Add SignedMath with math utilities for signed integers (#2686)

* add contract and tests

* avoid implicit cast

* add test cases

* fix test names

* modify avarage and add tests

* improve signed average formula

* fix lint

* better average formula

* refactor signed average testing

* add doc and changelog entry

* Update contracts/utils/math/SignedMath.sol

Co-authored-by: Francisco Giordano <frangio.1@gmail.com>

* remove ceilDiv

Co-authored-by: Hadrien Croubois <hadrien.croubois@gmail.com>
Co-authored-by: Francisco Giordano <frangio.1@gmail.com>

CHANGELOG.md

@@ -16,6 +16,7 @@
  * `ERC20`: reduce allowance before triggering transfer. ([#3056](https://github.com/OpenZeppelin/openzeppelin-contracts/pull/#3056))
  * `ERC20`: do not update allowance on `transferFrom` when allowance is `type(uint256).max`. ([#3085](https://github.com/OpenZeppelin/openzeppelin-contracts/pull/#3085))
  * `ERC777`: do not update allowance on `transferFrom` when allowance is `type(uint256).max`. ([#3085](https://github.com/OpenZeppelin/openzeppelin-contracts/pull/#3085))
+ * `SignedMath`: a new signed version of the Math library with `max`, `min`,  and `average`. ([#2686](https://github.com/OpenZeppelin/openzeppelin-contracts/pull/2686))
 
 ### Breaking change
 

contracts/mocks/SignedMathMock.sol

+// SPDX-License-Identifier: MIT
+
+pragma solidity ^0.8.0;
+
+import "../utils/math/SignedMath.sol";
+
+contract SignedMathMock {
+    function max(int256 a, int256 b) public pure returns (int256) {
+        return SignedMath.max(a, b);
+    }
+
+    function min(int256 a, int256 b) public pure returns (int256) {
+        return SignedMath.min(a, b);
+    }
+
+    function average(int256 a, int256 b) public pure returns (int256) {
+        return SignedMath.average(a, b);
+    }
+}

contracts/utils/README.adoc

@@ -27,6 +27,8 @@ Finally, {Create2} contains all necessary utilities to safely use the https://bl
 
 {{Math}}
 
+{{SignedMath}}
+
 {{SafeCast}}
 
 {{SafeMath}}

contracts/utils/math/SignedMath.sol

+// SPDX-License-Identifier: MIT
+
+pragma solidity ^0.8.0;
+
+/**
+ * @dev Standard signed math utilities missing in the Solidity language.
+ */
+library SignedMath {
+    /**
+     * @dev Returns the largest of two signed numbers.
+     */
+    function max(int256 a, int256 b) internal pure returns (int256) {
+        return a >= b ? a : b;
+    }
+
+    /**
+     * @dev Returns the smallest of two signed numbers.
+     */
+    function min(int256 a, int256 b) internal pure returns (int256) {
+        return a < b ? a : b;
+    }
+
+    /**
+     * @dev Returns the average of two signed numbers without overflow.
+     * The result is rounded towards zero.
+     */
+    function average(int256 a, int256 b) internal pure returns (int256) {
+        // Formula from the book "Hacker's Delight"
+        int256 x = (a & b) + ((a ^ b) >> 1);
+        return x + (int256(uint256(x) >> 255) & (a ^ b));
+    }
+}

test/utils/math/SignedMath.test.js

+const { BN, constants } = require('@openzeppelin/test-helpers');
+const { expect } = require('chai');
+const { MIN_INT256, MAX_INT256 } = constants;
+
+const SignedMathMock = artifacts.require('SignedMathMock');
+
+contract('SignedMath', function (accounts) {
+  const min = new BN('-1234');
+  const max = new BN('5678');
+
+  beforeEach(async function () {
+    this.math = await SignedMathMock.new();
+  });
+
+  describe('max', function () {
+    it('is correctly detected in first argument position', async function () {
+      expect(await this.math.max(max, min)).to.be.bignumber.equal(max);
+    });
+
+    it('is correctly detected in second argument position', async function () {
+      expect(await this.math.max(min, max)).to.be.bignumber.equal(max);
+    });
+  });
+
+  describe('min', function () {
+    it('is correctly detected in first argument position', async function () {
+      expect(await this.math.min(min, max)).to.be.bignumber.equal(min);
+    });
+
+    it('is correctly detected in second argument position', async function () {
+      expect(await this.math.min(max, min)).to.be.bignumber.equal(min);
+    });
+  });
+
+  describe('average', function () {
+    function bnAverage (a, b) {
+      return a.add(b).divn(2);
+    }
+
+    it('is correctly calculated with various input', async function () {
+      const valuesX = [
+        new BN('0'),
+        new BN('3'),
+        new BN('-3'),
+        new BN('4'),
+        new BN('-4'),
+        new BN('57417'),
+        new BN('-57417'),
+        new BN('42304'),
+        new BN('-42304'),
+        MIN_INT256,
+        MAX_INT256,
+      ];
+
+      const valuesY = [
+        new BN('0'),
+        new BN('5'),
+        new BN('-5'),
+        new BN('2'),
+        new BN('-2'),
+        new BN('57417'),
+        new BN('-57417'),
+        new BN('42304'),
+        new BN('-42304'),
+        MIN_INT256,
+        MAX_INT256,
+      ];
+
+      for (const x of valuesX) {
+        for (const y of valuesY) {
+          expect(await this.math.average(x, y))
+            .to.be.bignumber.equal(bnAverage(x, y), `Bad result for average(${x}, ${y})`);
+        }
+      }
+    });
+  });
+});