DRAFT: Gradual Dutch Auction Module

src/lib/FullMath.sol

@@ -0,0 +1,128 @@
+// SPDX-License-Identifier: MIT
+pragma solidity >=0.8.0;
+
+/// @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 = (type(uint256).max - 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 mulDivUp(
+        uint256 a,
+        uint256 b,
+        uint256 denominator
+    ) internal pure returns (uint256 result) {
+        result = mulDiv(a, b, denominator);
+        unchecked {
+            if (mulmod(a, b, denominator) > 0) {
+                require(result < type(uint256).max);
+                result++;
+            }
+        }
+    }
+}