{ "contractName": "GnosisMath", "abi": [ { "constant": true, "inputs": [], "name": "LN2", "outputs": [ { "name": "", "type": "uint256" } ], "payable": false, "stateMutability": "view", "type": "function" }, { "constant": true, "inputs": [], "name": "LOG2_E", "outputs": [ { "name": "", "type": "uint256" } ], "payable": false, "stateMutability": "view", "type": "function" }, { "constant": true, "inputs": [], "name": "ONE", "outputs": [ { "name": "", "type": "uint256" } ], "payable": false, "stateMutability": "view", "type": "function" }, { "constant": true, "inputs": [ { "name": "x", "type": "int256" } ], "name": "exp", "outputs": [ { "name": "", "type": "uint256" } ], "payable": false, "stateMutability": "pure", "type": "function" }, { "constant": true, "inputs": [ { "name": "x", "type": "uint256" } ], "name": "ln", "outputs": [ { "name": "", "type": "int256" } ], "payable": false, "stateMutability": "pure", "type": "function" }, { "constant": true, "inputs": [ { "name": "x", "type": "uint256" } ], "name": "floorLog2", "outputs": [ { "name": "lo", "type": "int256" } ], "payable": false, "stateMutability": "pure", "type": "function" }, { "constant": true, "inputs": [ { "name": "nums", "type": "int256[]" } ], "name": "max", "outputs": [ { "name": "maxNum", "type": "int256" } ], "payable": false, "stateMutability": "pure", "type": "function" } ], "bytecode": 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"source": "pragma solidity ^0.5.2;\n\n/// @title Math library - Allows calculation of logarithmic and exponential functions\n/// @author Alan Lu - \n/// @author Stefan George - \nlibrary GnosisMath {\n /*\n * Constants\n */\n // This is equal to 1 in our calculations\n uint public constant ONE = 0x10000000000000000;\n uint public constant LN2 = 0xb17217f7d1cf79ac;\n uint public constant LOG2_E = 0x171547652b82fe177;\n\n /*\n * Public functions\n */\n /// @dev Returns natural exponential function value of given x\n /// @param x x\n /// @return e**x\n function exp(int x) public pure returns (uint) {\n // revert if x is > MAX_POWER, where\n // MAX_POWER = int(mp.floor(mp.log(mpf(2**256 - 1) / ONE) * ONE))\n require(x <= 2454971259878909886679);\n // return 0 if exp(x) is tiny, using\n // MIN_POWER = int(mp.floor(mp.log(mpf(1) / ONE) * ONE))\n if (x < -818323753292969962227) return 0;\n // Transform so that e^x -> 2^x\n x = x * int(ONE) / int(LN2);\n // 2^x = 2^whole(x) * 2^frac(x)\n // ^^^^^^^^^^ is a bit shift\n // so Taylor expand on z = frac(x)\n int shift;\n uint z;\n if (x >= 0) {\n shift = x / int(ONE);\n z = uint(x % int(ONE));\n } else {\n shift = x / int(ONE) - 1;\n z = ONE - uint(-x % int(ONE));\n }\n // 2^x = 1 + (ln 2) x + (ln 2)^2/2! x^2 + ...\n //\n // Can generate the z coefficients using mpmath and the following lines\n // >>> from mpmath import mp\n // >>> mp.dps = 100\n // >>> ONE = 0x10000000000000000\n // >>> print('\\n'.join(hex(int(mp.log(2)**i / mp.factorial(i) * ONE)) for i in range(1, 7)))\n // 0xb17217f7d1cf79ab\n // 0x3d7f7bff058b1d50\n // 0xe35846b82505fc5\n // 0x276556df749cee5\n // 0x5761ff9e299cc4\n // 0xa184897c363c3\n uint zpow = z;\n uint result = ONE;\n result += 0xb17217f7d1cf79ab * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0x3d7f7bff058b1d50 * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0xe35846b82505fc5 * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0x276556df749cee5 * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0x5761ff9e299cc4 * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0xa184897c363c3 * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0xffe5fe2c4586 * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0x162c0223a5c8 * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0x1b5253d395e * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0x1e4cf5158b * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0x1e8cac735 * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0x1c3bd650 * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0x1816193 * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0x131496 * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0xe1b7 * zpow / ONE;\n zpow = zpow * z / ONE;\n result += 0x9c7 * zpow / ONE;\n if (shift >= 0) {\n if (result >> (256 - shift) > 0) return (2 ** 256 - 1);\n return result << shift;\n } else return result >> (-shift);\n }\n\n /// @dev Returns natural logarithm value of given x\n /// @param x x\n /// @return ln(x)\n function ln(uint x) public pure returns (int) {\n require(x > 0);\n // binary search for floor(log2(x))\n int ilog2 = floorLog2(x);\n int z;\n if (ilog2 < 0) z = int(x << uint(-ilog2));\n else z = int(x >> uint(ilog2));\n // z = x * 2^-⌊log₂x⌋\n // so 1 <= z < 2\n // and ln z = ln x - ⌊log₂x⌋/log₂e\n // so just compute ln z using artanh series\n // and calculate ln x from that\n int term = (z - int(ONE)) * int(ONE) / (z + int(ONE));\n int halflnz = term;\n int termpow = term * term / int(ONE) * term / int(ONE);\n halflnz += termpow / 3;\n termpow = termpow * term / int(ONE) * term / int(ONE);\n halflnz += termpow / 5;\n termpow = termpow * term / int(ONE) * term / int(ONE);\n halflnz += termpow / 7;\n termpow = termpow * term / int(ONE) * term / int(ONE);\n halflnz += termpow / 9;\n termpow = termpow * term / int(ONE) * term / int(ONE);\n halflnz += termpow / 11;\n termpow = termpow * term / int(ONE) * term / int(ONE);\n halflnz += termpow / 13;\n termpow = termpow * term / int(ONE) * term / int(ONE);\n halflnz += termpow / 15;\n termpow = termpow * term / int(ONE) * term / int(ONE);\n halflnz += termpow / 17;\n termpow = termpow * term / int(ONE) * term / int(ONE);\n halflnz += termpow / 19;\n termpow = termpow * term / int(ONE) * term / int(ONE);\n halflnz += termpow / 21;\n termpow = termpow * term / int(ONE) * term / int(ONE);\n halflnz += termpow / 23;\n termpow = termpow * term / int(ONE) * term / int(ONE);\n halflnz += termpow / 25;\n return (ilog2 * int(ONE)) * int(ONE) / int(LOG2_E) + 2 * halflnz;\n }\n\n /// @dev Returns base 2 logarithm value of given x\n /// @param x x\n /// @return logarithmic value\n function floorLog2(uint x) public pure returns (int lo) {\n lo = -64;\n int hi = 193;\n // I use a shift here instead of / 2 because it floors instead of rounding towards 0\n int mid = (hi + lo) >> 1;\n while ((lo + 1) < hi) {\n if (mid < 0 && x << uint(-mid) < ONE || mid >= 0 && x >> uint(mid) < ONE) hi = mid;\n else lo = mid;\n mid = (hi + lo) >> 1;\n }\n }\n\n /// @dev Returns maximum of an array\n /// @param nums Numbers to look through\n /// @return Maximum number\n function max(int[] memory nums) public pure returns (int maxNum) {\n require(nums.length > 0);\n maxNum = -2 ** 255;\n for (uint i = 0; i < nums.length; i++) if (nums[i] > maxNum) maxNum = nums[i];\n }\n\n /// @dev Returns whether an add operation causes an overflow\n /// @param a First addend\n /// @param b Second addend\n /// @return Did no overflow occur?\n function safeToAdd(uint a, uint b) internal pure returns (bool) {\n return a + b >= a;\n }\n\n /// @dev Returns whether a subtraction operation causes an underflow\n /// @param a Minuend\n /// @param b Subtrahend\n /// @return Did no underflow occur?\n function safeToSub(uint a, uint b) internal pure returns (bool) {\n return a >= b;\n }\n\n /// @dev Returns whether a multiply operation causes an overflow\n /// @param a First factor\n /// @param b Second factor\n /// @return Did no overflow occur?\n function safeToMul(uint a, uint b) 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