PRBMathSD59x18.sol

// SPDX-License-Identifier: Unlicense
pragma solidity 0.8.13;

import "./PRBMathCommon.sol";

/* solhint-disable */
/// @title PRBMathSD59x18
/// @author Paul Razvan Berg
/// @notice Smart contract library for advanced fixed-point math. It works with int256 numbers considered to have 18
/// trailing decimals. We call this number representation signed 59.18-decimal fixed-point, since the numbers can have
/// a sign and there can be up to 59 digits in the integer part and up to 18 decimals in the fractional part. The numbers
/// are bound by the minimum and the maximum values permitted by the Solidity type int256.
library PRBMathSD59x18 {
    /// @dev log2(e) as a signed 59.18-decimal fixed-point number.
    int256 internal constant LOG2_E = 1442695040888963407;

    /// @dev Half the SCALE number.
    int256 internal constant HALF_SCALE = 5e17;

    /// @dev The maximum value a signed 59.18-decimal fixed-point number can have.
    int256 internal constant MAX_SD59x18 = 57896044618658097711785492504343953926634992332820282019728792003956564819967;

    /// @dev How many trailing decimals can be represented.
    int256 internal constant SCALE = 1e18;

    /// INTERNAL FUNCTIONS ///


    /// @notice Calculates the natural exponent of x.
    ///
    /// @dev Based on the insight that e^x = 2^(x * log2(e)).
    ///
    /// Requirements:
    /// - All from "log2".
    /// - x must be less than 88722839111672999628.
    ///
    /// @param x The exponent as a signed 59.18-decimal fixed-point number.
    /// @return result The result as a signed 59.18-decimal fixed-point number.
    function exp(int256 x) internal pure returns (int256 result) {
        // Without this check, the value passed to "exp2" would be less than -59794705707972522261.
        if (x < -41446531673892822322) {
            return 0;
        }

        // Without this check, the value passed to "exp2" would be greater than 128e18.
        require(x < 88722839111672999628);

        // Do the fixed-point multiplication inline to save gas.
        unchecked {
            int256 doubleScaleProduct = x * LOG2_E;
            result = exp2((doubleScaleProduct + HALF_SCALE) / SCALE);
        }
    }

    /// @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 128e18 or less.
    /// - The result must fit within MAX_SD59x18.
    ///
    /// Caveats:
    /// - For any x less than -59794705707972522261, the result is zero.
    ///
    /// @param x The exponent as a signed 59.18-decimal fixed-point number.
    /// @return result The result as a signed 59.18-decimal fixed-point number.
    function exp2(int256 x) internal pure returns (int256 result) {
        // This works because 2^-x = 1/2^x.
        if (x < 0) {
            // 2**59.794705707972522262 is the maximum number whose inverse does not equal zero.
            if (x < -59794705707972522261) {
                return 0;
            }

            // Do the fixed-point inversion inline to save gas. The numerator is SCALE * SCALE.
            unchecked { result = 1e36 / exp2(-x); }
            return result;
        } else {
            // 2**128 doesn't fit within the 128.128-bit fixed-point representation.
            require(x < 128e18);

            unchecked {
                // Convert x to the 128.128-bit fixed-point format.
                uint256 x128x128 = (uint256(x) << 128) / uint256(SCALE);

                // Safe to convert the result to int256 directly because the maximum input allowed is 128e18.
                result = int256(PRBMathCommon.exp2(x128x128));
            }
        }
    }
}
/* solhint-enable */