{ "contractName": "Fixed192x64Math", "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": "int256" }, { "name": "estimationMode", "type": "Fixed192x64Math.EstimationMode" } ], "name": "pow2", "outputs": [ { "name": "", "type": "uint256" } ], "payable": false, "stateMutability": "pure", 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"source": "pragma solidity >=0.4.24 ^0.5.1;\n\n\n/// @title Fixed192x64Math library - Allows calculation of logarithmic and exponential functions\n/// @author Alan Lu - \n/// @author Stefan George - \nlibrary Fixed192x64Math {\n\n enum EstimationMode { LowerBound, UpperBound, Midpoint }\n\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)\n public\n pure\n returns (uint)\n {\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)\n return 0;\n\n // Transform so that e^x -> 2^x\n (uint lower, uint upper) = pow2Bounds(x * int(ONE) / int(LN2));\n return (upper - lower) / 2 + lower;\n }\n\n /// @dev Returns estimate of 2**x given x\n /// @param x exponent in fixed point\n /// @param estimationMode whether to return a lower bound, upper bound, or a midpoint\n /// @return estimate of 2**x in fixed point\n function pow2(int x, EstimationMode estimationMode)\n public\n pure\n returns (uint)\n {\n (uint lower, uint upper) = pow2Bounds(x);\n if(estimationMode == EstimationMode.LowerBound) {\n return lower;\n }\n if(estimationMode == EstimationMode.UpperBound) {\n return upper;\n }\n if(estimationMode == EstimationMode.Midpoint) {\n return (upper - lower) / 2 + lower;\n }\n revert();\n }\n\n /// @dev Returns bounds for value of 2**x given x\n /// @param x exponent in fixed point\n /// @return {\n /// \"lower\": \"lower bound of 2**x in fixed point\",\n /// \"upper\": \"upper bound of 2**x in fixed point\"\n /// }\n function pow2Bounds(int x)\n public\n pure\n returns (uint lower, uint upper)\n {\n // revert if x is > MAX_POWER, where\n // MAX_POWER = int(mp.floor(mp.log(mpf(2**256 - 1) / ONE, 2) * ONE))\n require(x <= 3541774862152233910271);\n // return 0 if exp(x) is tiny, using\n // MIN_POWER = int(mp.floor(mp.log(mpf(1) / ONE, 2) * ONE))\n if (x < -1180591620717411303424)\n return (0, 1);\n\n // 2^x = 2^(floor(x)) * 2^(x-floor(x))\n // ^^^^^^^^^^^^^^ is a bit shift of ceil(x)\n // so Taylor expand on z = x-floor(x), z in [0, 1)\n int shift;\n int z;\n if (x >= 0) {\n shift = x / int(ONE);\n z = x % int(ONE);\n }\n else {\n shift = (x+1) / int(ONE) - 1;\n z = x - (int(ONE) * shift);\n }\n assert(z >= 0);\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 // >>> coeffs = [mp.log(2)**i / mp.factorial(i) for i in range(1, 21)]\n // >>> shifts = [64 - int(mp.log(c, 2)) for c in coeffs]\n // >>> print('\\n'.join(hex(int(c * (1 << s))) + ', ' + str(s) for c, s in zip(coeffs, shifts)))\n int result = int(ONE) << 64;\n int zpow = z;\n result += 0xb17217f7d1cf79ab * zpow;\n zpow = zpow * z / int(ONE);\n result += 0xf5fdeffc162c7543 * zpow >> (66 - 64);\n zpow = zpow * z / int(ONE);\n result += 0xe35846b82505fc59 * zpow >> (68 - 64);\n zpow = zpow * z / int(ONE);\n result += 0x9d955b7dd273b94e * zpow >> (70 - 64);\n zpow = zpow * z / int(ONE);\n result += 0xaec3ff3c53398883 * zpow >> (73 - 64);\n zpow = zpow * z / int(ONE);\n result += 0xa184897c363c3b7a * zpow >> (76 - 64);\n zpow = zpow * z / int(ONE);\n result += 0xffe5fe2c45863435 * zpow >> (80 - 64);\n zpow = zpow * z / int(ONE);\n result += 0xb160111d2e411fec * zpow >> (83 - 64);\n zpow = zpow * z / int(ONE);\n result += 0xda929e9caf3e1ed2 * zpow >> (87 - 64);\n zpow = zpow * z / int(ONE);\n result += 0xf267a8ac5c764fb7 * zpow >> (91 - 64);\n zpow = zpow * z / int(ONE);\n result += 0xf465639a8dd92607 * zpow >> (95 - 64);\n zpow = zpow * z / int(ONE);\n result += 0xe1deb287e14c2f15 * zpow >> (99 - 64);\n zpow = zpow * z / int(ONE);\n result += 0xc0b0c98b3687cb14 * zpow >> (103 - 64);\n zpow = zpow * z / int(ONE);\n result += 0x98a4b26ac3c54b9f * zpow >> (107 - 64);\n zpow = zpow * z / int(ONE);\n result += 0xe1b7421d82010f33 * zpow >> (112 - 64);\n zpow = zpow * z / int(ONE);\n result += 0x9c744d73cfc59c91 * zpow >> (116 - 64);\n zpow = zpow * z / int(ONE);\n result += 0xcc2225a0e12d3eab * zpow >> (121 - 64);\n zpow = zpow * z / int(ONE);\n zpow = 0xfb8bb5eda1b4aeb9 * zpow >> (126 - 64);\n result += zpow;\n zpow = int(8 * ONE);\n\n shift -= 64;\n if (shift >= 0) {\n if (result >> (256-shift) == 0) {\n lower = uint(result) << shift;\n zpow <<= shift; // todo: is this safe?\n if (lower + uint(zpow) >= lower)\n upper = lower + uint(zpow);\n else\n upper = 2**256-1;\n return (lower, upper);\n }\n else\n return (2**256-1, 2**256-1);\n }\n zpow = (zpow >> (-shift)) + 1;\n lower = uint(result) >> 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"typeIdentifier": "t_type$_t_int256_$", "typeString": "type(int256)" }, "typeName": "int" }, "id": 6651, "isConstant": false, "isLValue": false, "isPure": false, "kind": "typeConversion", "lValueRequested": false, "names": [], "nodeType": "FunctionCall", "src": "7920:13:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "src": "7911:22:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "id": 6653, "nodeType": "ExpressionStatement", "src": "7911:22:28" }, { "expression": { "argumentTypes": null, "id": 6656, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftHandSide": { "argumentTypes": null, "id": 6654, "name": "y", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6588, "src": "7951:1:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "nodeType": "Assignment", "operator": "/=", "rightHandSide": { "argumentTypes": 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"internal" } ], "id": 6620, "initialValue": { "argumentTypes": null, "hexValue": "31", "id": 6619, "isConstant": false, "isLValue": false, "isPure": true, "kind": "number", "lValueRequested": false, "nodeType": "Literal", "src": "7752:1:28", "subdenomination": null, "typeDescriptions": { "typeIdentifier": "t_rational_1_by_1", "typeString": "int_const 1" }, "value": "1" }, "nodeType": "VariableDeclarationStatement", "src": "7744:9:28" }, "loopExpression": { "expression": { "argumentTypes": null, "id": 6625, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "nodeType": "UnaryOperation", "operator": "++", "prefix": false, "src": "7764:3:28", "subExpression": { "argumentTypes": null, "id": 6624, "name": "m", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6618, "src": "7764:1:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "id": 6626, "nodeType": "ExpressionStatement", "src": "7764:3:28" }, "nodeType": "ForStatement", "src": "7739:243:28" }, { "expression": { "argumentTypes": null, "components": [ { "argumentTypes": null, "id": 6662, "name": "lower", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6571, "src": "8000:5:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, { "argumentTypes": null, "commonType": { "typeIdentifier": "t_int256", "typeString": "int256" }, "id": 6665, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "id": 6663, "name": "lower", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6571, "src": "8007:5:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "nodeType": "BinaryOperation", "operator": "+", "rightExpression": { "argumentTypes": null, "hexValue": "34", "id": 6664, "isConstant": false, "isLValue": false, "isPure": true, "kind": "number", "lValueRequested": false, "nodeType": "Literal", "src": "8015:1:28", "subdenomination": null, "typeDescriptions": { "typeIdentifier": "t_rational_4_by_1", "typeString": "int_const 4" }, "value": "4" }, "src": "8007:9:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } } ], "id": 6666, "isConstant": false, "isInlineArray": false, "isLValue": false, "isPure": false, "lValueRequested": false, "nodeType": "TupleExpression", "src": "7999:18:28", "typeDescriptions": { "typeIdentifier": "t_tuple$_t_int256_$_t_int256_$", "typeString": "tuple(int256,int256)" } }, "functionReturnParameters": 6574, "id": 6667, "nodeType": "Return", "src": "7992:25:28" } ] }, "documentation": "@dev Returns bounds for value of binaryLog(x) given x\n @param x logarithm argument in fixed point\n @return {\n \"lower\": \"lower bound of binaryLog(x) in fixed point\",\n \"upper\": \"upper bound of binaryLog(x) in fixed point\"\n }", "id": 6669, "implemented": true, "kind": "function", "modifiers": [], "name": "log2Bounds", "nodeType": "FunctionDefinition", "parameters": { "id": 6569, "nodeType": "ParameterList", "parameters": [ { "constant": false, "id": 6568, "name": "x", "nodeType": "VariableDeclaration", "scope": 6669, "src": "7300:6:28", "stateVariable": false, "storageLocation": "default", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" }, "typeName": { "id": 6567, "name": "uint", "nodeType": "ElementaryTypeName", "src": "7300:4:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "value": null, "visibility": "internal" } ], "src": "7299:8:28" }, "returnParameters": { "id": 6574, "nodeType": "ParameterList", "parameters": [ { "constant": false, "id": 6571, "name": "lower", "nodeType": "VariableDeclaration", "scope": 6669, "src": "7353:9:28", "stateVariable": false, "storageLocation": "default", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" }, "typeName": { "id": 6570, "name": "int", "nodeType": "ElementaryTypeName", "src": "7353:3:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "value": null, "visibility": "internal" }, { "constant": false, "id": 6573, "name": "upper", "nodeType": "VariableDeclaration", "scope": 6669, "src": "7364:9:28", "stateVariable": false, "storageLocation": "default", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" }, "typeName": { "id": 6572, "name": "int", "nodeType": "ElementaryTypeName", "src": "7364:3:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "value": null, "visibility": "internal" } ], "src": "7352:22:28" }, "scope": 6793, "src": "7280:744:28", "stateMutability": "pure", "superFunction": null, "visibility": "public" }, { "body": { "id": 6744, "nodeType": "Block", "src": "8222:404:28", "statements": [ { "expression": { "argumentTypes": null, "id": 6679, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftHandSide": { "argumentTypes": null, "id": 6676, "name": "lo", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6674, "src": "8232:2:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "nodeType": "Assignment", "operator": "=", "rightHandSide": { "argumentTypes": null, "id": 6678, "isConstant": false, "isLValue": false, "isPure": true, "lValueRequested": false, "nodeType": "UnaryOperation", "operator": "-", "prefix": true, "src": "8237:3:28", "subExpression": { "argumentTypes": null, "hexValue": "3634", "id": 6677, "isConstant": false, "isLValue": false, "isPure": true, "kind": "number", "lValueRequested": false, "nodeType": "Literal", "src": "8238:2:28", "subdenomination": null, "typeDescriptions": { "typeIdentifier": "t_rational_64_by_1", "typeString": "int_const 64" }, "value": "64" }, "typeDescriptions": { "typeIdentifier": "t_rational_minus_64_by_1", "typeString": "int_const -64" } }, "src": "8232:8:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "id": 6680, "nodeType": "ExpressionStatement", "src": "8232:8:28" }, { "assignments": [ 6682 ], "declarations": [ { "constant": false, "id": 6682, "name": "hi", "nodeType": "VariableDeclaration", "scope": 6744, "src": "8250:6:28", "stateVariable": false, "storageLocation": "default", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" }, "typeName": { "id": 6681, "name": "int", "nodeType": "ElementaryTypeName", "src": "8250:3:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "value": null, "visibility": "internal" } ], "id": 6684, "initialValue": { "argumentTypes": null, "hexValue": "313933", "id": 6683, "isConstant": false, "isLValue": false, "isPure": true, "kind": "number", "lValueRequested": false, "nodeType": "Literal", "src": "8259:3:28", "subdenomination": null, "typeDescriptions": { "typeIdentifier": "t_rational_193_by_1", "typeString": "int_const 193" }, "value": "193" }, "nodeType": "VariableDeclarationStatement", "src": "8250:12:28" }, { "assignments": [ 6686 ], "declarations": [ { "constant": false, "id": 6686, "name": "mid", "nodeType": "VariableDeclaration", "scope": 6744, "src": "8365:7:28", "stateVariable": false, "storageLocation": "default", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" }, "typeName": { "id": 6685, "name": "int", "nodeType": "ElementaryTypeName", "src": "8365:3:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "value": null, "visibility": "internal" } ], "id": 6693, "initialValue": { "argumentTypes": null, "commonType": { "typeIdentifier": "t_int256", "typeString": "int256" }, "id": 6692, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "components": [ { "argumentTypes": null, "commonType": { "typeIdentifier": "t_int256", "typeString": "int256" }, "id": 6689, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "id": 6687, "name": "hi", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6682, "src": "8376:2:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "nodeType": "BinaryOperation", "operator": "+", "rightExpression": { "argumentTypes": null, "id": 6688, "name": "lo", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6674, "src": "8381:2:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "src": "8376:7:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } } ], "id": 6690, "isConstant": false, "isInlineArray": false, "isLValue": false, "isPure": false, "lValueRequested": false, "nodeType": "TupleExpression", "src": "8375:9:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "nodeType": "BinaryOperation", "operator": ">>", "rightExpression": { "argumentTypes": null, "hexValue": "31", "id": 6691, "isConstant": false, "isLValue": false, "isPure": true, "kind": "number", "lValueRequested": false, "nodeType": "Literal", "src": "8388:1:28", "subdenomination": null, "typeDescriptions": { "typeIdentifier": "t_rational_1_by_1", "typeString": "int_const 1" }, "value": "1" }, "src": "8375:14:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "nodeType": "VariableDeclarationStatement", "src": "8365:24:28" }, { "body": { "id": 6742, "nodeType": "Block", "src": "8420:200:28", "statements": [ { "condition": { "argumentTypes": null, "commonType": { "typeIdentifier": "t_bool", "typeString": "bool" }, "id": 6723, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "commonType": { "typeIdentifier": "t_bool", "typeString": "bool" }, "id": 6711, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "commonType": { "typeIdentifier": "t_int256", "typeString": "int256" }, "id": 6702, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "id": 6700, "name": "mid", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6686, "src": "8438:3:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "nodeType": "BinaryOperation", "operator": "<", "rightExpression": { "argumentTypes": null, "hexValue": "30", "id": 6701, "isConstant": false, "isLValue": false, "isPure": true, "kind": "number", "lValueRequested": false, "nodeType": "Literal", "src": "8444:1:28", "subdenomination": null, "typeDescriptions": { "typeIdentifier": "t_rational_0_by_1", "typeString": "int_const 0" }, "value": "0" }, "src": "8438:7:28", "typeDescriptions": { "typeIdentifier": "t_bool", "typeString": "bool" } }, "nodeType": "BinaryOperation", "operator": "&&", "rightExpression": { "argumentTypes": null, "commonType": { "typeIdentifier": "t_uint256", "typeString": "uint256" }, "id": 6710, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "commonType": { "typeIdentifier": "t_uint256", "typeString": "uint256" }, "id": 6708, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "id": 6703, "name": "x", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6671, "src": "8449:1:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "nodeType": "BinaryOperation", "operator": "<<", "rightExpression": { "argumentTypes": null, "arguments": [ { "argumentTypes": null, "id": 6706, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "nodeType": "UnaryOperation", "operator": "-", "prefix": true, "src": "8459:4:28", "subExpression": { "argumentTypes": null, "id": 6705, "name": "mid", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6686, "src": "8460:3:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } } ], "expression": { "argumentTypes": [ { "typeIdentifier": "t_int256", "typeString": "int256" } ], "id": 6704, "isConstant": false, "isLValue": false, "isPure": true, "lValueRequested": false, "nodeType": "ElementaryTypeNameExpression", "src": "8454:4:28", "typeDescriptions": { "typeIdentifier": "t_type$_t_uint256_$", "typeString": "type(uint256)" }, "typeName": "uint" }, "id": 6707, "isConstant": false, "isLValue": false, "isPure": false, "kind": 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false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "commonType": { "typeIdentifier": "t_int256", "typeString": "int256" }, "id": 6714, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "id": 6712, "name": "mid", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6686, "src": "8474:3:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "nodeType": "BinaryOperation", "operator": ">=", "rightExpression": { "argumentTypes": null, "hexValue": "30", "id": 6713, "isConstant": false, "isLValue": false, "isPure": true, "kind": "number", "lValueRequested": false, "nodeType": "Literal", "src": "8481:1:28", "subdenomination": null, "typeDescriptions": { "typeIdentifier": "t_rational_0_by_1", "typeString": "int_const 0" }, "value": "0" }, "src": "8474:8:28", "typeDescriptions": { "typeIdentifier": "t_bool", "typeString": "bool" } }, "nodeType": "BinaryOperation", "operator": "&&", "rightExpression": { "argumentTypes": null, "commonType": { "typeIdentifier": "t_uint256", "typeString": "uint256" }, "id": 6721, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "commonType": { "typeIdentifier": "t_uint256", "typeString": "uint256" }, "id": 6719, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "id": 6715, "name": "x", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6671, "src": "8486:1:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "nodeType": "BinaryOperation", "operator": ">>", "rightExpression": { "argumentTypes": null, "arguments": [ { "argumentTypes": null, "id": 6717, "name": "mid", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6686, "src": 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null, "typeDescriptions": { "typeIdentifier": "t_rational_256_by_1", "typeString": "int_const 256" }, "value": "256" }, "src": "5669:6:28", "typeDescriptions": { "typeIdentifier": "t_rational_115792089237316195423570985008687907853269984665640564039457584007913129639936_by_1", "typeString": "int_const 1157...(70 digits omitted)...9936" } }, "nodeType": "BinaryOperation", "operator": "-", "rightExpression": { "argumentTypes": null, "hexValue": "31", "id": 6421, "isConstant": false, "isLValue": false, "isPure": true, "kind": "number", "lValueRequested": false, "nodeType": "Literal", "src": "5676:1:28", "subdenomination": null, "typeDescriptions": { "typeIdentifier": "t_rational_1_by_1", "typeString": "int_const 1" }, "value": "1" }, "src": "5669:8:28", "typeDescriptions": { "typeIdentifier": "t_rational_115792089237316195423570985008687907853269984665640564039457584007913129639935_by_1", "typeString": "int_const 1157...(70 digits omitted)...9935" } }, "src": "5661:16:28", 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}, "id": 6464, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "arguments": [ { "argumentTypes": null, "id": 6459, "name": "result", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 5989, "src": "5864:6:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } } ], "expression": { "argumentTypes": [ { "typeIdentifier": "t_int256", "typeString": "int256" } ], "id": 6458, "isConstant": false, "isLValue": false, "isPure": true, "lValueRequested": false, "nodeType": "ElementaryTypeNameExpression", "src": "5859:4:28", "typeDescriptions": { "typeIdentifier": "t_type$_t_uint256_$", "typeString": "type(uint256)" }, "typeName": "uint" }, "id": 6460, "isConstant": false, "isLValue": false, "isPure": false, "kind": "typeConversion", "lValueRequested": false, "names": [], "nodeType": "FunctionCall", "src": "5859:12:28", "typeDescriptions": { "typeIdentifier": 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"typeIdentifier": "t_uint256", "typeString": "uint256" } }, "src": "5851:32:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "id": 6466, "nodeType": "ExpressionStatement", "src": "5851:32:28" }, { "expression": { "argumentTypes": null, "id": 6473, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftHandSide": { "argumentTypes": null, "id": 6467, "name": "upper", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 5913, "src": "5893:5:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "nodeType": "Assignment", "operator": "=", "rightHandSide": { "argumentTypes": null, "commonType": { "typeIdentifier": "t_uint256", "typeString": "uint256" }, "id": 6472, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "id": 6468, "name": "lower", "nodeType": "Identifier", 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false, "names": [], "nodeType": "FunctionCall", "src": "5909:10:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "src": "5901:18:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "src": "5893:26:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "id": 6474, "nodeType": "ExpressionStatement", "src": "5893:26:28" }, { "expression": { "argumentTypes": null, "components": [ { "argumentTypes": null, "id": 6475, "name": "lower", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 5911, "src": "5937:5:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, { "argumentTypes": null, "id": 6476, "name": "upper", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 5913, "src": "5944:5:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } } ], "id": 6477, "isConstant": false, "isInlineArray": false, "isLValue": false, "isPure": false, "lValueRequested": false, "nodeType": "TupleExpression", "src": "5936:14:28", "typeDescriptions": { "typeIdentifier": "t_tuple$_t_uint256_$_t_uint256_$", "typeString": "tuple(uint256,uint256)" } }, "functionReturnParameters": 5914, "id": 6478, "nodeType": "Return", "src": "5929:21:28" } ] }, "documentation": "@dev Returns bounds for value of 2**x given x\n @param x exponent in fixed point\n @return {\n \"lower\": \"lower bound of 2**x in fixed point\",\n \"upper\": \"upper bound of 2**x in fixed point\"\n }", "id": 6480, "implemented": true, "kind": "function", "modifiers": [], "name": "pow2Bounds", "nodeType": "FunctionDefinition", "parameters": { "id": 5909, "nodeType": "ParameterList", "parameters": [ { "constant": false, "id": 5908, "name": "x", "nodeType": "VariableDeclaration", "scope": 6480, "src": "2254:5:28", "stateVariable": false, "storageLocation": "default", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" }, "typeName": { "id": 5907, "name": "int", "nodeType": "ElementaryTypeName", "src": "2254:3:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "value": null, "visibility": "internal" } ], "src": "2253:7:28" }, "returnParameters": { "id": 5914, "nodeType": "ParameterList", "parameters": [ { "constant": false, "id": 5911, "name": "lower", "nodeType": "VariableDeclaration", "scope": 6480, "src": "2306:10:28", "stateVariable": false, "storageLocation": "default", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" }, "typeName": { "id": 5910, "name": "uint", "nodeType": "ElementaryTypeName", "src": "2306:4:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "value": null, "visibility": "internal" }, { "constant": false, "id": 5913, "name": "upper", "nodeType": "VariableDeclaration", "scope": 6480, "src": "2318:10:28", "stateVariable": false, "storageLocation": "default", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" }, "typeName": { "id": 5912, "name": "uint", "nodeType": "ElementaryTypeName", "src": "2318:4:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "value": null, "visibility": "internal" } ], "src": "2305:24:28" }, "scope": 6793, "src": "2234:3723:28", "stateMutability": "pure", "superFunction": null, "visibility": "public" }, { "body": { "id": 6513, "nodeType": "Block", "src": "6134:126:28", "statements": [ { "assignments": [ 6488, 6490 ], "declarations": [ { "constant": false, "id": 6488, "name": "lower", "nodeType": "VariableDeclaration", "scope": 6513, "src": "6145:9:28", "stateVariable": false, "storageLocation": "default", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" }, "typeName": { "id": 6487, "name": "int", "nodeType": "ElementaryTypeName", "src": "6145:3:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "value": null, "visibility": "internal" }, { "constant": false, "id": 6490, "name": "upper", "nodeType": "VariableDeclaration", "scope": 6513, "src": "6156:9:28", "stateVariable": false, "storageLocation": "default", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" }, "typeName": { "id": 6489, "name": "int", "nodeType": "ElementaryTypeName", "src": "6156:3:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "value": null, "visibility": "internal" } ], "id": 6494, "initialValue": { "argumentTypes": null, "arguments": [ { "argumentTypes": null, "id": 6492, "name": "x", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6482, "src": "6180:1:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } } ], "expression": { "argumentTypes": [ { "typeIdentifier": "t_uint256", "typeString": "uint256" } ], "id": 6491, "name": "log2Bounds", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6669, "src": "6169:10:28", "typeDescriptions": { "typeIdentifier": "t_function_internal_pure$_t_uint256_$returns$_t_int256_$_t_int256_$", "typeString": "function (uint256) pure returns (int256,int256)" } }, "id": 6493, "isConstant": false, "isLValue": false, "isPure": false, "kind": "functionCall", "lValueRequested": false, "names": [], "nodeType": "FunctionCall", "src": "6169:13:28", "typeDescriptions": { "typeIdentifier": "t_tuple$_t_int256_$_t_int256_$", "typeString": "tuple(int256,int256)" } }, "nodeType": "VariableDeclarationStatement", "src": "6144:38:28" }, { "expression": { "argumentTypes": null, "commonType": { "typeIdentifier": "t_int256", "typeString": "int256" }, "id": 6511, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "commonType": { "typeIdentifier": "t_int256", "typeString": "int256" }, "id": 6507, "isConstant": false, "isLValue": false, 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"typeString": "int256" } }, "functionReturnParameters": 6486, "id": 6512, "nodeType": "Return", "src": "6192:61:28" } ] }, "documentation": "@dev Returns natural logarithm value of given x\n @param x x\n @return ln(x)", "id": 6514, "implemented": true, "kind": "function", "modifiers": [], "name": "ln", "nodeType": "FunctionDefinition", "parameters": { "id": 6483, "nodeType": "ParameterList", "parameters": [ { "constant": false, "id": 6482, "name": "x", "nodeType": "VariableDeclaration", "scope": 6514, "src": "6072:6:28", "stateVariable": false, "storageLocation": "default", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" }, "typeName": { "id": 6481, "name": "uint", "nodeType": "ElementaryTypeName", "src": "6072:4:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "value": null, "visibility": "internal" } ], "src": "6071:8:28" }, "returnParameters": { "id": 6486, "nodeType": "ParameterList", "parameters": [ { "constant": 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"typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "id": 6616, "nodeType": "ExpressionStatement", "src": "7599:17:28" }, { "body": { "id": 6660, "nodeType": "Block", "src": "7769:213:28", "statements": [ { "condition": { "argumentTypes": null, "commonType": { "typeIdentifier": "t_uint256", "typeString": "uint256" }, "id": 6629, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "id": 6627, "name": "y", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6588, "src": "7786:1:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "nodeType": "BinaryOperation", "operator": "==", "rightExpression": { "argumentTypes": null, "id": 6628, "name": "ONE", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 5802, "src": "7791:3:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": 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"typeIdentifier": "t_bool", "typeString": "bool" } }, "falseBody": null, "id": 6659, "nodeType": "IfStatement", "src": "7876:96:28", "trueBody": { "id": 6658, "nodeType": "Block", "src": "7893:79:28", "statements": [ { "expression": { "argumentTypes": null, "id": 6652, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftHandSide": { "argumentTypes": null, "id": 6646, "name": "lower", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6571, "src": "7911:5:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "nodeType": "Assignment", "operator": "+=", "rightHandSide": { "argumentTypes": null, "arguments": [ { "argumentTypes": null, "commonType": { "typeIdentifier": "t_uint256", "typeString": "uint256" }, "id": 6650, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "id": 6648, "name": "ONE", "nodeType": "Identifier", 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"isPure": false, "kind": "typeConversion", "lValueRequested": false, "names": [], "nodeType": "FunctionCall", "src": "7920:13:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "src": "7911:22:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "id": 6653, "nodeType": "ExpressionStatement", "src": "7911:22:28" }, { "expression": { "argumentTypes": null, "id": 6656, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftHandSide": { "argumentTypes": null, "id": 6654, "name": "y", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6588, "src": "7951:1:28", "typeDescriptions": { "typeIdentifier": "t_uint256", "typeString": "uint256" } }, "nodeType": "Assignment", "operator": "/=", "rightHandSide": { "argumentTypes": null, "hexValue": "32", "id": 6655, "isConstant": false, "isLValue": false, "isPure": true, "kind": "number", "lValueRequested": false, "nodeType": 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"number", "lValueRequested": false, "nodeType": "Literal", "src": "7760:2:28", "subdenomination": null, "typeDescriptions": { "typeIdentifier": "t_rational_64_by_1", "typeString": "int_const 64" }, "value": "64" }, "src": "7755:7:28", "typeDescriptions": { "typeIdentifier": "t_bool", "typeString": "bool" } }, "id": 6661, "initializationExpression": { "assignments": [ 6618 ], "declarations": [ { "constant": false, "id": 6618, "name": "m", "nodeType": "VariableDeclaration", "scope": 6661, "src": "7744:5:28", "stateVariable": false, "storageLocation": "default", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" }, "typeName": { "id": 6617, "name": "int", "nodeType": "ElementaryTypeName", "src": "7744:3:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "value": null, "visibility": "internal" } ], "id": 6620, "initialValue": { "argumentTypes": null, "hexValue": "31", "id": 6619, "isConstant": false, "isLValue": false, "isPure": true, "kind": "number", "lValueRequested": false, "nodeType": "Literal", "src": "7752:1:28", "subdenomination": null, "typeDescriptions": { "typeIdentifier": "t_rational_1_by_1", "typeString": "int_const 1" }, "value": "1" }, "nodeType": "VariableDeclarationStatement", "src": "7744:9:28" }, "loopExpression": { "expression": { "argumentTypes": null, "id": 6625, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "nodeType": "UnaryOperation", "operator": "++", "prefix": false, "src": "7764:3:28", "subExpression": { "argumentTypes": null, "id": 6624, "name": "m", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6618, "src": "7764:1:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "id": 6626, "nodeType": "ExpressionStatement", "src": "7764:3:28" }, "nodeType": "ForStatement", "src": "7739:243:28" }, { "expression": { "argumentTypes": null, "components": [ { "argumentTypes": null, "id": 6662, "name": "lower", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6571, "src": "8000:5:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, { "argumentTypes": null, "commonType": { "typeIdentifier": "t_int256", "typeString": "int256" }, "id": 6665, "isConstant": false, "isLValue": false, "isPure": false, "lValueRequested": false, "leftExpression": { "argumentTypes": null, "id": 6663, "name": "lower", "nodeType": "Identifier", "overloadedDeclarations": [], "referencedDeclaration": 6571, "src": "8007:5:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } }, "nodeType": "BinaryOperation", "operator": "+", "rightExpression": { "argumentTypes": null, "hexValue": "34", "id": 6664, "isConstant": false, "isLValue": false, "isPure": true, "kind": "number", "lValueRequested": false, "nodeType": "Literal", "src": "8015:1:28", "subdenomination": null, "typeDescriptions": { "typeIdentifier": "t_rational_4_by_1", "typeString": "int_const 4" }, "value": "4" }, "src": "8007:9:28", "typeDescriptions": { "typeIdentifier": "t_int256", "typeString": "int256" } } ], "id": 6666, "isConstant": false, "isInlineArray": false, "isLValue": false, "isPure": false, "lValueRequested": false, "nodeType": "TupleExpression", "src": "7999:18:28", "typeDescriptions": { "typeIdentifier": "t_tuple$_t_int256_$_t_int256_$", "typeString": "tuple(int256,int256)" } }, "functionReturnParameters": 6574, "id": 6667, "nodeType": "Return", "src": "7992:25:28" } ] }, "documentation": "@dev Returns bounds for value of binaryLog(x) given x\n @param x logarithm argument in fixed point\n @return {\n \"lower\": \"lower bound of binaryLog(x) in fixed point\",\n \"upper\": \"upper bound of binaryLog(x) in fixed point\"\n }", "id": 6669, "implemented": true, "kind": "function", "modifiers": [], "name": "log2Bounds", "nodeType": 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"log2Bounds(uint256)": { "details": "Returns bounds for value of binaryLog(x) given x", "params": { "x": "logarithm argument in fixed point" }, "return": "{ \"lower\": \"lower bound of binaryLog(x) in fixed point\", \"upper\": \"upper bound of binaryLog(x) in fixed point\" }" }, "max(int256[])": { "details": "Returns maximum of an array", "params": { "nums": "Numbers to look through" }, "return": "Maximum number" }, "pow2(int256,Fixed192x64Math.EstimationMode)": { "details": "Returns estimate of 2**x given x", "params": { "estimationMode": "whether to return a lower bound, upper bound, or a midpoint", "x": "exponent in fixed point" }, "return": "estimate of 2**x in fixed point" }, "pow2Bounds(int256)": { "details": "Returns bounds for value of 2**x given x", "params": { "x": "exponent in fixed point" }, "return": "{ \"lower\": \"lower bound of 2**x in fixed point\", \"upper\": \"upper bound of 2**x in fixed point\" }" } }, "title": "Fixed192x64Math library - Allows calculation of logarithmic and exponential functions" }, "userdoc": { "methods": {} } }