// SPDX-License-Identifier: MIT
pragma solidity >=0.8.4 <0.9.0;
/// @title Contains 512-bit math functions
/// @notice Facilitates multiplication and division that can have overflow of an intermediate value without any loss of precision
/// @dev Handles "phantom overflow" i.e., allows multiplication and division where an intermediate value overflows 256 bits
library FullMath {
/// @notice Calculates floor(a×b÷denominator) with full precision. Throws if result overflows a uint256 or denominator == 0
/// @param a The multiplicand
/// @param b The multiplier
/// @param denominator The divisor
/// @return result The 256-bit result
/// @dev Credit to Remco Bloemen under MIT license https://xn--2-umb.com/21/muldiv
function mulDiv(
uint256 a,
uint256 b,
uint256 denominator
) internal pure returns (uint256 result) {
unchecked {
// 512-bit multiply [prod1 prod0] = a * b
// Compute the product mod 2**256 and mod 2**256 - 1
// then use the Chinese Remainder Theorem to reconstruct
// the 512 bit result. The result is stored in two 256
// variables such that product = prod1 * 2**256 + prod0
uint256 prod0; // Least significant 256 bits of the product
uint256 prod1; // Most significant 256 bits of the product
assembly {
let mm := mulmod(a, b, not(0))
prod0 := mul(a, b)
prod1 := sub(sub(mm, prod0), lt(mm, prod0))
}
// Handle non-overflow cases, 256 by 256 division
if (prod1 == 0) {
require(denominator > 0);
assembly {
result := div(prod0, denominator)
}
return result;
}
// Make sure the result is less than 2**256.
// Also prevents denominator == 0
require(denominator > prod1);
///////////////////////////////////////////////
// 512 by 256 division.
///////////////////////////////////////////////
// Make division exact by subtracting the remainder from [prod1 prod0]
// Compute remainder using mulmod
uint256 remainder;
assembly {
remainder := mulmod(a, b, denominator)
}
// Subtract 256 bit number from 512 bit number
assembly {
prod1 := sub(prod1, gt(remainder, prod0))
prod0 := sub(prod0, remainder)
}
// Factor powers of two out of denominator
// Compute largest power of two divisor of denominator.
// Always >= 1.
uint256 twos = (~denominator + 1) & denominator;
// Divide denominator by power of two
assembly {
denominator := div(denominator, twos)
}
// Divide [prod1 prod0] by the factors of two
assembly {
prod0 := div(prod0, twos)
}
// Shift in bits from prod1 into prod0. For this we need
// to flip `twos` such that it is 2**256 / twos.
// If twos is zero, then it becomes one
assembly {
twos := add(div(sub(0, twos), twos), 1)
}
prod0 |= prod1 * twos;
// Invert denominator mod 2**256
// Now that denominator is an odd number, it has an inverse
// modulo 2**256 such that denominator * inv = 1 mod 2**256.
// Compute the inverse by starting with a seed that is correct
// correct for four bits. That is, denominator * inv = 1 mod 2**4
uint256 inv = (3 * denominator) ^ 2;
// Now use Newton-Raphson iteration to improve the precision.
// Thanks to Hensel's lifting lemma, this also works in modular
// arithmetic, doubling the correct bits in each step.
inv *= 2 - denominator * inv; // inverse mod 2**8
inv *= 2 - denominator * inv; // inverse mod 2**16
inv *= 2 - denominator * inv; // inverse mod 2**32
inv *= 2 - denominator * inv; // inverse mod 2**64
inv *= 2 - denominator * inv; // inverse mod 2**128
inv *= 2 - denominator * inv; // inverse mod 2**256
// Because the division is now exact we can divide by multiplying
// with the modular inverse of denominator. This will give us the
// correct result modulo 2**256. Since the precoditions guarantee
// that the outcome is less than 2**256, this is the final result.
// We don't need to compute the high bits of the result and prod1
// is no longer required.
result = prod0 * inv;
return result;
}
}
/// @notice Calculates ceil(a×b÷denominator) with full precision. Throws if result overflows a uint256 or denominator == 0
/// @param a The multiplicand
/// @param b The multiplier
/// @param denominator The divisor
/// @return result The 256-bit result
function mulDivRoundingUp(
uint256 a,
uint256 b,
uint256 denominator
) internal pure returns (uint256 result) {
unchecked {
result = mulDiv(a, b, denominator);
if (mulmod(a, b, denominator) > 0) {
require(result < type(uint256).max);
result++;
}
}
}
}
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