pragma solidity >0.5.4; import "openzeppelin-solidity/contracts/math/SafeMath.sol"; contract BancorFormula { using SafeMath for uint256; uint16 public version = 6; uint256 private constant ONE = 1; uint32 private constant MAX_RATIO = 1000000; uint8 private constant MIN_PRECISION = 32; uint8 private constant MAX_PRECISION = 127; /** * Auto-generated via 'PrintIntScalingFactors.py' */ uint256 private constant FIXED_1 = 0x080000000000000000000000000000000; uint256 private constant FIXED_2 = 0x100000000000000000000000000000000; uint256 private constant MAX_NUM = 0x200000000000000000000000000000000; /** * Auto-generated via 'PrintLn2ScalingFactors.py' */ uint256 private constant LN2_NUMERATOR = 0x3f80fe03f80fe03f80fe03f80fe03f8; uint256 private constant LN2_DENOMINATOR = 0x5b9de1d10bf4103d647b0955897ba80; /** * Auto-generated via 'PrintFunctionOptimalLog.py' and 'PrintFunctionOptimalExp.py' */ uint256 private constant OPT_LOG_MAX_VAL = 0x15bf0a8b1457695355fb8ac404e7a79e3; uint256 private constant OPT_EXP_MAX_VAL = 0x800000000000000000000000000000000; /** * Auto-generated via 'PrintFunctionConstructor.py' */ uint256[128] private maxExpArray; constructor() public { // maxExpArray[ 0] = 0x6bffffffffffffffffffffffffffffffff; // maxExpArray[ 1] = 0x67ffffffffffffffffffffffffffffffff; // maxExpArray[ 2] = 0x637fffffffffffffffffffffffffffffff; // maxExpArray[ 3] = 0x5f6fffffffffffffffffffffffffffffff; // maxExpArray[ 4] = 0x5b77ffffffffffffffffffffffffffffff; // maxExpArray[ 5] = 0x57b3ffffffffffffffffffffffffffffff; // maxExpArray[ 6] = 0x5419ffffffffffffffffffffffffffffff; // maxExpArray[ 7] = 0x50a2ffffffffffffffffffffffffffffff; // maxExpArray[ 8] = 0x4d517fffffffffffffffffffffffffffff; // maxExpArray[ 9] = 0x4a233fffffffffffffffffffffffffffff; // maxExpArray[ 10] = 0x47165fffffffffffffffffffffffffffff; // maxExpArray[ 11] = 0x4429afffffffffffffffffffffffffffff; // maxExpArray[ 12] = 0x415bc7ffffffffffffffffffffffffffff; // maxExpArray[ 13] = 0x3eab73ffffffffffffffffffffffffffff; // maxExpArray[ 14] = 0x3c1771ffffffffffffffffffffffffffff; // maxExpArray[ 15] = 0x399e96ffffffffffffffffffffffffffff; // maxExpArray[ 16] = 0x373fc47fffffffffffffffffffffffffff; // maxExpArray[ 17] = 0x34f9e8ffffffffffffffffffffffffffff; // maxExpArray[ 18] = 0x32cbfd5fffffffffffffffffffffffffff; // maxExpArray[ 19] = 0x30b5057fffffffffffffffffffffffffff; // maxExpArray[ 20] = 0x2eb40f9fffffffffffffffffffffffffff; // maxExpArray[ 21] = 0x2cc8340fffffffffffffffffffffffffff; // maxExpArray[ 22] = 0x2af09481ffffffffffffffffffffffffff; // maxExpArray[ 23] = 0x292c5bddffffffffffffffffffffffffff; // maxExpArray[ 24] = 0x277abdcdffffffffffffffffffffffffff; // maxExpArray[ 25] = 0x25daf6657fffffffffffffffffffffffff; // maxExpArray[ 26] = 0x244c49c65fffffffffffffffffffffffff; // maxExpArray[ 27] = 0x22ce03cd5fffffffffffffffffffffffff; // maxExpArray[ 28] = 0x215f77c047ffffffffffffffffffffffff; // maxExpArray[ 29] = 0x1fffffffffffffffffffffffffffffffff; // maxExpArray[ 30] = 0x1eaefdbdabffffffffffffffffffffffff; // maxExpArray[ 31] = 0x1d6bd8b2ebffffffffffffffffffffffff; maxExpArray[32] = 0x1c35fedd14ffffffffffffffffffffffff; maxExpArray[33] = 0x1b0ce43b323fffffffffffffffffffffff; maxExpArray[34] = 0x19f0028ec1ffffffffffffffffffffffff; maxExpArray[35] = 0x18ded91f0e7fffffffffffffffffffffff; maxExpArray[36] = 0x17d8ec7f0417ffffffffffffffffffffff; maxExpArray[37] = 0x16ddc6556cdbffffffffffffffffffffff; maxExpArray[38] = 0x15ecf52776a1ffffffffffffffffffffff; maxExpArray[39] = 0x15060c256cb2ffffffffffffffffffffff; maxExpArray[40] = 0x1428a2f98d72ffffffffffffffffffffff; maxExpArray[41] = 0x13545598e5c23fffffffffffffffffffff; maxExpArray[42] = 0x1288c4161ce1dfffffffffffffffffffff; maxExpArray[43] = 0x11c592761c666fffffffffffffffffffff; maxExpArray[44] = 0x110a688680a757ffffffffffffffffffff; maxExpArray[45] = 0x1056f1b5bedf77ffffffffffffffffffff; maxExpArray[46] = 0x0faadceceeff8bffffffffffffffffffff; maxExpArray[47] = 0x0f05dc6b27edadffffffffffffffffffff; maxExpArray[48] = 0x0e67a5a25da4107fffffffffffffffffff; maxExpArray[49] = 0x0dcff115b14eedffffffffffffffffffff; maxExpArray[50] = 0x0d3e7a392431239fffffffffffffffffff; maxExpArray[51] = 0x0cb2ff529eb71e4fffffffffffffffffff; maxExpArray[52] = 0x0c2d415c3db974afffffffffffffffffff; maxExpArray[53] = 0x0bad03e7d883f69bffffffffffffffffff; maxExpArray[54] = 0x0b320d03b2c343d5ffffffffffffffffff; maxExpArray[55] = 0x0abc25204e02828dffffffffffffffffff; maxExpArray[56] = 0x0a4b16f74ee4bb207fffffffffffffffff; maxExpArray[57] = 0x09deaf736ac1f569ffffffffffffffffff; maxExpArray[58] = 0x0976bd9952c7aa957fffffffffffffffff; maxExpArray[59] = 0x09131271922eaa606fffffffffffffffff; maxExpArray[60] = 0x08b380f3558668c46fffffffffffffffff; maxExpArray[61] = 0x0857ddf0117efa215bffffffffffffffff; maxExpArray[62] = 0x07ffffffffffffffffffffffffffffffff; maxExpArray[63] = 0x07abbf6f6abb9d087fffffffffffffffff; maxExpArray[64] = 0x075af62cbac95f7dfa7fffffffffffffff; maxExpArray[65] = 0x070d7fb7452e187ac13fffffffffffffff; maxExpArray[66] = 0x06c3390ecc8af379295fffffffffffffff; maxExpArray[67] = 0x067c00a3b07ffc01fd6fffffffffffffff; maxExpArray[68] = 0x0637b647c39cbb9d3d27ffffffffffffff; maxExpArray[69] = 0x05f63b1fc104dbd39587ffffffffffffff; maxExpArray[70] = 0x05b771955b36e12f7235ffffffffffffff; maxExpArray[71] = 0x057b3d49dda84556d6f6ffffffffffffff; maxExpArray[72] = 0x054183095b2c8ececf30ffffffffffffff; maxExpArray[73] = 0x050a28be635ca2b888f77fffffffffffff; maxExpArray[74] = 0x04d5156639708c9db33c3fffffffffffff; maxExpArray[75] = 0x04a23105873875bd52dfdfffffffffffff; maxExpArray[76] = 0x0471649d87199aa990756fffffffffffff; maxExpArray[77] = 0x04429a21a029d4c1457cfbffffffffffff; maxExpArray[78] = 0x0415bc6d6fb7dd71af2cb3ffffffffffff; maxExpArray[79] = 0x03eab73b3bbfe282243ce1ffffffffffff; maxExpArray[80] = 0x03c1771ac9fb6b4c18e229ffffffffffff; maxExpArray[81] = 0x0399e96897690418f785257fffffffffff; maxExpArray[82] = 0x0373fc456c53bb779bf0ea9fffffffffff; maxExpArray[83] = 0x034f9e8e490c48e67e6ab8bfffffffffff; maxExpArray[84] = 0x032cbfd4a7adc790560b3337ffffffffff; maxExpArray[85] = 0x030b50570f6e5d2acca94613ffffffffff; maxExpArray[86] = 0x02eb40f9f620fda6b56c2861ffffffffff; maxExpArray[87] = 0x02cc8340ecb0d0f520a6af58ffffffffff; maxExpArray[88] = 0x02af09481380a0a35cf1ba02ffffffffff; maxExpArray[89] = 0x0292c5bdd3b92ec810287b1b3fffffffff; maxExpArray[90] = 0x0277abdcdab07d5a77ac6d6b9fffffffff; maxExpArray[91] = 0x025daf6654b1eaa55fd64df5efffffffff; maxExpArray[92] = 0x0244c49c648baa98192dce88b7ffffffff; maxExpArray[93] = 0x022ce03cd5619a311b2471268bffffffff; maxExpArray[94] = 0x0215f77c045fbe885654a44a0fffffffff; maxExpArray[95] = 0x01ffffffffffffffffffffffffffffffff; maxExpArray[96] = 0x01eaefdbdaaee7421fc4d3ede5ffffffff; maxExpArray[97] = 0x01d6bd8b2eb257df7e8ca57b09bfffffff; maxExpArray[98] = 0x01c35fedd14b861eb0443f7f133fffffff; maxExpArray[99] = 0x01b0ce43b322bcde4a56e8ada5afffffff; maxExpArray[100] = 0x019f0028ec1fff007f5a195a39dfffffff; maxExpArray[101] = 0x018ded91f0e72ee74f49b15ba527ffffff; maxExpArray[102] = 0x017d8ec7f04136f4e5615fd41a63ffffff; maxExpArray[103] = 0x016ddc6556cdb84bdc8d12d22e6fffffff; maxExpArray[104] = 0x015ecf52776a1155b5bd8395814f7fffff; maxExpArray[105] = 0x015060c256cb23b3b3cc3754cf40ffffff; maxExpArray[106] = 0x01428a2f98d728ae223ddab715be3fffff; maxExpArray[107] = 0x013545598e5c23276ccf0ede68034fffff; maxExpArray[108] = 0x01288c4161ce1d6f54b7f61081194fffff; maxExpArray[109] = 0x011c592761c666aa641d5a01a40f17ffff; maxExpArray[110] = 0x0110a688680a7530515f3e6e6cfdcdffff; maxExpArray[111] = 0x01056f1b5bedf75c6bcb2ce8aed428ffff; maxExpArray[112] = 0x00faadceceeff8a0890f3875f008277fff; maxExpArray[113] = 0x00f05dc6b27edad306388a600f6ba0bfff; maxExpArray[114] = 0x00e67a5a25da41063de1495d5b18cdbfff; maxExpArray[115] = 0x00dcff115b14eedde6fc3aa5353f2e4fff; maxExpArray[116] = 0x00d3e7a3924312399f9aae2e0f868f8fff; maxExpArray[117] = 0x00cb2ff529eb71e41582cccd5a1ee26fff; maxExpArray[118] = 0x00c2d415c3db974ab32a51840c0b67edff; maxExpArray[119] = 0x00bad03e7d883f69ad5b0a186184e06bff; maxExpArray[120] = 0x00b320d03b2c343d4829abd6075f0cc5ff; maxExpArray[121] = 0x00abc25204e02828d73c6e80bcdb1a95bf; maxExpArray[122] = 0x00a4b16f74ee4bb2040a1ec6c15fbbf2df; maxExpArray[123] = 0x009deaf736ac1f569deb1b5ae3f36c130f; maxExpArray[124] = 0x00976bd9952c7aa957f5937d790ef65037; maxExpArray[125] = 0x009131271922eaa6064b73a22d0bd4f2bf; maxExpArray[126] = 0x008b380f3558668c46c91c49a2f8e967b9; maxExpArray[127] = 0x00857ddf0117efa215952912839f6473e6; } /** * @dev given a token supply, reserve balance, ratio and a deposit amount (in the reserve token), * calculates the return for a given conversion (in the main token) * * Formula: * Return = _supply * ((1 + _depositAmount / _reserveBalance) ^ (_reserveRatio / 1000000) - 1) * * @param _supply token total supply * @param _reserveBalance total reserve balance * @param _reserveRatio reserve ratio, represented in ppm, 1-1000000 * @param _depositAmount deposit amount, in reserve token * * @return purchase return amount */ function calculatePurchaseReturn( uint256 _supply, uint256 _reserveBalance, uint32 _reserveRatio, uint256 _depositAmount ) public view returns (uint256) { // validate input require( _supply > 0 && _reserveBalance > 0 && _reserveRatio > 0 && _reserveRatio <= MAX_RATIO ); // special case for 0 deposit amount if (_depositAmount == 0) return 0; // special case if the ratio = 100% if (_reserveRatio == MAX_RATIO) return _supply.mul(_depositAmount) / _reserveBalance; uint256 result; uint8 precision; uint256 baseN = _depositAmount.add(_reserveBalance); (result, precision) = power(baseN, _reserveBalance, _reserveRatio, MAX_RATIO); uint256 temp = _supply.mul(result) >> precision; return temp - _supply; } /** * @dev given a token supply, reserve balance, ratio and a sell amount (in the main token), * calculates the return for a given conversion (in the reserve token) * * Formula: * Return = _reserveBalance * (1 - (1 - _sellAmount / _supply) ^ (1000000 / _reserveRatio)) * * @param _supply token total supply * @param _reserveBalance total reserve * @param _reserveRatio constant reserve Ratio, represented in ppm, 1-1000000 * @param _sellAmount sell amount, in the token itself * * @return sale return amount */ function calculateSaleReturn( uint256 _supply, uint256 _reserveBalance, uint32 _reserveRatio, uint256 _sellAmount ) public view returns (uint256) { // validate input require( _supply > 0 && _reserveBalance > 0 && _reserveRatio > 0 && _reserveRatio <= MAX_RATIO && _sellAmount <= _supply ); // special case for 0 sell amount if (_sellAmount == 0) return 0; // special case for selling the entire supply if (_sellAmount == _supply) return _reserveBalance; // special case if the ratio = 100% if (_reserveRatio == MAX_RATIO) return _reserveBalance.mul(_sellAmount) / _supply; uint256 result; uint8 precision; uint256 baseD = _supply - _sellAmount; (result, precision) = power(_supply, baseD, MAX_RATIO, _reserveRatio); uint256 temp1 = _reserveBalance.mul(result); uint256 temp2 = _reserveBalance << precision; return (temp1 - temp2) / result; } /** * @dev given two reserve balances/ratios and a sell amount (in the first reserve token), * calculates the return for a conversion from the first reserve token to the second reserve token (in the second reserve token) * note that prior to version 4, you should use 'calculateCrossConnectorReturn' instead * * Formula: * Return = _toReserveBalance * (1 - (_fromReserveBalance / (_fromReserveBalance + _amount)) ^ (_fromReserveRatio / _toReserveRatio)) * * @param _fromReserveBalance input reserve balance * @param _fromReserveRatio input reserve ratio, represented in ppm, 1-1000000 * @param _toReserveBalance output reserve balance * @param _toReserveRatio output reserve ratio, represented in ppm, 1-1000000 * @param _amount input reserve amount * * @return second reserve amount */ function calculateCrossReserveReturn( uint256 _fromReserveBalance, uint32 _fromReserveRatio, uint256 _toReserveBalance, uint32 _toReserveRatio, uint256 _amount ) public view returns (uint256) { // validate input require( _fromReserveBalance > 0 && _fromReserveRatio > 0 && _fromReserveRatio <= MAX_RATIO && _toReserveBalance > 0 && _toReserveRatio > 0 && _toReserveRatio <= MAX_RATIO ); // special case for equal ratios if (_fromReserveRatio == _toReserveRatio) return _toReserveBalance.mul(_amount) / _fromReserveBalance.add(_amount); uint256 result; uint8 precision; uint256 baseN = _fromReserveBalance.add(_amount); (result, precision) = power( baseN, _fromReserveBalance, _fromReserveRatio, _toReserveRatio ); uint256 temp1 = _toReserveBalance.mul(result); uint256 temp2 = _toReserveBalance << precision; return (temp1 - temp2) / result; } /** * @dev given a smart token supply, reserve balance, total ratio and an amount of requested smart tokens, * calculates the amount of reserve tokens required for purchasing the given amount of smart tokens * * Formula: * Return = _reserveBalance * (((_supply + _amount) / _supply) ^ (MAX_RATIO / _totalRatio) - 1) * * @param _supply smart token supply * @param _reserveBalance reserve token balance * @param _totalRatio total ratio, represented in ppm, 2-2000000 * @param _amount requested amount of smart tokens * * @return amount of reserve tokens */ function calculateFundCost( uint256 _supply, uint256 _reserveBalance, uint32 _totalRatio, uint256 _amount ) public view returns (uint256) { // validate input require( _supply > 0 && _reserveBalance > 0 && _totalRatio > 1 && _totalRatio <= MAX_RATIO * 2 ); // special case for 0 amount if (_amount == 0) return 0; // special case if the total ratio = 100% if (_totalRatio == MAX_RATIO) return (_amount.mul(_reserveBalance) - 1) / _supply + 1; uint256 result; uint8 precision; uint256 baseN = _supply.add(_amount); (result, precision) = power(baseN, _supply, MAX_RATIO, _totalRatio); uint256 temp = ((_reserveBalance.mul(result) - 1) >> precision) + 1; return temp - _reserveBalance; } /** * @dev given a smart token supply, reserve balance, total ratio and an amount of smart tokens to liquidate, * calculates the amount of reserve tokens received for selling the given amount of smart tokens * * Formula: * Return = _reserveBalance * (1 - ((_supply - _amount) / _supply) ^ (MAX_RATIO / _totalRatio)) * * @param _supply smart token supply * @param _reserveBalance reserve token balance * @param _totalRatio total ratio, represented in ppm, 2-2000000 * @param _amount amount of smart tokens to liquidate * * @return amount of reserve tokens */ function calculateLiquidateReturn( uint256 _supply, uint256 _reserveBalance, uint32 _totalRatio, uint256 _amount ) public view returns (uint256) { // validate input require( _supply > 0 && _reserveBalance > 0 && _totalRatio > 1 && _totalRatio <= MAX_RATIO * 2 && _amount <= _supply ); // special case for 0 amount if (_amount == 0) return 0; // special case for liquidating the entire supply if (_amount == _supply) return _reserveBalance; // special case if the total ratio = 100% if (_totalRatio == MAX_RATIO) return _amount.mul(_reserveBalance) / _supply; uint256 result; uint8 precision; uint256 baseD = _supply - _amount; (result, precision) = power(_supply, baseD, MAX_RATIO, _totalRatio); uint256 temp1 = _reserveBalance.mul(result); uint256 temp2 = _reserveBalance << precision; return (temp1 - temp2) / result; } /** * @dev General Description: * Determine a value of precision. * Calculate an integer approximation of (_baseN / _baseD) ^ (_expN / _expD) * 2 ^ precision. * Return the result along with the precision used. * * Detailed Description: * Instead of calculating "base ^ exp", we calculate "e ^ (log(base) * exp)". * The value of "log(base)" is represented with an integer slightly smaller than "log(base) * 2 ^ precision". * The larger "precision" is, the more accurately this value represents the real value. * However, the larger "precision" is, the more bits are required in order to store this value. * And the exponentiation function, which takes "x" and calculates "e ^ x", is limited to a maximum exponent (maximum value of "x"). * This maximum exponent depends on the "precision" used, and it is given by "maxExpArray[precision] >> (MAX_PRECISION - precision)". * Hence we need to determine the highest precision which can be used for the given input, before calling the exponentiation function. * This allows us to compute "base ^ exp" with maximum accuracy and without exceeding 256 bits in any of the intermediate computations. * This functions assumes that "_expN < 2 ^ 256 / log(MAX_NUM - 1)", otherwise the multiplication should be replaced with a "safeMul". * Since we rely on unsigned-integer arithmetic and "base < 1" ==> "log(base) < 0", this function does not support "_baseN < _baseD". */ function power( uint256 _baseN, uint256 _baseD, uint32 _expN, uint32 _expD ) internal view returns (uint256, uint8) { require(_baseN < MAX_NUM); uint256 baseLog; uint256 base = (_baseN * FIXED_1) / _baseD; if (base < OPT_LOG_MAX_VAL) { baseLog = optimalLog(base); } else { baseLog = generalLog(base); } uint256 baseLogTimesExp = (baseLog * _expN) / _expD; if (baseLogTimesExp < OPT_EXP_MAX_VAL) { return (optimalExp(baseLogTimesExp), MAX_PRECISION); } else { uint8 precision = findPositionInMaxExpArray(baseLogTimesExp); return ( generalExp(baseLogTimesExp >> (MAX_PRECISION - precision), precision), precision ); } } /** * @dev computes log(x / FIXED_1) * FIXED_1. * This functions assumes that "x >= FIXED_1", because the output would be negative otherwise. */ function generalLog(uint256 x) internal pure returns (uint256) { uint256 res = 0; // If x >= 2, then we compute the integer part of log2(x), which is larger than 0. if (x >= FIXED_2) { uint8 count = floorLog2(x / FIXED_1); x >>= count; // now x < 2 res = count * FIXED_1; } // If x > 1, then we compute the fraction part of log2(x), which is larger than 0. if (x > FIXED_1) { for (uint8 i = MAX_PRECISION; i > 0; --i) { x = (x * x) / FIXED_1; // now 1 < x < 4 if (x >= FIXED_2) { x >>= 1; // now 1 < x < 2 res += ONE << (i - 1); } } } return (res * LN2_NUMERATOR) / LN2_DENOMINATOR; } /** * @dev computes the largest integer smaller than or equal to the binary logarithm of the input. */ function floorLog2(uint256 _n) internal pure returns (uint8) { uint8 res = 0; if (_n < 256) { // At most 8 iterations while (_n > 1) { _n >>= 1; res += 1; } } else { // Exactly 8 iterations for (uint8 s = 128; s > 0; s >>= 1) { if (_n >= (ONE << s)) { _n >>= s; res |= s; } } } return res; } /** * @dev the global "maxExpArray" is sorted in descending order, and therefore the following statements are equivalent: * - This function finds the position of [the smallest value in "maxExpArray" larger than or equal to "x"] * - This function finds the highest position of [a value in "maxExpArray" larger than or equal to "x"] */ function findPositionInMaxExpArray(uint256 _x) internal view returns (uint8) { uint8 lo = MIN_PRECISION; uint8 hi = MAX_PRECISION; while (lo + 1 < hi) { uint8 mid = (lo + hi) / 2; if (maxExpArray[mid] >= _x) lo = mid; else hi = mid; } if (maxExpArray[hi] >= _x) return hi; if (maxExpArray[lo] >= _x) return lo; require(false); return 0; } /** * @dev this function can be auto-generated by the script 'PrintFunctionGeneralExp.py'. * it approximates "e ^ x" via maclaurin summation: "(x^0)/0! + (x^1)/1! + ... + (x^n)/n!". * it returns "e ^ (x / 2 ^ precision) * 2 ^ precision", that is, the result is upshifted for accuracy. * the global "maxExpArray" maps each "precision" to "((maximumExponent + 1) << (MAX_PRECISION - precision)) - 1". * the maximum permitted value for "x" is therefore given by "maxExpArray[precision] >> (MAX_PRECISION - precision)". */ function generalExp(uint256 _x, uint8 _precision) internal pure returns (uint256) { uint256 xi = _x; uint256 res = 0; xi = (xi * _x) >> _precision; res += xi * 0x3442c4e6074a82f1797f72ac0000000; // add x^02 * (33! / 02!) xi = (xi * _x) >> _precision; res += xi * 0x116b96f757c380fb287fd0e40000000; // add x^03 * (33! / 03!) xi = (xi * _x) >> _precision; res += xi * 0x045ae5bdd5f0e03eca1ff4390000000; // add x^04 * (33! / 04!) xi = (xi * _x) >> _precision; res += xi * 0x00defabf91302cd95b9ffda50000000; // add x^05 * (33! / 05!) xi = (xi * _x) >> _precision; res += xi * 0x002529ca9832b22439efff9b8000000; // add x^06 * (33! / 06!) xi = (xi * _x) >> _precision; res += xi * 0x00054f1cf12bd04e516b6da88000000; // add x^07 * (33! / 07!) xi = (xi * _x) >> _precision; res += xi * 0x0000a9e39e257a09ca2d6db51000000; // add x^08 * (33! / 08!) xi = (xi * _x) >> _precision; res += xi * 0x000012e066e7b839fa050c309000000; // add x^09 * (33! / 09!) xi = (xi * _x) >> _precision; res += xi * 0x000001e33d7d926c329a1ad1a800000; // add x^10 * (33! / 10!) xi = (xi * _x) >> _precision; res += xi * 0x0000002bee513bdb4a6b19b5f800000; // add x^11 * (33! / 11!) xi = (xi * _x) >> _precision; res += xi * 0x00000003a9316fa79b88eccf2a00000; // add x^12 * (33! / 12!) xi = (xi * _x) >> _precision; res += xi * 0x0000000048177ebe1fa812375200000; // add x^13 * (33! / 13!) xi = (xi * _x) >> _precision; res += xi * 0x0000000005263fe90242dcbacf00000; // add x^14 * (33! / 14!) xi = (xi * _x) >> _precision; res += xi * 0x000000000057e22099c030d94100000; // add x^15 * (33! / 15!) xi = (xi * _x) >> _precision; res += xi * 0x0000000000057e22099c030d9410000; // add x^16 * (33! / 16!) xi = (xi * _x) >> _precision; res += xi * 0x00000000000052b6b54569976310000; // add x^17 * (33! / 17!) xi = (xi * _x) >> _precision; res += xi * 0x00000000000004985f67696bf748000; // add x^18 * (33! / 18!) xi = (xi * _x) >> _precision; res += xi * 0x000000000000003dea12ea99e498000; // add x^19 * (33! / 19!) xi = (xi * _x) >> _precision; res += xi * 0x00000000000000031880f2214b6e000; // add x^20 * (33! / 20!) xi = (xi * _x) >> _precision; res += xi * 0x000000000000000025bcff56eb36000; // add x^21 * (33! / 21!) xi = (xi * _x) >> _precision; res += xi * 0x000000000000000001b722e10ab1000; // add x^22 * (33! / 22!) xi = (xi * _x) >> _precision; res += xi * 0x0000000000000000001317c70077000; // add x^23 * (33! / 23!) xi = (xi * _x) >> _precision; res += xi * 0x00000000000000000000cba84aafa00; // add x^24 * (33! / 24!) xi = (xi * _x) >> _precision; res += xi * 0x00000000000000000000082573a0a00; // add x^25 * (33! / 25!) xi = (xi * _x) >> _precision; res += xi * 0x00000000000000000000005035ad900; // add x^26 * (33! / 26!) xi = (xi * _x) >> _precision; res += xi * 0x000000000000000000000002f881b00; // add x^27 * (33! / 27!) xi = (xi * _x) >> _precision; res += xi * 0x0000000000000000000000001b29340; // add x^28 * (33! / 28!) xi = (xi * _x) >> _precision; res += xi * 0x00000000000000000000000000efc40; // add x^29 * (33! / 29!) xi = (xi * _x) >> _precision; res += xi * 0x0000000000000000000000000007fe0; // add x^30 * (33! / 30!) xi = (xi * _x) >> _precision; res += xi * 0x0000000000000000000000000000420; // add x^31 * (33! / 31!) xi = (xi * _x) >> _precision; res += xi * 0x0000000000000000000000000000021; // add x^32 * (33! / 32!) xi = (xi * _x) >> _precision; res += xi * 0x0000000000000000000000000000001; // add x^33 * (33! / 33!) return res / 0x688589cc0e9505e2f2fee5580000000 + _x + (ONE << _precision); // divide by 33! and then add x^1 / 1! + x^0 / 0! } /** * @dev computes log(x / FIXED_1) * FIXED_1 * Input range: FIXED_1 <= x <= LOG_EXP_MAX_VAL - 1 * Auto-generated via 'PrintFunctionOptimalLog.py' * Detailed description: * - Rewrite the input as a product of natural exponents and a single residual r, such that 1 < r < 2 * - The natural logarithm of each (pre-calculated) exponent is the degree of the exponent * - The natural logarithm of r is calculated via Taylor series for log(1 + x), where x = r - 1 * - The natural logarithm of the input is calculated by summing up the intermediate results above * - For example: log(250) = log(e^4 * e^1 * e^0.5 * 1.021692859) = 4 + 1 + 0.5 + log(1 + 0.021692859) */ function optimalLog(uint256 x) internal pure returns (uint256) { uint256 res = 0; uint256 y; uint256 z; uint256 w; if (x >= 0xd3094c70f034de4b96ff7d5b6f99fcd8) { res += 0x40000000000000000000000000000000; x = (x * FIXED_1) / 0xd3094c70f034de4b96ff7d5b6f99fcd8; } // add 1 / 2^1 if (x >= 0xa45af1e1f40c333b3de1db4dd55f29a7) { res += 0x20000000000000000000000000000000; x = (x * FIXED_1) / 0xa45af1e1f40c333b3de1db4dd55f29a7; } // add 1 / 2^2 if (x >= 0x910b022db7ae67ce76b441c27035c6a1) { res += 0x10000000000000000000000000000000; x = (x * FIXED_1) / 0x910b022db7ae67ce76b441c27035c6a1; } // add 1 / 2^3 if (x >= 0x88415abbe9a76bead8d00cf112e4d4a8) { res += 0x08000000000000000000000000000000; x = (x * FIXED_1) / 0x88415abbe9a76bead8d00cf112e4d4a8; } // add 1 / 2^4 if (x >= 0x84102b00893f64c705e841d5d4064bd3) { res += 0x04000000000000000000000000000000; x = (x * FIXED_1) / 0x84102b00893f64c705e841d5d4064bd3; } // add 1 / 2^5 if (x >= 0x8204055aaef1c8bd5c3259f4822735a2) { res += 0x02000000000000000000000000000000; x = (x * FIXED_1) / 0x8204055aaef1c8bd5c3259f4822735a2; } // add 1 / 2^6 if (x >= 0x810100ab00222d861931c15e39b44e99) { res += 0x01000000000000000000000000000000; x = (x * FIXED_1) / 0x810100ab00222d861931c15e39b44e99; } // add 1 / 2^7 if (x >= 0x808040155aabbbe9451521693554f733) { res += 0x00800000000000000000000000000000; x = (x * FIXED_1) / 0x808040155aabbbe9451521693554f733; } // add 1 / 2^8 z = y = x - FIXED_1; w = (y * y) / FIXED_1; res += (z * (0x100000000000000000000000000000000 - y)) / 0x100000000000000000000000000000000; z = (z * w) / FIXED_1; // add y^01 / 01 - y^02 / 02 res += (z * (0x0aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa - y)) / 0x200000000000000000000000000000000; z = (z * w) / FIXED_1; // add y^03 / 03 - y^04 / 04 res += (z * (0x099999999999999999999999999999999 - y)) / 0x300000000000000000000000000000000; z = (z * w) / FIXED_1; // add y^05 / 05 - y^06 / 06 res += (z * (0x092492492492492492492492492492492 - y)) / 0x400000000000000000000000000000000; z = (z * w) / FIXED_1; // add y^07 / 07 - y^08 / 08 res += (z * (0x08e38e38e38e38e38e38e38e38e38e38e - y)) / 0x500000000000000000000000000000000; z = (z * w) / FIXED_1; // add y^09 / 09 - y^10 / 10 res += (z * (0x08ba2e8ba2e8ba2e8ba2e8ba2e8ba2e8b - y)) / 0x600000000000000000000000000000000; z = (z * w) / FIXED_1; // add y^11 / 11 - y^12 / 12 res += (z * (0x089d89d89d89d89d89d89d89d89d89d89 - y)) / 0x700000000000000000000000000000000; z = (z * w) / FIXED_1; // add y^13 / 13 - y^14 / 14 res += (z * (0x088888888888888888888888888888888 - y)) / 0x800000000000000000000000000000000; // add y^15 / 15 - y^16 / 16 return res; } /** * @dev computes e ^ (x / FIXED_1) * FIXED_1 * input range: 0 <= x <= OPT_EXP_MAX_VAL - 1 * auto-generated via 'PrintFunctionOptimalExp.py' * Detailed description: * - Rewrite the input as a sum of binary exponents and a single residual r, as small as possible * - The exponentiation of each binary exponent is given (pre-calculated) * - The exponentiation of r is calculated via Taylor series for e^x, where x = r * - The exponentiation of the input is calculated by multiplying the intermediate results above * - For example: e^5.521692859 = e^(4 + 1 + 0.5 + 0.021692859) = e^4 * e^1 * e^0.5 * e^0.021692859 */ function optimalExp(uint256 x) internal pure returns (uint256) { uint256 res = 0; uint256 y; uint256 z; z = y = x % 0x10000000000000000000000000000000; // get the input modulo 2^(-3) z = (z * y) / FIXED_1; res += z * 0x10e1b3be415a0000; // add y^02 * (20! / 02!) z = (z * y) / FIXED_1; res += z * 0x05a0913f6b1e0000; // add y^03 * (20! / 03!) z = (z * y) / FIXED_1; res += z * 0x0168244fdac78000; // add y^04 * (20! / 04!) z = (z * y) / FIXED_1; res += z * 0x004807432bc18000; // add y^05 * (20! / 05!) z = (z * y) / FIXED_1; res += z * 0x000c0135dca04000; // add y^06 * (20! / 06!) z = (z * y) / FIXED_1; res += z * 0x0001b707b1cdc000; // add y^07 * (20! / 07!) z = (z * y) / FIXED_1; res += z * 0x000036e0f639b800; // add y^08 * (20! / 08!) z = (z * y) / FIXED_1; res += z * 0x00000618fee9f800; // add y^09 * (20! / 09!) z = (z * y) / FIXED_1; res += z * 0x0000009c197dcc00; // add y^10 * (20! / 10!) z = (z * y) / FIXED_1; res += z * 0x0000000e30dce400; // add y^11 * (20! / 11!) z = (z * y) / FIXED_1; res += z * 0x000000012ebd1300; // add y^12 * (20! / 12!) z = (z * y) / FIXED_1; res += z * 0x0000000017499f00; // add y^13 * (20! / 13!) z = (z * y) / FIXED_1; res += z * 0x0000000001a9d480; // add y^14 * (20! / 14!) z = (z * y) / FIXED_1; res += z * 0x00000000001c6380; // add y^15 * (20! / 15!) z = (z * y) / FIXED_1; res += z * 0x000000000001c638; // add y^16 * (20! / 16!) z = (z * y) / FIXED_1; res += z * 0x0000000000001ab8; // add y^17 * (20! / 17!) z = (z * y) / FIXED_1; res += z * 0x000000000000017c; // add y^18 * (20! / 18!) z = (z * y) / FIXED_1; res += z * 0x0000000000000014; // add y^19 * (20! / 19!) z = (z * y) / FIXED_1; res += z * 0x0000000000000001; // add y^20 * (20! / 20!) res = res / 0x21c3677c82b40000 + y + FIXED_1; // divide by 20! and then add y^1 / 1! + y^0 / 0! if ((x & 0x010000000000000000000000000000000) != 0) res = (res * 0x1c3d6a24ed82218787d624d3e5eba95f9) / 0x18ebef9eac820ae8682b9793ac6d1e776; // multiply by e^2^(-3) if ((x & 0x020000000000000000000000000000000) != 0) res = (res * 0x18ebef9eac820ae8682b9793ac6d1e778) / 0x1368b2fc6f9609fe7aceb46aa619baed4; // multiply by e^2^(-2) if ((x & 0x040000000000000000000000000000000) != 0) res = (res * 0x1368b2fc6f9609fe7aceb46aa619baed5) / 0x0bc5ab1b16779be3575bd8f0520a9f21f; // multiply by e^2^(-1) if ((x & 0x080000000000000000000000000000000) != 0) res = (res * 0x0bc5ab1b16779be3575bd8f0520a9f21e) / 0x0454aaa8efe072e7f6ddbab84b40a55c9; // multiply by e^2^(+0) if ((x & 0x100000000000000000000000000000000) != 0) res = (res * 0x0454aaa8efe072e7f6ddbab84b40a55c5) / 0x00960aadc109e7a3bf4578099615711ea; // multiply by e^2^(+1) if ((x & 0x200000000000000000000000000000000) != 0) res = (res * 0x00960aadc109e7a3bf4578099615711d7) / 0x0002bf84208204f5977f9a8cf01fdce3d; // multiply by e^2^(+2) if ((x & 0x400000000000000000000000000000000) != 0) res = (res * 0x0002bf84208204f5977f9a8cf01fdc307) / 0x0000003c6ab775dd0b95b4cbee7e65d11; // multiply by e^2^(+3) return res; } /** * @dev deprecated, backward compatibility */ function calculateCrossConnectorReturn( uint256 _fromConnectorBalance, uint32 _fromConnectorWeight, uint256 _toConnectorBalance, uint32 _toConnectorWeight, uint256 _amount ) public view returns (uint256) { return calculateCrossReserveReturn( _fromConnectorBalance, _fromConnectorWeight, _toConnectorBalance, _toConnectorWeight, _amount ); } }