diff --git a/contracts/utils/math/Math.sol b/contracts/utils/math/Math.sol index a8851f8eb..8edbf1ef8 100644 --- a/contracts/utils/math/Math.sol +++ b/contracts/utils/math/Math.sol @@ -127,9 +127,9 @@ library Math { */ function mulDiv(uint256 x, uint256 y, uint256 denominator) internal pure returns (uint256 result) { unchecked { - // 512-bit multiply [prod1 prod0] = x * y. Compute the product mod 2^256 and mod 2^256 - 1, then use + // 512-bit multiply [prod1 prod0] = x * y. Compute the product mod 2²⁵⁶ and mod 2²⁵⁶ - 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. + // variables such that product = prod1 * 2²⁵⁶ + prod0. uint256 prod0 = x * y; // Least significant 256 bits of the product uint256 prod1; // Most significant 256 bits of the product assembly { @@ -145,7 +145,7 @@ library Math { return prod0 / denominator; } - // Make sure the result is less than 2^256. Also prevents denominator == 0. + // Make sure the result is less than 2²⁵⁶. Also prevents denominator == 0. if (denominator <= prod1) { Panic.panic(denominator == 0 ? Panic.DIVISION_BY_ZERO : Panic.UNDER_OVERFLOW); } @@ -176,30 +176,30 @@ library Math { // Divide [prod1 prod0] by twos. prod0 := div(prod0, twos) - // Flip twos such that it is 2^256 / twos. If twos is zero, then it becomes one. + // Flip twos such that it is 2²⁵⁶ / twos. If twos is zero, then it becomes one. twos := add(div(sub(0, twos), twos), 1) } // Shift in bits from prod1 into prod0. 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 for - // four bits. That is, denominator * inv = 1 mod 2^4. + // Invert denominator mod 2²⁵⁶. Now that denominator is an odd number, it has an inverse modulo 2²⁵⁶ such + // that denominator * inv ≡ 1 mod 2²⁵⁶. Compute the inverse by starting with a seed that is correct for + // four bits. That is, denominator * inv ≡ 1 mod 2⁴. 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 + inverse *= 2 - denominator * inverse; // inverse mod 2⁸ + inverse *= 2 - denominator * inverse; // inverse mod 2¹⁶ + inverse *= 2 - denominator * inverse; // inverse mod 2³² + inverse *= 2 - denominator * inverse; // inverse mod 2⁶⁴ + inverse *= 2 - denominator * inverse; // inverse mod 2¹²⁸ + inverse *= 2 - denominator * inverse; // inverse mod 2²⁵⁶ // 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 + // This will give us the correct result modulo 2²⁵⁶. Since the preconditions guarantee that the outcome is + // less than 2²⁵⁶, 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; return result;