// SPDX-License-Identifier: GPL-3.0-or-later // This program is free software: you can redistribute it and/or modify // it under the terms of the GNU General Public License as published by // the Free Software Foundation, either version 3 of the License, or // (at your option) any later version. // This program is distributed in the hope that it will be useful, // but WITHOUT ANY WARRANTY; without even the implied warranty of // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the // GNU General Public License for more details. // You should have received a copy of the GNU General Public License // along with this program. If not, see . pragma solidity ^0.7.0; import "./LogExpMath.sol"; import "../helpers/BalancerErrors.sol"; /* solhint-disable private-vars-leading-underscore */ library FixedPoint { uint256 internal constant ONE = 1e18; // 18 decimal places uint256 internal constant MAX_POW_RELATIVE_ERROR = 10000; // 10^(-14) // Minimum base for the power function when the exponent is 'free' (larger than ONE). uint256 internal constant MIN_POW_BASE_FREE_EXPONENT = 0.7e18; function add(uint256 a, uint256 b) internal pure returns (uint256) { // Fixed Point addition is the same as regular checked addition uint256 c = a + b; _require(c >= a, Errors.ADD_OVERFLOW); return c; } function sub(uint256 a, uint256 b) internal pure returns (uint256) { // Fixed Point addition is the same as regular checked addition _require(b <= a, Errors.SUB_OVERFLOW); uint256 c = a - b; return c; } function mulDown(uint256 a, uint256 b) internal pure returns (uint256) { uint256 product = a * b; _require(a == 0 || product / a == b, Errors.MUL_OVERFLOW); return product / ONE; } function mulUp(uint256 a, uint256 b) internal pure returns (uint256) { uint256 product = a * b; _require(a == 0 || product / a == b, Errors.MUL_OVERFLOW); if (product == 0) { return 0; } else { // The traditional divUp formula is: // divUp(x, y) := (x + y - 1) / y // To avoid intermediate overflow in the addition, we distribute the division and get: // divUp(x, y) := (x - 1) / y + 1 // Note that this requires x != 0, which we already tested for. return ((product - 1) / ONE) + 1; } } function divDown(uint256 a, uint256 b) internal pure returns (uint256) { _require(b != 0, Errors.ZERO_DIVISION); if (a == 0) { return 0; } else { uint256 aInflated = a * ONE; _require(aInflated / a == ONE, Errors.DIV_INTERNAL); // mul overflow return aInflated / b; } } function divUp(uint256 a, uint256 b) internal pure returns (uint256) { _require(b != 0, Errors.ZERO_DIVISION); if (a == 0) { return 0; } else { uint256 aInflated = a * ONE; _require(aInflated / a == ONE, Errors.DIV_INTERNAL); // mul overflow // The traditional divUp formula is: // divUp(x, y) := (x + y - 1) / y // To avoid intermediate overflow in the addition, we distribute the division and get: // divUp(x, y) := (x - 1) / y + 1 // Note that this requires x != 0, which we already tested for. return ((aInflated - 1) / b) + 1; } } /** * @dev Returns x^y, assuming both are fixed point numbers, rounding down. The result is guaranteed to not be above * the true value (that is, the error function expected - actual is always positive). */ function powDown(uint256 x, uint256 y) internal pure returns (uint256) { uint256 raw = LogExpMath.pow(x, y); uint256 maxError = add(mulUp(raw, MAX_POW_RELATIVE_ERROR), 1); if (raw < maxError) { return 0; } else { return sub(raw, maxError); } } /** * @dev Returns x^y, assuming both are fixed point numbers, rounding up. The result is guaranteed to not be below * the true value (that is, the error function expected - actual is always negative). */ function powUp(uint256 x, uint256 y) internal pure returns (uint256) { uint256 raw = LogExpMath.pow(x, y); uint256 maxError = add(mulUp(raw, MAX_POW_RELATIVE_ERROR), 1); return add(raw, maxError); } /** * @dev Returns the complement of a value (1 - x), capped to 0 if x is larger than 1. * * Useful when computing the complement for values with some level of relative error, as it strips this error and * prevents intermediate negative values. */ function complement(uint256 x) internal pure returns (uint256) { return (x < ONE) ? (ONE - x) : 0; } }