000000000 { result := mul(uUNIT, 28) } case 100000000000000000000000000000000000000000000000 { result := mul(uUNIT, 29) } case 1000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 30) } case 10000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 31) } case 100000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 32) } case 1000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 33) } case 10000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 34) } case 100000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 35) } case 1000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 36) } case 10000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 37) } case 100000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 38) } case 1000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 39) } case 10000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 40) } case 100000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 41) } case 1000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 42) } case 10000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 43) } case 100000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 44) } case 1000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 45) } case 10000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 46) } case 100000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 47) } case 1000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 48) } case 10000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 49) } case 100000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 50) } case 1000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 51) } case 10000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 52) } case 100000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 53) } case 1000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 54) } case 10000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 55) } case 100000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 56) } case 1000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 57) } case 10000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 58) } case 100000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 59) } default { result := uMAX_UD60x18 } } if (unwrap(result) == uMAX_UD60x18) { unchecked { // Do the fixed-point division inline to save gas. result = wrap((unwrap(log2(x)) * uUNIT) / uLOG2_10); } } } /// @notice Calculates the binary logarithm of x. /// /// @dev Based on the iterative approximation algorithm. /// https://en.wikipedia.org/wiki/Binary_logarithm#Iterative_approximation /// /// Requirements: /// - x must be greater than or equal to UNIT, otherwise the result would be negative. /// /// Caveats: /// - The results are nor perfectly accurate to the last decimal, due to the lossy precision of the iterative approximation. /// /// @param x The UD60x18 number for which to calculate the binary logarithm. /// @return result The binary logarithm as an UD60x18 number. function log2(UD60x18 x) pure returns (UD60x18 result) { uint256 xUint = unwrap(x); if (xUint < uUNIT) { revert PRBMath_UD60x18_Log_InputTooSmall(x); } unchecked { // Calculate the integer part of the logarithm, add it to the result and finally calculate y = x * 2^(-n). uint256 n = msb(xUint / uUNIT); // This is the integer part of the logarithm as an UD60x18 number. The operation can't overflow because n // n is maximum 255 and UNIT is 1e18. uint256 resultUint = n * uUNIT; // This is $y = x * 2^{-n}$. uint256 y = xUint >> n; // If y is 1, the fractional part is zero. if (y == uUNIT) { return wrap(resultUint); } // Calculate the fractional part via the iterative approximation. // The "delta.rshift(1)" part is equivalent to "delta /= 2", but shifting bits is faster. uint256 DOUBLE_UNIT = 2e18; for (uint256 delta = uHALF_UNIT; delta > 0; delta >>= 1) { y = (y * y) / uUNIT; // Is y^2 > 2 and so in the range [2,4)? if (y >= DOUBLE_UNIT) { // Add the 2^{-m} factor to the logarithm. resultUint += delta; // Corresponds to z/2 on Wikipedia. y >>= 1; } } result = wrap(resultUint); } } /// @notice Multiplies two UD60x18 numbers together, returning a new UD60x18 number. /// @dev See the documentation for the `Common.mulDiv18` function. /// @param x The multiplicand as an UD60x18 number. /// @param y The multiplier as an UD60x18 number. /// @return result The product as an UD60x18 number. function mul(UD60x18 x, UD60x18 y) pure returns (UD60x18 result) { result = wrap(mulDiv18(unwrap(x), unwrap(y))); } /// @notice Raises x to the power of y. /// /// @dev Based on the formula: /// /// $$ /// x^y = 2^{log_2{x} * y} /// $$ /// /// Requirements: /// - All from `exp2`, `log2` and `mul`. /// /// Caveats: /// - All from `exp2`, `log2` and `mul`. /// - Assumes 0^0 is 1. /// /// @param x Number to raise to given power y, as an UD60x18 number. /// @param y Exponent to raise x to, as an UD60x18 number. /// @return result x raised to power y, as an UD60x18 number. function pow(UD60x18 x, UD60x18 y) pure returns (UD60x18 result) { uint256 xUint = unwrap(x); uint256 yUint = unwrap(y); if (xUint == 0) { result = yUint == 0 ? UNIT : ZERO; } else { if (yUint == uUNIT) { result = x; } else { result = exp2(mul(log2(x), y)); } } } /// @notice Raises x (an UD60x18 number) to the power y (unsigned basic integer) using the famous algorithm /// "exponentiation by squaring". /// /// @dev See https://en.wikipedia.org/wiki/Exponentiation_by_squaring /// /// Requirements: /// - The result must fit within `MAX_UD60x18`. /// /// Caveats: /// - All from "Common.mulDiv18". /// - Assumes 0^0 is 1. /// /// @param x The base as an UD60x18 number. /// @param y The exponent as an uint256. /// @return result The result as an UD60x18 number. function powu(UD60x18 x, uint256 y) pure returns (UD60x18 result) { // Calculate the first iteration of the loop in advance. uint256 xUint = unwrap(x); uint256 resultUint = y & 1 > 0 ? xUint : uUNIT; // Equivalent to "for(y /= 2; y > 0; y /= 2)" but faster. for (y >>= 1; y > 0; y >>= 1) { xUint = mulDiv18(xUint, xUint); // Equivalent to "y % 2 == 1" but faster. if (y & 1 > 0) { resultUint = mulDiv18(resultUint, xUint); } } result = wrap(resultUint); } /// @notice Calculates the square root of x, rounding down. /// @dev Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method. /// /// Requirements: /// - x must be less than `MAX_UD60x18` divided by `UNIT`. /// /// @param x The UD60x18 number for which to calculate the square root. /// @return result The result as an UD60x18 number. function sqrt(UD60x18 x) pure returns (UD60x18 result) { uint256 xUint = unwrap(x); unchecked { if (xUint > uMAX_UD60x18 / uUNIT) { revert PRBMath_UD60x18_Sqrt_Overflow(x); } // Multiply x by `UNIT` to account for the factor of `UNIT` that is picked up when multiplying two UD60x18 // numbers together (in this case, the two numbers are both the square root). result = wrap(prbSqrt(xUint * uUNIT)); } }
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import "./Casting.sol" as C; import "./Helpers.sol" as H; import "./Math.sol" as M; /// @notice The unsigned 60.18-decimal fixed-point number representation, which can have up to 60 digits and up to 18 decimals. /// The values of this are bound by the minimum and the maximum values permitted by the Solidity type uint256. /// @dev The value type is defined here so it can be imported in all other files. type UD60x18 is uint256; /*////////////////////////////////////////////////////////////////////////// CASTING //////////////////////////////////////////////////////////////////////////*/ using { C.intoSD1x18, C.intoUD2x18, C.intoSD59x18, C.intoUint128, C.intoUint256, C.intoUint40, C.unwrap } for UD60x18 global; /*////////////////////////////////////////////////////////////////////////// MATHEMATICAL FUNCTIONS //////////////////////////////////////////////////////////////////////////*/ /// The global "using for" directive makes the functions in this library callable on the UD60x18 type. using { M.avg, M.ceil, M.div, M.exp, M.exp2, M.floor, M.frac, M.gm, M.inv, M.ln, M.log10, M.log2, M.mul, M.pow, M.powu, M.sqrt } for UD60x18 global; /*////////////////////////////////////////////////////////////////////////// HELPER FUNCTIONS //////////////////////////////////////////////////////////////////////////*/ /// The global "using for" directive makes the functions in this library callable on the UD60x18 type. using { H.add, H.and, H.eq, H.gt, H.gte, H.isZero, H.lshift, H.lt, H.lte, H.mod, H.neq, H.or, H.rshift, H.sub, H.uncheckedAdd, H.uncheckedSub, H.xor } for UD60x18 global;
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; /// Common mathematical functions used in both SD59x18 and UD60x18. Note that these global functions do not /// always operate with SD59x18 and UD60x18 numbers. /*////////////////////////////////////////////////////////////////////////// CUSTOM ERRORS //////////////////////////////////////////////////////////////////////////*/ /// @notice Emitted when the ending result in the fixed-point version of `mulDiv` would overflow uint256. error PRBMath_MulDiv18_Overflow(uint256 x, uint256 y); /// @notice Emitted when the ending result in `mulDiv` would overflow uint256. error PRBMath_MulDiv_Overflow(uint256 x, uint256 y, uint256 denominator); /// @notice Emitted when attempting to run `mulDiv` with one of the inputs `type(int256).min`. error PRBMath_MulDivSigned_InputTooSmall(); /// @notice Emitted when the ending result in the signed version of `mulDiv` would overflow int256. error PRBMath_MulDivSigned_Overflow(int256 x, int256 y); /*////////////////////////////////////////////////////////////////////////// CONSTANTS //////////////////////////////////////////////////////////////////////////*/ /// @dev The maximum value an uint128 number can have. uint128 constant MAX_UINT128 = type(uint128).max; /// @dev The maximum value an uint40 number can have. uint40 constant MAX_UINT40 = type(uint40).max; /// @dev How many trailing decimals can be represented. uint256 constant UNIT = 1e18; /// @dev Largest power of two that is a divisor of `UNIT`. uint256 constant UNIT_LPOTD = 262144; /// @dev The `UNIT` number inverted mod 2^256. uint256 constant UNIT_INVERSE = 78156646155174841979727994598816262306175212592076161876661_508869554232690281; /*////////////////////////////////////////////////////////////////////////// FUNCTIONS //////////////////////////////////////////////////////////////////////////*/ /// @notice Finds the zero-based index of the first one in the binary representation of x. /// @dev See the note on msb in the "Find First Set" Wikipedia article https://en.wikipedia.org/wiki/Find_first_set /// /// Each of the steps in this implementation is equivalent to this high-level code: /// /// ```solidity /// if (x >= 2 ** 128) { /// x >>= 128; /// result += 128; /// } /// ``` /// /// Where 128 is swapped with each respective power of two factor. See the full high-level implementation here: /// https://gist.github.com/PaulRBerg/f932f8693f2733e30c4d479e8e980948 /// /// A list of the Yul instructions used below: /// - "gt" is "greater than" /// - "or" is the OR bitwise operator /// - "shl" is "shift left" /// - "shr" is "shift right" /// /// @param x The uint256 number for which to find the index of the most significant bit. /// @return result The index of the most significant bit as an uint256. function msb(uint256 x) pure returns (uint256 result) { // 2^128 assembly ("memory-safe") { let factor := shl(7, gt(x, 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)) x := shr(factor, x) result := or(result, factor) } // 2^64 assembly ("memory-safe") { let factor := shl(6, gt(x, 0xFFFFFFFFFFFFFFFF)) x := shr(factor, x) result := or(result, factor) } // 2^32 assembly ("memory-safe") { let factor := shl(5, gt(x, 0xFFFFFFFF)) x := shr(factor, x) result := or(result, factor) } // 2^16 assembly ("memory-safe") { let factor := shl(4, gt(x, 0xFFFF)) x := shr(factor, x) result := or(result, factor) } // 2^8 assembly ("memory-safe") { let factor := shl(3, gt(x, 0xFF)) x := shr(factor, x) result := or(result, factor) } // 2^4 assembly ("memory-safe") { let factor := shl(2, gt(x, 0xF)) x := shr(factor, x) result := or(result, factor) } // 2^2 assembly ("memory-safe") { let factor := shl(1, gt(x, 0x3)) x := shr(factor, x) result := or(result, factor) } // 2^1 // No need to shift x any more. assembly ("memory-safe") { let factor := gt(x, 0x1) result := or(result, factor) } } /// @notice Calculates floor(x*y÷denominator) with full precision. /// /// @dev Credits to Remco Bloemen under MIT license https://xn--2-umb.com/21/muldiv. /// /// Requirements: /// - The denominator cannot be zero. /// - The result must fit within uint256. /// /// Caveats: /// - This function does not work with fixed-point numbers. /// /// @param x The multiplicand as an uint256. /// @param y The multiplier as an uint256. /// @param denominator The divisor as an uint256. /// @return result The result as an uint256. function mulDiv(uint256 x, uint256 y, uint256 denominator) pure returns (uint256 result) { // 512-bit multiply [prod1 prod0] = x * y. Compute the product mod 2^256 and mod 2^256 - 1, then use // 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 ("memory-safe") { let mm := mulmod(x, y, not(0)) prod0 := mul(x, y) prod1 := sub(sub(mm, prod0), lt(mm, prod0)) } // Handle non-overflow cases, 256 by 256 division. if (prod1 == 0) { unchecked { return prod0 / denominator; } } // Make sure the result is less than 2^256. Also prevents denominator == 0. if (prod1 >= denominator) { revert PRBMath_MulDiv_Overflow(x, y, denominator); } /////////////////////////////////////////////// // 512 by 256 division. /////////////////////////////////////////////// // Make division exact by subtracting the remainder from [prod1 prod0]. uint256 remainder; assembly ("memory-safe") { // Compute remainder using the mulmod Yul instruction. remainder := mulmod(x, y, denominator) // Subtract 256 bit number from 512 bit number. prod1 := sub(prod1, gt(remainder, prod0)) prod0 := sub(prod0, remainder) } // Factor powers of two out of denominator and compute largest power of two divisor of denominator. Always >= 1. // See https://cs.stackexchange.com/q/138556/92363. unchecked { // Does not overflow because the denominator cannot be zero at this stage in the function. uint256 lpotdod = denominator & (~denominator + 1); assembly ("memory-safe") { // Divide denominator by lpotdod. denominator := div(denominator, lpotdod) // Divide [prod1 prod0] by lpotdod. prod0 := div(prod0, lpotdod) // Flip lpotdod such that it is 2^256 / lpotdod. If lpotdod is zero, then it becomes one. lpotdod := add(div(sub(0, lpotdod), lpotdod), 1) } // Shift in bits from prod1 into prod0. prod0 |= prod1 * lpotdod; // 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 for // four bits. That is, denominator * inv = 1 mod 2^4. uint256 inverse = (3 * denominator) ^ 2; // Use the 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. inverse *= 2 - denominator * inverse; // inverse mod 2^8 inverse *= 2 - denominator * inverse; // inverse mod 2^16 inverse *= 2 - denominator * inverse; // inverse mod 2^32 inverse *= 2 - denominator * inverse; // inverse mod 2^64 inverse *= 2 - denominator * inverse; // inverse mod 2^128 inverse *= 2 - denominator * inverse; // 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 preconditions 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 * inverse; } } /// @notice Calculates floor(x*y÷1e18) with full precision. /// /// @dev Variant of `mulDiv` with constant folding, i.e. in which the denominator is always 1e18. Before returning the /// final result, we add 1 if `(x * y) % UNIT >= HALF_UNIT`. Without this adjustment, 6.6e-19 would be truncated to 0 /// instead of being rounded to 1e-18. See "Listing 6" and text above it at https://accu.org/index.php/journals/1717. /// /// Requirements: /// - The result must fit within uint256. /// /// Caveats: /// - The body is purposely left uncommented; to understand how this works, see the NatSpec comments in `mulDiv`. /// - It is assumed that the result can never be `type(uint256).max` when x and y solve the following two equations: /// 1. x * y = type(uint256).max * UNIT /// 2. (x * y) % UNIT >= UNIT / 2 /// /// @param x The multiplicand as an unsigned 60.18-decimal fixed-point number. /// @param y The multiplier as an unsigned 60.18-decimal fixed-point number. /// @return result The result as an unsigned 60.18-decimal fixed-point number. function mulDiv18(uint256 x, uint256 y) pure returns (uint256 result) { uint256 prod0; uint256 prod1; assembly ("memory-safe") { let mm := mulmod(x, y, not(0)) prod0 := mul(x, y) prod1 := sub(sub(mm, prod0), lt(mm, prod0)) } if (prod1 >= UNIT) { revert PRBMath_MulDiv18_Overflow(x, y); } uint256 remainder; assembly ("memory-safe") { remainder := mulmod(x, y, UNIT) } if (prod1 == 0) { unchecked { return prod0 / UNIT; } } assembly ("memory-safe") { result := mul( or( div(sub(prod0, remainder), UNIT_LPOTD), mul(sub(prod1, gt(remainder, prod0)), add(div(sub(0, UNIT_LPOTD), UNIT_LPOTD), 1)) ), UNIT_INVERSE ) } } /// @notice Calculates floor(x*y÷denominator) with full precision. /// /// @dev An extension of `mulDiv` for signed numbers. Works by computing the signs and the absolute values separately. /// /// Requirements: /// - None of the inputs can be `type(int256).min`. /// - The result must fit within int256. /// /// @param x The multiplicand as an int256. /// @param y The multiplier as an int256. /// @param denominator The divisor as an int256. /// @return result The result as an int256. function mulDivSigned(int256 x, int256 y, int256 denominator) pure returns (int256 result) { if (x == type(int256).min || y == type(int256).min || denominator == type(int256).min) { revert PRBMath_MulDivSigned_InputTooSmall(); } // Get hold of the absolute values of x, y and the denominator. uint256 absX; uint256 absY; uint256 absD; unchecked { absX = x < 0 ? uint256(-x) : uint256(x); absY = y < 0 ? uint256(-y) : uint256(y); absD = denominator < 0 ? uint256(-denominator) : uint256(denominator); } // Compute the absolute value of (x*y)÷denominator. The result must fit within int256. uint256 rAbs = mulDiv(absX, absY, absD); if (rAbs > uint256(type(int256).max)) { revert PRBMath_MulDivSigned_Overflow(x, y); } // Get the signs of x, y and the denominator. uint256 sx; uint256 sy; uint256 sd; assembly ("memory-safe") { // This works thanks to two's complement. // "sgt" stands for "signed greater than" and "sub(0,1)" is max uint256. sx := sgt(x, sub(0, 1)) sy := sgt(y, sub(0, 1)) sd := sgt(denominator, sub(0, 1)) } // XOR over sx, sy and sd. What this does is to check whether there are 1 or 3 negative signs in the inputs. // If there are, the result should be negative. Otherwise, it should be positive. unchecked { result = sx ^ sy ^ sd == 0 ? -int256(rAbs) : int256(rAbs); } } /// @notice Calculates the binary exponent of x using the binary fraction method. /// @dev Has to use 192.64-bit fixed-point numbers. /// See https://ethereum.stackexchange.com/a/96594/24693. /// @param x The exponent as an unsigned 192.64-bit fixed-point number. /// @return result The result as an unsigned 60.18-decimal fixed-point number. function prbExp2(uint256 x) pure returns (uint256 result) { unchecked { // Start from 0.5 in the 192.64-bit fixed-point format. result = 0x800000000000000000000000000000000000000000000000; // Multiply the result by root(2, 2^-i) when the bit at position i is 1. None of the intermediary results overflows // because the initial result is 2^191 and all magic factors are less than 2^65. if (x & 0xFF00000000000000 > 0) { if (x & 0x8000000000000000 > 0) { result = (result * 0x16A09E667F3BCC909) >> 64; } if (x & 0x4000000000000000 > 0) { result = (result * 0x1306FE0A31B7152DF) >> 64; } if (x & 0x2000000000000000 > 0) { result = (result * 0x1172B83C7D517ADCE) >> 64; } if (x & 0x1000000000000000 > 0) { result = (result * 0x10B5586CF9890F62A) >> 64; } if (x & 0x800000000000000 > 0) { result = (result * 0x1059B0D31585743AE) >> 64; } if (x & 0x400000000000000 > 0) { result = (result * 0x102C9A3E778060EE7) >> 64; } if (x & 0x200000000000000 > 0) { result = (result * 0x10163DA9FB33356D8) >> 64; } if (x & 0x100000000000000 > 0) { result = (result * 0x100B1AFA5ABCBED61) >> 64; } } if (x & 0xFF000000000000 > 0) { if (x & 0x80000000000000 > 0) { result = (result * 0x10058C86DA1C09EA2) >> 64; } if (x & 0x40000000000000 > 0) { result = (result * 0x1002C605E2E8CEC50) >> 64; } if (x & 0x20000000000000 > 0) { result = (result * 0x100162F3904051FA1) >> 64; } if (x & 0x10000000000000 > 0) { result = (result * 0x1000B175EFFDC76BA) >> 64; } if (x & 0x8000000000000 > 0) { result = (result * 0x100058BA01FB9F96D) >> 64; } if (x & 0x4000000000000 > 0) { result = (result * 0x10002C5CC37DA9492) >> 64; } if (x & 0x2000000000000 > 0) { result = (result * 0x1000162E525EE0547) >> 64; } if (x & 0x1000000000000 > 0) { result = (result * 0x10000B17255775C04) >> 64; } } if (x & 0xFF0000000000 > 0) { if (x & 0x800000000000 > 0) { result = (result * 0x1000058B91B5BC9AE) >> 64; } if (x & 0x400000000000 > 0) { result = (result * 0x100002C5C89D5EC6D) >> 64; } if (x & 0x200000000000 > 0) { result = (result * 0x10000162E43F4F831) >> 64; } if (x & 0x100000000000 > 0) { result = (result * 0x100000B1721BCFC9A) >> 64; } if (x & 0x80000000000 > 0) { result = (result * 0x10000058B90CF1E6E) >> 64; } if (x & 0x40000000000 > 0) { result = (result * 0x1000002C5C863B73F) >> 64; } if (x & 0x20000000000 > 0) { result = (result * 0x100000162E430E5A2) >> 64; } if (x & 0x10000000000 > 0) { result = (result * 0x1000000B172183551) >> 64; } } if (x & 0xFF00000000 > 0) { if (x & 0x8000000000 > 0) { result = (result * 0x100000058B90C0B49) >> 64; } if (x & 0x4000000000 > 0) { result = (result * 0x10000002C5C8601CC) >> 64; } if (x & 0x2000000000 > 0) { result = (result * 0x1000000162E42FFF0) >> 64; } if (x & 0x1000000000 > 0) { result = (result * 0x10000000B17217FBB) >> 64; } if (x & 0x800000000 > 0) { result = (result * 0x1000000058B90BFCE) >> 64; } if (x & 0x400000000 > 0) { result = (result * 0x100000002C5C85FE3) >> 64; } if (x & 0x200000000 > 0) { result = (result * 0x10000000162E42FF1) >> 64; } if (x & 0x100000000 > 0) { result = (result * 0x100000000B17217F8) >> 64; } } if (x & 0xFF00000000 > 0) { if (x & 0x80000000 > 0) { result = (result * 0x10000000058B90BFC) >> 64; } if (x & 0x40000000 > 0) { result = (result * 0x1000000002C5C85FE) >> 64; } if (x & 0x20000000 > 0) { result = (result * 0x100000000162E42FF) >> 64; } if (x & 0x10000000 > 0) { result = (result * 0x1000000000B17217F) >> 64; } if (x & 0x8000000 > 0) { result = (result * 0x100000000058B90C0) >> 64; } if (x & 0x4000000 > 0) { result = (result * 0x10000000002C5C860) >> 64; } if (x & 0x2000000 > 0) { result = (result * 0x1000000000162E430) >> 64; } if (x & 0x1000000 > 0) { result = (result * 0x10000000000B17218) >> 64; } } if (x & 0xFF0000 > 0) { if (x & 0x800000 > 0) { result = (result * 0x1000000000058B90C) >> 64; } if (x & 0x400000 > 0) { result = (result * 0x100000000002C5C86) >> 64; } if (x & 0x200000 > 0) { result = (result * 0x10000000000162E43) >> 64; } if (x & 0x100000 > 0) { result = (result * 0x100000000000B1721) >> 64; } if (x & 0x80000 > 0) { result = (result * 0x10000000000058B91) >> 64; } if (x & 0x40000 > 0) { result = (result * 0x1000000000002C5C8) >> 64; } if (x & 0x20000 > 0) { result = (result * 0x100000000000162E4) >> 64; } if (x & 0x10000 > 0) { result = (result * 0x1000000000000B172) >> 64; } } if (x & 0xFF00 > 0) { if (x & 0x8000 > 0) { result = (result * 0x100000000000058B9) >> 64; } if (x & 0x4000 > 0) { result = (result * 0x10000000000002C5D) >> 64; } if (x & 0x2000 > 0) { result = (result * 0x1000000000000162E) >> 64; } if (x & 0x1000 > 0) { result = (result * 0x10000000000000B17) >> 64; } if (x & 0x800 > 0) { result = (result * 0x1000000000000058C) >> 64; } if (x & 0x400 > 0) { result = (result * 0x100000000000002C6) >> 64; } if (x & 0x200 > 0) { result = (result * 0x10000000000000163) >> 64; } if (x & 0x100 > 0) { result = (result * 0x100000000000000B1) >> 64; } } if (x & 0xFF > 0) { if (x & 0x80 > 0) { result = (result * 0x10000000000000059) >> 64; } if (x & 0x40 > 0) { result = (result * 0x1000000000000002C) >> 64; } if (x & 0x20 > 0) { result = (result * 0x10000000000000016) >> 64; } if (x & 0x10 > 0) { result = (result * 0x1000000000000000B) >> 64; } if (x & 0x8 > 0) { result = (result * 0x10000000000000006) >> 64; } if (x & 0x4 > 0) { result = (result * 0x10000000000000003) >> 64; } if (x & 0x2 > 0) { result = (result * 0x10000000000000001) >> 64; } if (x & 0x1 > 0) { result = (result * 0x10000000000000001) >> 64; } } // We're doing two things at the same time: // // 1. Multiply the result by 2^n + 1, where "2^n" is the integer part and the one is added to account for // the fact that we initially set the result to 0.5. This is accomplished by subtracting from 191 // rather than 192. // 2. Convert the result to the unsigned 60.18-decimal fixed-point format. // // This works because 2^(191-ip) = 2^ip / 2^191, where "ip" is the integer part "2^n". result *= UNIT; result >>= (191 - (x >> 64)); } } /// @notice Calculates the square root of x, rounding down if x is not a perfect square. /// @dev Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method. /// Credits to OpenZeppelin for the explanations in code comments below. /// /// Caveats: /// - This function does not work with fixed-point numbers. /// /// @param x The uint256 number for which to calculate the square root. /// @return result The result as an uint256. function prbSqrt(uint256 x) pure returns (uint256 result) { if (x == 0) { return 0; } // For our first guess, we get the biggest power of 2 which is smaller than the square root of x. // // We know that the "msb" (most significant bit) of x is a power of 2 such that we have: // // $$ // msb(x) <= x <= 2*msb(x)$ // $$ // // We write $msb(x)$ as $2^k$ and we get: // // $$ // k = log_2(x) // $$ // // Thus we can write the initial inequality as: // // $$ // 2^{log_2(x)} <= x <= 2*2^{log_2(x)+1} \\ // sqrt(2^k) <= sqrt(x) < sqrt(2^{k+1}) \\ // 2^{k/2} <= sqrt(x) < 2^{(k+1)/2} <= 2^{(k/2)+1} // $$ // // Consequently, $2^{log_2(x) /2}` is a good first approximation of sqrt(x) with at least one correct bit. uint256 xAux = uint256(x); result = 1; if (xAux >= 2 ** 128) { xAux >>= 128; result <<= 64; } if (xAux >= 2 ** 64) { xAux >>= 64; result <<= 32; } if (xAux >= 2 ** 32) { xAux >>= 32; result <<= 16; } if (xAux >= 2 ** 16) { xAux >>= 16; result <<= 8; } if (xAux >= 2 ** 8) { xAux >>= 8; result <<= 4; } if (xAux >= 2 ** 4) { xAux >>= 4; result <<= 2; } if (xAux >= 2 ** 2) { result <<= 1; } // At this point, `result` is an estimation with at least one bit of precision. We know the true value has at // most 128 bits, since it is the square root of a uint256. Newton's method converges quadratically (precision // doubles at every iteration). We thus need at most 7 iteration to turn our partial result with one bit of // precision into the expected uint128 result. unchecked { result = (result + x / result) >> 1; result = (result + x / result) >> 1; result = (result + x / result) >> 1; result = (result + x / result) >> 1; result = (result + x / result) >> 1; result = (result + x / result) >> 1; result = (result + x / result) >> 1; // Round down the result in case x is not a perfect square. uint256 roundedDownResult = x / result; if (result >= roundedDownResult) { result = roundedDownResult; } } }
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { SD1x18 } from "./ValueType.sol"; /// @dev Euler's number as an SD1x18 number. SD1x18 constant E = SD1x18.wrap(2_718281828459045235); /// @dev The maximum value an SD1x18 number can have. int64 constant uMAX_SD1x18 = 9_223372036854775807; SD1x18 constant MAX_SD1x18 = SD1x18.wrap(uMAX_SD1x18); /// @dev The maximum value an SD1x18 number can have. int64 constant uMIN_SD1x18 = -9_223372036854775808; SD1x18 constant MIN_SD1x18 = SD1x18.wrap(uMIN_SD1x18); /// @dev PI as an SD1x18 number. SD1x18 constant PI = SD1x18.wrap(3_141592653589793238); /// @dev The unit amount that implies how many trailing decimals can be represented. SD1x18 constant UNIT = SD1x18.wrap(1e18); int256 constant uUNIT = 1e18;
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import "./Casting.sol" as C; /// @notice The signed 1.18-decimal fixed-point number representation, which can have up to 1 digit and up to 18 decimals. /// The values of this are bound by the minimum and the maximum values permitted by the underlying Solidity type int64. /// This is useful when end users want to use int64 to save gas, e.g. with tight variable packing in contract storage. type SD1x18 is int64; /*////////////////////////////////////////////////////////////////////////// CASTING //////////////////////////////////////////////////////////////////////////*/ using { C.intoSD59x18, C.intoUD2x18, C.intoUD60x18, C.intoUint256, C.intoUint128, C.intoUint40, C.unwrap } for SD1x18 global;
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { SD59x18 } from "./ValueType.sol"; /// NOTICE: the "u" prefix stands for "unwrapped". /// @dev Euler's number as an SD59x18 number. SD59x18 constant E = SD59x18.wrap(2_718281828459045235); /// @dev Half the UNIT number. int256 constant uHALF_UNIT = 0.5e18; SD59x18 constant HALF_UNIT = SD59x18.wrap(uHALF_UNIT); /// @dev log2(10) as an SD59x18 number. int256 constant uLOG2_10 = 3_321928094887362347; SD59x18 constant LOG2_10 = SD59x18.wrap(uLOG2_10); /// @dev log2(e) as an SD59x18 number. int256 constant uLOG2_E = 1_442695040888963407; SD59x18 constant LOG2_E = SD59x18.wrap(uLOG2_E); /// @dev The maximum value an SD59x18 number can have. int256 constant uMAX_SD59x18 = 57896044618658097711785492504343953926634992332820282019728_792003956564819967; SD59x18 constant MAX_SD59x18 = SD59x18.wrap(uMAX_SD59x18); /// @dev The maximum whole value an SD59x18 number can have. int256 constant uMAX_WHOLE_SD59x18 = 57896044618658097711785492504343953926634992332820282019728_000000000000000000; SD59x18 constant MAX_WHOLE_SD59x18 = SD59x18.wrap(uMAX_WHOLE_SD59x18); /// @dev The minimum value an SD59x18 number can have. int256 constant uMIN_SD59x18 = -57896044618658097711785492504343953926634992332820282019728_792003956564819968; SD59x18 constant MIN_SD59x18 = SD59x18.wrap(uMIN_SD59x18); /// @dev The minimum whole value an SD59x18 number can have. int256 constant uMIN_WHOLE_SD59x18 = -57896044618658097711785492504343953926634992332820282019728_000000000000000000; SD59x18 constant MIN_WHOLE_SD59x18 = SD59x18.wrap(uMIN_WHOLE_SD59x18); /// @dev PI as an SD59x18 number. SD59x18 constant PI = SD59x18.wrap(3_141592653589793238); /// @dev The unit amount that implies how many trailing decimals can be represented. int256 constant uUNIT = 1e18; SD59x18 constant UNIT = SD59x18.wrap(1e18); /// @dev Zero as an SD59x18 number. SD59x18 constant ZERO = SD59x18.wrap(0);
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import "./Casting.sol" as C; import "./Helpers.sol" as H; import "./Math.sol" as M; /// @notice The signed 59.18-decimal fixed-point number representation, which can have up to 59 digits and up to 18 decimals. /// The values of this are bound by the minimum and the maximum values permitted by the underlying Solidity type int256. type SD59x18 is int256; /*////////////////////////////////////////////////////////////////////////// CASTING //////////////////////////////////////////////////////////////////////////*/ using { C.intoInt256, C.intoSD1x18, C.intoUD2x18, C.intoUD60x18, C.intoUint256, C.intoUint128, C.intoUint40, C.unwrap } for SD59x18 global; /*////////////////////////////////////////////////////////////////////////// MATHEMATICAL FUNCTIONS //////////////////////////////////////////////////////////////////////////*/ using { M.abs, M.avg, M.ceil, M.div, M.exp, M.exp2, M.floor, M.frac, M.gm, M.inv, M.log10, M.log2, M.ln, M.mul, M.pow, M.powu, M.sqrt } for SD59x18 global; /*////////////////////////////////////////////////////////////////////////// HELPER FUNCTIONS //////////////////////////////////////////////////////////////////////////*/ using { H.add, H.and, H.eq, H.gt, H.gte, H.isZero, H.lshift, H.lt, H.lte, H.mod, H.neq, H.or, H.rshift, H.sub, H.uncheckedAdd, H.uncheckedSub, H.uncheckedUnary, H.xor } for SD59x18 global;
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { UD2x18 } from "./ValueType.sol"; /// @dev Euler's number as an UD2x18 number. UD2x18 constant E = UD2x18.wrap(2_718281828459045235); /// @dev The maximum value an UD2x18 number can have. uint64 constant uMAX_UD2x18 = 18_446744073709551615; UD2x18 constant MAX_UD2x18 = UD2x18.wrap(uMAX_UD2x18); /// @dev PI as an UD2x18 number. UD2x18 constant PI = UD2x18.wrap(3_141592653589793238); /// @dev The unit amount that implies how many trailing decimals can be represented. uint256 constant uUNIT = 1e18; UD2x18 constant UNIT = UD2x18.wrap(1e18);
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import "./Casting.sol" as C; /// @notice The unsigned 2.18-decimal fixed-point number representation, which can have up to 2 digits and up to 18 decimals. /// The values of this are bound by the minimum and the maximum values permitted by the underlying Solidity type uint64. /// This is useful when end users want to use uint64 to save gas, e.g. with tight variable packing in contract storage. type UD2x18 is uint64; /*////////////////////////////////////////////////////////////////////////// CASTING //////////////////////////////////////////////////////////////////////////*/ using { C.intoSD1x18, C.intoSD59x18, C.intoUD60x18, C.intoUint256, C.intoUint128, C.intoUint40, C.unwrap } for UD2x18 global;
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { MAX_UINT40 } from "../Common.sol"; import { SD59x18 } from "../sd59x18/ValueType.sol"; import { UD2x18 } from "../ud2x18/ValueType.sol"; import { UD60x18 } from "../ud60x18/ValueType.sol"; import { PRBMath_SD1x18_ToUD2x18_Underflow, PRBMath_SD1x18_ToUD60x18_Underflow, PRBMath_SD1x18_ToUint128_Underflow, PRBMath_SD1x18_ToUint256_Underflow, PRBMath_SD1x18_ToUint40_Overflow, PRBMath_SD1x18_ToUint40_Underflow } from "./Errors.sol"; import { SD1x18 } from "./ValueType.sol"; /// @notice Casts an SD1x18 number into SD59x18. /// @dev There is no overflow check because the domain of SD1x18 is a subset of SD59x18. function intoSD59x18(SD1x18 x) pure returns (SD59x18 result) { result = SD59x18.wrap(int256(SD1x18.unwrap(x))); } /// @notice Casts an SD1x18 number into UD2x18. /// - x must be positive. function intoUD2x18(SD1x18 x) pure returns (UD2x18 result) { int64 xInt = SD1x18.unwrap(x); if (xInt < 0) { revert PRBMath_SD1x18_ToUD2x18_Underflow(x); } result = UD2x18.wrap(uint64(xInt)); } /// @notice Casts an SD1x18 number into UD60x18. /// @dev Requirements: /// - x must be positive. function intoUD60x18(SD1x18 x) pure returns (UD60x18 result) { int64 xInt = SD1x18.unwrap(x); if (xInt < 0) { revert PRBMath_SD1x18_ToUD60x18_Underflow(x); } result = UD60x18.wrap(uint64(xInt)); } /// @notice Casts an SD1x18 number into uint256. /// @dev Requirements: /// - x must be positive. function intoUint256(SD1x18 x) pure returns (uint256 result) { int64 xInt = SD1x18.unwrap(x); if (xInt < 0) { revert PRBMath_SD1x18_ToUint256_Underflow(x); } result = uint256(uint64(xInt)); } /// @notice Casts an SD1x18 number into uint128. /// @dev Requirements: /// - x must be positive. function intoUint128(SD1x18 x) pure returns (uint128 result) { int64 xInt = SD1x18.unwrap(x); if (xInt < 0) { revert PRBMath_SD1x18_ToUint128_Underflow(x); } result = uint128(uint64(xInt)); } /// @notice Casts an SD1x18 number into uint40. /// @dev Requirements: /// - x must be positive. /// - x must be less than or equal to `MAX_UINT40`. function intoUint40(SD1x18 x) pure returns (uint40 result) { int64 xInt = SD1x18.unwrap(x); if (xInt < 0) { revert PRBMath_SD1x18_ToUint40_Underflow(x); } if (xInt > int64(uint64(MAX_UINT40))) { revert PRBMath_SD1x18_ToUint40_Overflow(x); } result = uint40(uint64(xInt)); } /// @notice Alias for the `wrap` function. function sd1x18(int64 x) pure returns (SD1x18 result) { result = SD1x18.wrap(x); } /// @notice Unwraps an SD1x18 number into int64. function unwrap(SD1x18 x) pure returns (int64 result) { result = SD1x18.unwrap(x); } /// @notice Wraps an int64 number into the SD1x18 value type. function wrap(int64 x) pure returns (SD1x18 result) { result = SD1x18.wrap(x); }
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { MAX_UINT128, MAX_UINT40 } from "../Common.sol"; import { uMAX_SD1x18, uMIN_SD1x18 } from "../sd1x18/Constants.sol"; import { SD1x18 } from "../sd1x18/ValueType.sol"; import { uMAX_UD2x18 } from "../ud2x18/Constants.sol"; import { UD2x18 } from "../ud2x18/ValueType.sol"; import { UD60x18 } from "../ud60x18/ValueType.sol"; import { PRBMath_SD59x18_IntoSD1x18_Overflow, PRBMath_SD59x18_IntoSD1x18_Underflow, PRBMath_SD59x18_IntoUD2x18_Overflow, PRBMath_SD59x18_IntoUD2x18_Underflow, PRBMath_SD59x18_IntoUD60x18_Underflow, PRBMath_SD59x18_IntoUint128_Overflow, PRBMath_SD59x18_IntoUint128_Underflow, PRBMath_SD59x18_IntoUint256_Underflow, PRBMath_SD59x18_IntoUint40_Overflow, PRBMath_SD59x18_IntoUint40_Underflow } from "./Errors.sol"; import { SD59x18 } from "./ValueType.sol"; /// @notice Casts an SD59x18 number into int256. /// @dev This is basically a functional alias for the `unwrap` function. function intoInt256(SD59x18 x) pure returns (int256 result) { result = SD59x18.unwrap(x); } /// @notice Casts an SD59x18 number into SD1x18. /// @dev Requirements: /// - x must be greater than or equal to `uMIN_SD1x18`. /// - x must be less than or equal to `uMAX_SD1x18`. function intoSD1x18(SD59x18 x) pure returns (SD1x18 result) { int256 xInt = SD59x18.unwrap(x); if (xInt < uMIN_SD1x18) { revert PRBMath_SD59x18_IntoSD1x18_Underflow(x); } if (xInt > uMAX_SD1x18) { revert PRBMath_SD59x18_IntoSD1x18_Overflow(x); } result = SD1x18.wrap(int64(xInt)); } /// @notice Casts an SD59x18 number into UD2x18. /// @dev Requirements: /// - x must be positive. /// - x must be less than or equal to `uMAX_UD2x18`. function intoUD2x18(SD59x18 x) pure returns (UD2x18 result) { int256 xInt = SD59x18.unwrap(x); if (xInt < 0) { revert PRBMath_SD59x18_IntoUD2x18_Underflow(x); } if (xInt > int256(uint256(uMAX_UD2x18))) { revert PRBMath_SD59x18_IntoUD2x18_Overflow(x); } result = UD2x18.wrap(uint64(uint256(xInt))); } /// @notice Casts an SD59x18 number into UD60x18. /// @dev Requirements: /// - x must be positive. function intoUD60x18(SD59x18 x) pure returns (UD60x18 result) { int256 xInt = SD59x18.unwrap(x); if (xInt < 0) { revert PRBMath_SD59x18_IntoUD60x18_Underflow(x); } result = UD60x18.wrap(uint256(xInt)); } /// @notice Casts an SD59x18 number into uint256. /// @dev Requirements: /// - x must be positive. function intoUint256(SD59x18 x) pure returns (uint256 result) { int256 xInt = SD59x18.unwrap(x); if (xInt < 0) { revert PRBMath_SD59x18_IntoUint256_Underflow(x); } result = uint256(xInt); } /// @notice Casts an SD59x18 number into uint128. /// @dev Requirements: /// - x must be positive. /// - x must be less than or equal to `uMAX_UINT128`. function intoUint128(SD59x18 x) pure returns (uint128 result) { int256 xInt = SD59x18.unwrap(x); if (xInt < 0) { revert PRBMath_SD59x18_IntoUint128_Underflow(x); } if (xInt > int256(uint256(MAX_UINT128))) { revert PRBMath_SD59x18_IntoUint128_Overflow(x); } result = uint128(uint256(xInt)); } /// @notice Casts an SD59x18 number into uint40. /// @dev Requirements: /// - x must be positive. /// - x must be less than or equal to `MAX_UINT40`. function intoUint40(SD59x18 x) pure returns (uint40 result) { int256 xInt = SD59x18.unwrap(x); if (xInt < 0) { revert PRBMath_SD59x18_IntoUint40_Underflow(x); } if (xInt > int256(uint256(MAX_UINT40))) { revert PRBMath_SD59x18_IntoUint40_Overflow(x); } result = uint40(uint256(xInt)); } /// @notice Alias for the `wrap` function. function sd(int256 x) pure returns (SD59x18 result) { result = SD59x18.wrap(x); } /// @notice Alias for the `wrap` function. function sd59x18(int256 x) pure returns (SD59x18 result) { result = SD59x18.wrap(x); } /// @notice Unwraps an SD59x18 number into int256. function unwrap(SD59x18 x) pure returns (int256 result) { result = SD59x18.unwrap(x); } /// @notice Wraps an int256 number into the SD59x18 value type. function wrap(int256 x) pure returns (SD59x18 result) { result = SD59x18.wrap(x); }
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { unwrap, wrap } from "./Casting.sol"; import { SD59x18 } from "./ValueType.sol"; /// @notice Implements the checked addition operation (+) in the SD59x18 type. function add(SD59x18 x, SD59x18 y) pure returns (SD59x18 result) { return wrap(unwrap(x) + unwrap(y)); } /// @notice Implements the AND (&) bitwise operation in the SD59x18 type. function and(SD59x18 x, int256 bits) pure returns (SD59x18 result) { return wrap(unwrap(x) & bits); } /// @notice Implements the equal (=) operation in the SD59x18 type. function eq(SD59x18 x, SD59x18 y) pure returns (bool result) { result = unwrap(x) == unwrap(y); } /// @notice Implements the greater than operation (>) in the SD59x18 type. function gt(SD59x18 x, SD59x18 y) pure returns (bool result) { result = unwrap(x) > unwrap(y); } /// @notice Implements the greater than or equal to operation (>=) in the SD59x18 type. function gte(SD59x18 x, SD59x18 y) pure returns (bool result) { result = unwrap(x) >= unwrap(y); } /// @notice Implements a zero comparison check function in the SD59x18 type. function isZero(SD59x18 x) pure returns (bool result) { result = unwrap(x) == 0; } /// @notice Implements the left shift operation (<<) in the SD59x18 type. function lshift(SD59x18 x, uint256 bits) pure returns (SD59x18 result) { result = wrap(unwrap(x) << bits); } /// @notice Implements the lower than operation (<) in the SD59x18 type. function lt(SD59x18 x, SD59x18 y) pure returns (bool result) { result = unwrap(x) < unwrap(y); } /// @notice Implements the lower than or equal to operation (<=) in the SD59x18 type. function lte(SD59x18 x, SD59x18 y) pure returns (bool result) { result = unwrap(x) <= unwrap(y); } /// @notice Implements the unchecked modulo operation (%) in the SD59x18 type. function mod(SD59x18 x, SD59x18 y) pure returns (SD59x18 result) { result = wrap(unwrap(x) % unwrap(y)); } /// @notice Implements the not equal operation (!=) in the SD59x18 type. function neq(SD59x18 x, SD59x18 y) pure returns (bool result) { result = unwrap(x) != unwrap(y); } /// @notice Implements the OR (|) bitwise operation in the SD59x18 type. function or(SD59x18 x, SD59x18 y) pure returns (SD59x18 result) { result = wrap(unwrap(x) | unwrap(y)); } /// @notice Implements the right shift operation (>>) in the SD59x18 type. function rshift(SD59x18 x, uint256 bits) pure returns (SD59x18 result) { result = wrap(unwrap(x) >> bits); } /// @notice Implements the checked subtraction operation (-) in the SD59x18 type. function sub(SD59x18 x, SD59x18 y) pure returns (SD59x18 result) { result = wrap(unwrap(x) - unwrap(y)); } /// @notice Implements the unchecked addition operation (+) in the SD59x18 type. function uncheckedAdd(SD59x18 x, SD59x18 y) pure returns (SD59x18 result) { unchecked { result = wrap(unwrap(x) + unwrap(y)); } } /// @notice Implements the unchecked subtraction operation (-) in the SD59x18 type. function uncheckedSub(SD59x18 x, SD59x18 y) pure returns (SD59x18 result) { unchecked { result = wrap(unwrap(x) - unwrap(y)); } } /// @notice Implements the unchecked unary minus operation (-) in the SD59x18 type. function uncheckedUnary(SD59x18 x) pure returns (SD59x18 result) { unchecked { result = wrap(-unwrap(x)); } } /// @notice Implements the XOR (^) bitwise operation in the SD59x18 type. function xor(SD59x18 x, SD59x18 y) pure returns (SD59x18 result) { result = wrap(unwrap(x) ^ unwrap(y)); }
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { MAX_UINT128, MAX_UINT40, msb, mulDiv, mulDiv18, prbExp2, prbSqrt } from "../Common.sol"; import { uHALF_UNIT, uLOG2_10, uLOG2_E, uMAX_SD59x18, uMAX_WHOLE_SD59x18, uMIN_SD59x18, uMIN_WHOLE_SD59x18, UNIT, uUNIT, ZERO } from "./Constants.sol"; import { PRBMath_SD59x18_Abs_MinSD59x18, PRBMath_SD59x18_Ceil_Overflow, PRBMath_SD59x18_Div_InputTooSmall, PRBMath_SD59x18_Div_Overflow, PRBMath_SD59x18_Exp_InputTooBig, PRBMath_SD59x18_Exp2_InputTooBig, PRBMath_SD59x18_Floor_Underflow, PRBMath_SD59x18_Gm_Overflow, PRBMath_SD59x18_Gm_NegativeProduct, PRBMath_SD59x18_Log_InputTooSmall, PRBMath_SD59x18_Mul_InputTooSmall, PRBMath_SD59x18_Mul_Overflow, PRBMath_SD59x18_Powu_Overflow, PRBMath_SD59x18_Sqrt_NegativeInput, PRBMath_SD59x18_Sqrt_Overflow } from "./Errors.sol"; import { unwrap, wrap } from "./Helpers.sol"; import { SD59x18 } from "./ValueType.sol"; /// @notice Calculate the absolute value of x. /// /// @dev Requirements: /// - x must be greater than `MIN_SD59x18`. /// /// @param x The SD59x18 number for which to calculate the absolute value. /// @param result The absolute value of x as an SD59x18 number. function abs(SD59x18 x) pure returns (SD59x18 result) { int256 xInt = unwrap(x); if (xInt == uMIN_SD59x18) { revert PRBMath_SD59x18_Abs_MinSD59x18(); } result = xInt < 0 ? wrap(-xInt) : x; } /// @notice Calculates the arithmetic average of x and y, rounding towards zero. /// @param x The first operand as an SD59x18 number. /// @param y The second operand as an SD59x18 number. /// @return result The arithmetic average as an SD59x18 number. function avg(SD59x18 x, SD59x18 y) pure returns (SD59x18 result) { int256 xInt = unwrap(x); int256 yInt = unwrap(y); unchecked { // This is equivalent to "x / 2 + y / 2" but faster. // This operation can never overflow. int256 sum = (xInt >> 1) + (yInt >> 1); if (sum < 0) { // If at least one of x and y is odd, we add 1 to the result, since shifting negative numbers to the right rounds // down to infinity. The right part is equivalent to "sum + (x % 2 == 1 || y % 2 == 1)" but faster. assembly ("memory-safe") { result := add(sum, and(or(xInt, yInt), 1)) } } else { // We need to add 1 if both x and y are odd to account for the double 0.5 remainder that is truncated after shifting. result = wrap(sum + (xInt & yInt & 1)); } } } /// @notice Yields the smallest whole SD59x18 number greater than or equal to x. /// /// @dev Optimized for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts. /// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions. /// /// Requirements: /// - x must be less than or equal to `MAX_WHOLE_SD59x18`. /// /// @param x The SD59x18 number to ceil. /// @param result The least number greater than or equal to x, as an SD59x18 number. function ceil(SD59x18 x) pure returns (SD59x18 result) { int256 xInt = unwrap(x); if (xInt > uMAX_WHOLE_SD59x18) { revert PRBMath_SD59x18_Ceil_Overflow(x); } int256 remainder = xInt % uUNIT; if (remainder == 0) { result = x; } else { unchecked { // Solidity uses C fmod style, which returns a modulus with the same sign as x. int256 resultInt = xInt - remainder; if (xInt > 0) { resultInt += uUNIT; } result = wrap(resultInt); } } } /// @notice Divides two SD59x18 numbers, returning a new SD59x18 number. Rounds towards zero. /// /// @dev This is a variant of `mulDiv` that works with signed numbers. Works by computing the signs and the absolute values /// separately. /// /// Requirements: /// - All from `Common.mulDiv`. /// - None of the inputs can be `MIN_SD59x18`. /// - The denominator cannot be zero. /// - The result must fit within int256. /// /// Caveats: /// - All from `Common.mulDiv`. /// /// @param x The numerator as an SD59x18 number. /// @param y The denominator as an SD59x18 number. /// @param result The quotient as an SD59x18 number. function div(SD59x18 x, SD59x18 y) pure returns (SD59x18 result) { int256 xInt = unwrap(x); int256 yInt = unwrap(y); if (xInt == uMIN_SD59x18 || yInt == uMIN_SD59x18) { revert PRBMath_SD59x18_Div_InputTooSmall(); } // Get hold of the absolute values of x and y. uint256 xAbs; uint256 yAbs; unchecked { xAbs = xInt < 0 ? uint256(-xInt) : uint256(xInt); yAbs = yInt < 0 ? uint256(-yInt) : uint256(yInt); } // Compute the absolute value (x*UNIT)÷y. The resulting value must fit within int256. uint256 resultAbs = mulDiv(xAbs, uint256(uUNIT), yAbs); if (resultAbs > uint256(uMAX_SD59x18)) { revert PRBMath_SD59x18_Div_Overflow(x, y); } // Check if x and y have the same sign. This works thanks to two's complement; the left-most bit is the sign bit. bool sameSign = (xInt ^ yInt) > -1; // If the inputs don't have the same sign, the result should be negative. Otherwise, it should be positive. unchecked { result = wrap(sameSign ? int256(resultAbs) : -int256(resultAbs)); } } /// @notice Calculates the natural exponent of x. /// /// @dev Based on the formula: /// /// $$ /// e^x = 2^{x * log_2{e}} /// $$ /// /// Requirements: /// - All from `log2`. /// - x must be less than 133.084258667509499441. /// /// Caveats: /// - All from `exp2`. /// - For any x less than -41.446531673892822322, the result is zero. /// /// @param x The exponent as an SD59x18 number. /// @return result The result as an SD59x18 number. function exp(SD59x18 x) pure returns (SD59x18 result) { int256 xInt = unwrap(x); // Without this check, the value passed to `exp2` would be less than -59.794705707972522261. if (xInt < -41_446531673892822322) { return ZERO; } // Without this check, the value passed to `exp2` would be greater than 192. if (xInt >= 133_084258667509499441) { revert PRBMath_SD59x18_Exp_InputTooBig(x); } unchecked { // Do the fixed-point multiplication inline to save gas. int256 doubleUnitProduct = xInt * uLOG2_E; result = exp2(wrap(doubleUnitProduct / uUNIT)); } } /// @notice Calculates the binary exponent of x using the binary fraction method. /// /// @dev Based on the formula: /// /// $$ /// 2^{-x} = \frac{1}{2^x} /// $$ /// /// See https://ethereum.stackexchange.com/q/79903/24693. /// /// Requirements: /// - x must be 192 or less. /// - The result must fit within `MAX_SD59x18`. /// /// Caveats: /// - For any x less than -59.794705707972522261, the result is zero. /// /// @param x The exponent as an SD59x18 number. /// @return result The result as an SD59x18 number. function exp2(SD59x18 x) pure returns (SD59x18 result) { int256 xInt = unwrap(x); if (xInt < 0) { // 2^59.794705707972522262 is the maximum number whose inverse does not truncate down to zero. if (xInt < -59_794705707972522261) { return ZERO; } unchecked { // Do the fixed-point inversion $1/2^x$ inline to save gas. 1e36 is UNIT * UNIT. result = wrap(1e36 / unwrap(exp2(wrap(-xInt)))); } } else { // 2^192 doesn't fit within the 192.64-bit format used internally in this function. if (xInt >= 192e18) { revert PRBMath_SD59x18_Exp2_InputTooBig(x); } unchecked { // Convert x to the 192.64-bit fixed-point format. uint256 x_192x64 = uint256((xInt << 64) / uUNIT); // It is safe to convert the result to int256 with no checks because the maximum input allowed in this function is 192. result = wrap(int256(prbExp2(x_192x64))); } } } /// @notice Yields the greatest whole SD59x18 number less than or equal to x. /// /// @dev Optimized for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts. /// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions. /// /// Requirements: /// - x must be greater than or equal to `MIN_WHOLE_SD59x18`. /// /// @param x The SD59x18 number to floor. /// @param result The greatest integer less than or equal to x, as an SD59x18 number. function floor(SD59x18 x) pure returns (SD59x18 result) { int256 xInt = unwrap(x); if (xInt < uMIN_WHOLE_SD59x18) { revert PRBMath_SD59x18_Floor_Underflow(x); } int256 remainder = xInt % uUNIT; if (remainder == 0) { result = x; } else { unchecked { // Solidity uses C fmod style, which returns a modulus with the same sign as x. int256 resultInt = xInt - remainder; if (xInt < 0) { resultInt -= uUNIT; } result = wrap(resultInt); } } } /// @notice Yields the excess beyond the floor of x for positive numbers and the part of the number to the right. /// of the radix point for negative numbers. /// @dev Based on the odd function definition. https://en.wikipedia.org/wiki/Fractional_part /// @param x The SD59x18 number to get the fractional part of. /// @param result The fractional part of x as an SD59x18 number. function frac(SD59x18 x) pure returns (SD59x18 result) { result = wrap(unwrap(x) % uUNIT); } /// @notice Calculates the geometric mean of x and y, i.e. sqrt(x * y), rounding down. /// /// @dev Requirements: /// - x * y must fit within `MAX_SD59x18`, lest it overflows. /// - x * y must not be negative, since this library does not handle complex numbers. /// /// @param x The first operand as an SD59x18 number. /// @param y The second operand as an SD59x18 number. /// @return result The result as an SD59x18 number. function gm(SD59x18 x, SD59x18 y) pure returns (SD59x18 result) { int256 xInt = unwrap(x); int256 yInt = unwrap(y); if (xInt == 0 || yInt == 0) { return ZERO; } unchecked { // Equivalent to "xy / x != y". Checking for overflow this way is faster than letting Solidity do it. int256 xyInt = xInt * yInt; if (xyInt / xInt != yInt) { revert PRBMath_SD59x18_Gm_Overflow(x, y); } // The product must not be negative, since this library does not handle complex numbers. if (xyInt < 0) { revert PRBMath_SD59x18_Gm_NegativeProduct(x, y); } // We don't need to multiply the result by `UNIT` here because the x*y product had picked up a factor of `UNIT` // during multiplication. See the comments within the `prbSqrt` function. uint256 resultUint = prbSqrt(uint256(xyInt)); result = wrap(int256(resultUint)); } } /// @notice Calculates 1 / x, rounding toward zero. /// /// @dev Requirements: /// - x cannot be zero. /// /// @param x The SD59x18 number for which to calculate the inverse. /// @return result The inverse as an SD59x18 number. function inv(SD59x18 x) pure returns (SD59x18 result) { // 1e36 is UNIT * UNIT. result = wrap(1e36 / unwrap(x)); } /// @notice Calculates the natural logarithm of x. /// /// @dev Based on the formula: /// /// $$ /// ln{x} = log_2{x} / log_2{e}$$. /// $$ /// /// Requirements: /// - All from `log2`. /// /// Caveats: /// - All from `log2`. /// - This doesn't return exactly 1 for 2.718281828459045235, for that more fine-grained precision is needed. /// /// @param x The SD59x18 number for which to calculate the natural logarithm. /// @return result The natural logarithm as an SD59x18 number. function ln(SD59x18 x) pure returns (SD59x18 result) { // Do the fixed-point multiplication inline to save gas. This is overflow-safe because the maximum value that log2(x) // can return is 195.205294292027477728. result = wrap((unwrap(log2(x)) * uUNIT) / uLOG2_E); } /// @notice Calculates the common logarithm of x. /// /// @dev First checks if x is an exact power of ten and it stops if yes. If it's not, calculates the common /// logarithm based on the formula: /// /// $$ /// log_{10}{x} = log_2{x} / log_2{10} /// $$ /// /// Requirements: /// - All from `log2`. /// /// Caveats: /// - All from `log2`. /// /// @param x The SD59x18 number for which to calculate the common logarithm. /// @return result The common logarithm as an SD59x18 number. function log10(SD59x18 x) pure returns (SD59x18 result) { int256 xInt = unwrap(x); if (xInt < 0) { revert PRBMath_SD59x18_Log_InputTooSmall(x); } // Note that the `mul` in this block is the assembly mul operation, not the SD59x18 `mul`. // prettier-ignore assembly ("memory-safe") { switch x case 1 { result := mul(uUNIT, sub(0, 18)) } case 10 { result := mul(uUNIT, sub(1, 18)) } case 100 { result := mul(uUNIT, sub(2, 18)) } case 1000 { result := mul(uUNIT, sub(3, 18)) } case 10000 { result := mul(uUNIT, sub(4, 18)) } case 100000 { result := mul(uUNIT, sub(5, 18)) } case 1000000 { result := mul(uUNIT, sub(6, 18)) } case 10000000 { result := mul(uUNIT, sub(7, 18)) } case 100000000 { result := mul(uUNIT, sub(8, 18)) } case 1000000000 { result := mul(uUNIT, sub(9, 18)) } case 10000000000 { result := mul(uUNIT, sub(10, 18)) } case 100000000000 { result := mul(uUNIT, sub(11, 18)) } case 1000000000000 { result := mul(uUNIT, sub(12, 18)) } case 10000000000000 { result := mul(uUNIT, sub(13, 18)) } case 100000000000000 { result := mul(uUNIT, sub(14, 18)) } case 1000000000000000 { result := mul(uUNIT, sub(15, 18)) } case 10000000000000000 { result := mul(uUNIT, sub(16, 18)) } case 100000000000000000 { result := mul(uUNIT, sub(17, 18)) } case 1000000000000000000 { result := 0 } case 10000000000000000000 { result := uUNIT } case 100000000000000000000 { result := mul(uUNIT, 2) } case 1000000000000000000000 { result := mul(uUNIT, 3) } case 10000000000000000000000 { result := mul(uUNIT, 4) } case 100000000000000000000000 { result := mul(uUNIT, 5) } case 1000000000000000000000000 { result := mul(uUNIT, 6) } case 10000000000000000000000000 { result := mul(uUNIT, 7) } case 100000000000000000000000000 { result := mul(uUNIT, 8) } case 1000000000000000000000000000 { result := mul(uUNIT, 9) } case 10000000000000000000000000000 { result := mul(uUNIT, 10) } case 100000000000000000000000000000 { result := mul(uUNIT, 11) } case 1000000000000000000000000000000 { result := mul(uUNIT, 12) } case 10000000000000000000000000000000 { result := mul(uUNIT, 13) } case 100000000000000000000000000000000 { result := mul(uUNIT, 14) } case 1000000000000000000000000000000000 { result := mul(uUNIT, 15) } case 10000000000000000000000000000000000 { result := mul(uUNIT, 16) } case 100000000000000000000000000000000000 { result := mul(uUNIT, 17) } case 1000000000000000000000000000000000000 { result := mul(uUNIT, 18) } case 10000000000000000000000000000000000000 { result := mul(uUNIT, 19) } case 100000000000000000000000000000000000000 { result := mul(uUNIT, 20) } case 1000000000000000000000000000000000000000 { result := mul(uUNIT, 21) } case 10000000000000000000000000000000000000000 { result := mul(uUNIT, 22) } case 100000000000000000000000000000000000000000 { result := mul(uUNIT, 23) } case 1000000000000000000000000000000000000000000 { result := mul(uUNIT, 24) } case 10000000000000000000000000000000000000000000 { result := mul(uUNIT, 25) } case 100000000000000000000000000000000000000000000 { result := mul(uUNIT, 26) } case 1000000000000000000000000000000000000000000000 { result := mul(uUNIT, 27) } case 10000000000000000000000000000000000000000000000 { result := mul(uUNIT, 28) } case 100000000000000000000000000000000000000000000000 { result := mul(uUNIT, 29) } case 1000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 30) } case 10000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 31) } case 100000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 32) } case 1000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 33) } case 10000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 34) } case 100000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 35) } case 1000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 36) } case 10000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 37) } case 100000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 38) } case 1000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 39) } case 10000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 40) } case 100000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 41) } case 1000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 42) } case 10000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 43) } case 100000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 44) } case 1000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 45) } case 10000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 46) } case 100000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 47) } case 1000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 48) } case 10000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 49) } case 100000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 50) } case 1000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 51) } case 10000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 52) } case 100000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 53) } case 1000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 54) } case 10000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 55) } case 100000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 56) } case 1000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 57) } case 10000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(uUNIT, 58) } default { result := uMAX_SD59x18 } } if (unwrap(result) == uMAX_SD59x18) { unchecked { // Do the fixed-point division inline to save gas. result = wrap((unwrap(log2(x)) * uUNIT) / uLOG2_10); } } } /// @notice Calculates the binary logarithm of x. /// /// @dev Based on the iterative approximation algorithm. /// https://en.wikipedia.org/wiki/Binary_logarithm#Iterative_approximation /// /// Requirements: /// - x must be greater than zero. /// /// Caveats: /// - The results are not perfectly accurate to the last decimal, due to the lossy precision of the iterative approximation. /// /// @param x The SD59x18 number for which to calculate the binary logarithm. /// @return result The binary logarithm as an SD59x18 number. function log2(SD59x18 x) pure returns (SD59x18 result) { int256 xInt = unwrap(x); if (xInt <= 0) { revert PRBMath_SD59x18_Log_InputTooSmall(x); } unchecked { // This works because of: // // $$ // log_2{x} = -log_2{\frac{1}{x}} // $$ int256 sign; if (xInt >= uUNIT) { sign = 1; } else { sign = -1; // Do the fixed-point inversion inline to save gas. The numerator is UNIT * UNIT. xInt = 1e36 / xInt; } // Calculate the integer part of the logarithm and add it to the result and finally calculate $y = x * 2^(-n)$. uint256 n = msb(uint256(xInt / uUNIT)); // This is the integer part of the logarithm as an SD59x18 number. The operation can't overflow // because n is maximum 255, UNIT is 1e18 and sign is either 1 or -1. int256 resultInt = int256(n) * uUNIT; // This is $y = x * 2^{-n}$. int256 y = xInt >> n; // If y is 1, the fractional part is zero. if (y == uUNIT) { return wrap(resultInt * sign); } // Calculate the fractional part via the iterative approximation. // The "delta >>= 1" part is equivalent to "delta /= 2", but shifting bits is faster. int256 DOUBLE_UNIT = 2e18; for (int256 delta = uHALF_UNIT; delta > 0; delta >>= 1) { y = (y * y) / uUNIT; // Is $y^2 > 2$ and so in the range [2,4)? if (y >= DOUBLE_UNIT) { // Add the 2^{-m} factor to the logarithm. resultInt = resultInt + delta; // Corresponds to z/2 on Wikipedia. y >>= 1; } } resultInt *= sign; result = wrap(resultInt); } } /// @notice Multiplies two SD59x18 numbers together, returning a new SD59x18 number. /// /// @dev This is a variant of `mulDiv` that works with signed numbers and employs constant folding, i.e. the denominator /// is always 1e18. /// /// Requirements: /// - All from `Common.mulDiv18`. /// - None of the inputs can be `MIN_SD59x18`. /// - The result must fit within `MAX_SD59x18`. /// /// Caveats: /// - To understand how this works in detail, see the NatSpec comments in `Common.mulDivSigned`. /// /// @param x The multiplicand as an SD59x18 number. /// @param y The multiplier as an SD59x18 number. /// @return result The product as an SD59x18 number. function mul(SD59x18 x, SD59x18 y) pure returns (SD59x18 result) { int256 xInt = unwrap(x); int256 yInt = unwrap(y); if (xInt == uMIN_SD59x18 || yInt == uMIN_SD59x18) { revert PRBMath_SD59x18_Mul_InputTooSmall(); } // Get hold of the absolute values of x and y. uint256 xAbs; uint256 yAbs; unchecked { xAbs = xInt < 0 ? uint256(-xInt) : uint256(xInt); yAbs = yInt < 0 ? uint256(-yInt) : uint256(yInt); } uint256 resultAbs = mulDiv18(xAbs, yAbs); if (resultAbs > uint256(uMAX_SD59x18)) { revert PRBMath_SD59x18_Mul_Overflow(x, y); } // Check if x and y have the same sign. This works thanks to two's complement; the left-most bit is the sign bit. bool sameSign = (xInt ^ yInt) > -1; // If the inputs have the same sign, the result should be negative. Otherwise, it should be positive. unchecked { result = wrap(sameSign ? int256(resultAbs) : -int256(resultAbs)); } } /// @notice Raises x to the power of y. /// /// @dev Based on the formula: /// /// $$ /// x^y = 2^{log_2{x} * y} /// $$ /// /// Requirements: /// - All from `exp2`, `log2` and `mul`. /// - x cannot be zero. /// /// Caveats: /// - All from `exp2`, `log2` and `mul`. /// - Assumes 0^0 is 1. /// /// @param x Number to raise to given power y, as an SD59x18 number. /// @param y Exponent to raise x to, as an SD59x18 number /// @return result x raised to power y, as an SD59x18 number. function pow(SD59x18 x, SD59x18 y) pure returns (SD59x18 result) { int256 xInt = unwrap(x); int256 yInt = unwrap(y); if (xInt == 0) { result = yInt == 0 ? UNIT : ZERO; } else { if (yInt == uUNIT) { result = x; } else { result = exp2(mul(log2(x), y)); } } } /// @notice Raises x (an SD59x18 number) to the power y (unsigned basic integer) using the famous algorithm /// algorithm "exponentiation by squaring". /// /// @dev See https://en.wikipedia.org/wiki/Exponentiation_by_squaring /// /// Requirements: /// - All from `abs` and `Common.mulDiv18`. /// - The result must fit within `MAX_SD59x18`. /// /// Caveats: /// - All from `Common.mulDiv18`. /// - Assumes 0^0 is 1. /// /// @param x The base as an SD59x18 number. /// @param y The exponent as an uint256. /// @return result The result as an SD59x18 number. function powu(SD59x18 x, uint256 y) pure returns (SD59x18 result) { uint256 xAbs = uint256(unwrap(abs(x))); // Calculate the first iteration of the loop in advance. uint256 resultAbs = y & 1 > 0 ? xAbs : uint256(uUNIT); // Equivalent to "for(y /= 2; y > 0; y /= 2)" but faster. uint256 yAux = y; for (yAux >>= 1; yAux > 0; yAux >>= 1) { xAbs = mulDiv18(xAbs, xAbs); // Equivalent to "y % 2 == 1" but faster. if (yAux & 1 > 0) { resultAbs = mulDiv18(resultAbs, xAbs); } } // The result must fit within `MAX_SD59x18`. if (resultAbs > uint256(uMAX_SD59x18)) { revert PRBMath_SD59x18_Powu_Overflow(x, y); } unchecked { // Is the base negative and the exponent an odd number? int256 resultInt = int256(resultAbs); bool isNegative = unwrap(x) < 0 && y & 1 == 1; if (isNegative) { resultInt = -resultInt; } result = wrap(resultInt); } } /// @notice Calculates the square root of x, rounding down. Only the positive root is returned. /// @dev Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method. /// /// Requirements: /// - x cannot be negative, since this library does not handle complex numbers. /// - x must be less than `MAX_SD59x18` divided by `UNIT`. /// /// @param x The SD59x18 number for which to calculate the square root. /// @return result The result as an SD59x18 number. function sqrt(SD59x18 x) pure returns (SD59x18 result) { int256 xInt = unwrap(x); if (xInt < 0) { revert PRBMath_SD59x18_Sqrt_NegativeInput(x); } if (xInt > uMAX_SD59x18 / uUNIT) { revert PRBMath_SD59x18_Sqrt_Overflow(x); } unchecked { // Multiply x by `UNIT` to account for the factor of `UNIT` that is picked up when multiplying two SD59x18 // numbers together (in this case, the two numbers are both the square root). uint256 resultUint = prbSqrt(uint256(xInt * uUNIT)); result = wrap(int256(resultUint)); } }
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { MAX_UINT40 } from "../Common.sol"; import { uMAX_SD1x18 } from "../sd1x18/Constants.sol"; import { SD1x18 } from "../sd1x18/ValueType.sol"; import { SD59x18 } from "../sd59x18/ValueType.sol"; import { UD2x18 } from "../ud2x18/ValueType.sol"; import { UD60x18 } from "../ud60x18/ValueType.sol"; import { PRBMath_UD2x18_IntoSD1x18_Overflow, PRBMath_UD2x18_IntoUint40_Overflow } from "./Errors.sol"; import { UD2x18 } from "./ValueType.sol"; /// @notice Casts an UD2x18 number into SD1x18. /// - x must be less than or equal to `uMAX_SD1x18`. function intoSD1x18(UD2x18 x) pure returns (SD1x18 result) { uint64 xUint = UD2x18.unwrap(x); if (xUint > uint64(uMAX_SD1x18)) { revert PRBMath_UD2x18_IntoSD1x18_Overflow(x); } result = SD1x18.wrap(int64(xUint)); } /// @notice Casts an UD2x18 number into SD59x18. /// @dev There is no overflow check because the domain of UD2x18 is a subset of SD59x18. function intoSD59x18(UD2x18 x) pure returns (SD59x18 result) { result = SD59x18.wrap(int256(uint256(UD2x18.unwrap(x)))); } /// @notice Casts an UD2x18 number into UD60x18. /// @dev There is no overflow check because the domain of UD2x18 is a subset of UD60x18. function intoUD60x18(UD2x18 x) pure returns (UD60x18 result) { result = UD60x18.wrap(UD2x18.unwrap(x)); } /// @notice Casts an UD2x18 number into uint128. /// @dev There is no overflow check because the domain of UD2x18 is a subset of uint128. function intoUint128(UD2x18 x) pure returns (uint128 result) { result = uint128(UD2x18.unwrap(x)); } /// @notice Casts an UD2x18 number into uint256. /// @dev There is no overflow check because the domain of UD2x18 is a subset of uint256. function intoUint256(UD2x18 x) pure returns (uint256 result) { result = uint256(UD2x18.unwrap(x)); } /// @notice Casts an UD2x18 number into uint40. /// @dev Requirements: /// - x must be less than or equal to `MAX_UINT40`. function intoUint40(UD2x18 x) pure returns (uint40 result) { uint64 xUint = UD2x18.unwrap(x); if (xUint > uint64(MAX_UINT40)) { revert PRBMath_UD2x18_IntoUint40_Overflow(x); } result = uint40(xUint); } /// @notice Alias for the `wrap` function. function ud2x18(uint64 x) pure returns (UD2x18 result) { result = UD2x18.wrap(x); } /// @notice Unwrap an UD2x18 number into uint64. function unwrap(UD2x18 x) pure returns (uint64 result) { result = UD2x18.unwrap(x); } /// @notice Wraps an uint64 number into the UD2x18 value type. function wrap(uint64 x) pure returns (UD2x18 result) { result = UD2x18.wrap(x); }
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { SD1x18 } from "./ValueType.sol"; /// @notice Emitted when trying to cast a SD1x18 number that doesn't fit in UD2x18. error PRBMath_SD1x18_ToUD2x18_Underflow(SD1x18 x); /// @notice Emitted when trying to cast a SD1x18 number that doesn't fit in UD60x18. error PRBMath_SD1x18_ToUD60x18_Underflow(SD1x18 x); /// @notice Emitted when trying to cast a SD1x18 number that doesn't fit in uint128. error PRBMath_SD1x18_ToUint128_Underflow(SD1x18 x); /// @notice Emitted when trying to cast a SD1x18 number that doesn't fit in uint256. error PRBMath_SD1x18_ToUint256_Underflow(SD1x18 x); /// @notice Emitted when trying to cast a SD1x18 number that doesn't fit in uint40. error PRBMath_SD1x18_ToUint40_Overflow(SD1x18 x); /// @notice Emitted when trying to cast a SD1x18 number that doesn't fit in uint40. error PRBMath_SD1x18_ToUint40_Underflow(SD1x18 x);
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { SD59x18 } from "./ValueType.sol"; /// @notice Emitted when taking the absolute value of `MIN_SD59x18`. error PRBMath_SD59x18_Abs_MinSD59x18(); /// @notice Emitted when ceiling a number overflows SD59x18. error PRBMath_SD59x18_Ceil_Overflow(SD59x18 x); /// @notice Emitted when converting a basic integer to the fixed-point format overflows SD59x18. error PRBMath_SD59x18_Convert_Overflow(int256 x); /// @notice Emitted when converting a basic integer to the fixed-point format underflows SD59x18. error PRBMath_SD59x18_Convert_Underflow(int256 x); /// @notice Emitted when dividing two numbers and one of them is `MIN_SD59x18`. error PRBMath_SD59x18_Div_InputTooSmall(); /// @notice Emitted when dividing two numbers and one of the intermediary unsigned results overflows SD59x18. error PRBMath_SD59x18_Div_Overflow(SD59x18 x, SD59x18 y); /// @notice Emitted when taking the natural exponent of a base greater than 133.084258667509499441. error PRBMath_SD59x18_Exp_InputTooBig(SD59x18 x); /// @notice Emitted when taking the binary exponent of a base greater than 192. error PRBMath_SD59x18_Exp2_InputTooBig(SD59x18 x); /// @notice Emitted when flooring a number underflows SD59x18. error PRBMath_SD59x18_Floor_Underflow(SD59x18 x); /// @notice Emitted when taking the geometric mean of two numbers and their product is negative. error PRBMath_SD59x18_Gm_NegativeProduct(SD59x18 x, SD59x18 y); /// @notice Emitted when taking the geometric mean of two numbers and multiplying them overflows SD59x18. error PRBMath_SD59x18_Gm_Overflow(SD59x18 x, SD59x18 y); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in SD1x18. error PRBMath_SD59x18_IntoSD1x18_Overflow(SD59x18 x); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in SD1x18. error PRBMath_SD59x18_IntoSD1x18_Underflow(SD59x18 x); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in UD2x18. error PRBMath_SD59x18_IntoUD2x18_Overflow(SD59x18 x); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in UD2x18. error PRBMath_SD59x18_IntoUD2x18_Underflow(SD59x18 x); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in UD60x18. error PRBMath_SD59x18_IntoUD60x18_Underflow(SD59x18 x); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in uint128. error PRBMath_SD59x18_IntoUint128_Overflow(SD59x18 x); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in uint128. error PRBMath_SD59x18_IntoUint128_Underflow(SD59x18 x); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in uint256. error PRBMath_SD59x18_IntoUint256_Underflow(SD59x18 x); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in uint40. error PRBMath_SD59x18_IntoUint40_Overflow(SD59x18 x); /// @notice Emitted when trying to cast an UD60x18 number that doesn't fit in uint40. error PRBMath_SD59x18_IntoUint40_Underflow(SD59x18 x); /// @notice Emitted when taking the logarithm of a number less than or equal to zero. error PRBMath_SD59x18_Log_InputTooSmall(SD59x18 x); /// @notice Emitted when multiplying two numbers and one of the inputs is `MIN_SD59x18`. error PRBMath_SD59x18_Mul_InputTooSmall(); /// @notice Emitted when multiplying two numbers and the intermediary absolute result overflows SD59x18. error PRBMath_SD59x18_Mul_Overflow(SD59x18 x, SD59x18 y); /// @notice Emitted when raising a number to a power and hte intermediary absolute result overflows SD59x18. error PRBMath_SD59x18_Powu_Overflow(SD59x18 x, uint256 y); /// @notice Emitted when taking the square root of a negative number. error PRBMath_SD59x18_Sqrt_NegativeInput(SD59x18 x); /// @notice Emitted when the calculating the square root overflows SD59x18. error PRBMath_SD59x18_Sqrt_Overflow(SD59x18 x);
// SPDX-License-Identifier: MIT pragma solidity >=0.8.13; import { UD2x18 } from "./ValueType.sol"; /// @notice Emitted when trying to cast a UD2x18 number that doesn't fit in SD1x18. error PRBMath_UD2x18_IntoSD1x18_Overflow(UD2x18 x); /// @notice Emitted when trying to cast a UD2x18 number that doesn't fit in uint40. error PRBMath_UD2x18_IntoUint40_Overflow(UD2x18 x);
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Contract Security Audit
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[{"inputs":[{"internalType":"uint256","name":"x","type":"uint256"},{"internalType":"uint256","name":"y","type":"uint256"}],"name":"PRBMath_MulDiv18_Overflow","type":"error"},{"inputs":[{"internalType":"uint256","name":"x","type":"uint256"},{"internalType":"uint256","name":"y","type":"uint256"},{"internalType":"uint256","name":"denominator","type":"uint256"}],"name":"PRBMath_MulDiv_Overflow","type":"error"},{"inputs":[{"internalType":"uint256","name":"x","type":"uint256"}],"name":"PRBMath_UD60x18_Convert_Overflow","type":"error"},{"inputs":[{"internalType":"UD60x18","name":"x","type":"uint256"}],"name":"PRBMath_UD60x18_Exp2_InputTooBig","type":"error"},{"inputs":[{"internalType":"UD60x18","name":"x","type":"uint256"}],"name":"PRBMath_UD60x18_Log_InputTooSmall","type":"error"},{"inputs":[],"name":"MIN_PRICE","outputs":[{"internalType":"uint256","name":"","type":"uint256"}],"stateMutability":"view","type":"function"},{"inputs":[{"internalType":"uint128","name":"spotPrice","type":"uint128"},{"internalType":"uint128","name":"delta","type":"uint128"},{"internalType":"uint256","name":"numItems","type":"uint256"},{"internalType":"uint256","name":"feeMultiplier","type":"uint256"},{"internalType":"uint256","name":"protocolFeeMultiplier","type":"uint256"}],"name":"getBuyInfo","outputs":[{"internalType":"enum CurveErrorCodes.Error","name":"error","type":"uint8"},{"internalType":"uint128","name":"newSpotPrice","type":"uint128"},{"internalType":"uint128","name":"newDelta","type":"uint128"},{"internalType":"uint256","name":"inputValue","type":"uint256"},{"internalType":"uint256","name":"tradeFee","type":"uint256"},{"internalType":"uint256","name":"protocolFee","type":"uint256"}],"stateMutability":"view","type":"function"},{"inputs":[{"internalType":"uint128","name":"spotPrice","type":"uint128"},{"internalType":"uint128","name":"delta","type":"uint128"},{"internalType":"uint256","name":"numItems","type":"uint256"},{"internalType":"uint256","name":"feeMultiplier","type":"uint256"},{"internalType":"uint256","name":"protocolFeeMultiplier","type":"uint256"}],"name":"getSellInfo","outputs":[{"internalType":"enum CurveErrorCodes.Error","name":"error","type":"uint8"},{"internalType":"uint128","name":"newSpotPrice","type":"uint128"},{"internalType":"uint128","name":"newDelta","type":"uint128"},{"internalType":"uint256","name":"outputValue","type":"uint256"},{"internalType":"uint256","name":"tradeFee","type":"uint256"},{"internalType":"uint256","name":"protocolFee","type":"uint256"}],"stateMutability":"view","type":"function"},{"inputs":[{"internalType":"uint128","name":"delta","type":"uint128"}],"name":"validateDelta","outputs":[{"internalType":"bool","name":"","type":"bool"}],"stateMutability":"pure","type":"function"},{"inputs":[{"internalType":"uint128","name":"newSpotPrice","type":"uint128"}],"name":"validateSpotPrice","outputs":[{"internalType":"bool","name":"","type":"bool"}],"stateMutability":"pure","type":"function"}]
Contract Creation Code
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Deployed Bytecode
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Multichain Portfolio | 30 Chains
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A contract address hosts a smart contract, which is a set of code stored on the blockchain that runs when predetermined conditions are met. Learn more about addresses in our Knowledge Base.