pragma solidity ^0.4.23; import './interfaces/IBancorFormula.sol'; import '../utility/Utils.sol'; contract BancorFormula is IBancorFormula, Utils { string public version = '0.3'; uint256 private constant ONE = 1; uint32 private constant MAX_WEIGHT = 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 'PrintFunctionBancorFormula.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, connector balance, weight and a deposit amount (in the connector token), calculates the return for a given conversion (in the main token) Formula: Return = _supply * ((1 + _depositAmount / _connectorBalance) ^ (_connectorWeight / 1000000) - 1) @param _supply token total supply @param _connectorBalance total connector balance @param _connectorWeight connector weight, represented in ppm, 1-1000000 @param _depositAmount deposit amount, in connector token @return purchase return amount */ function calculatePurchaseReturn(uint256 _supply, uint256 _connectorBalance, uint32 _connectorWeight, uint256 _depositAmount) public view returns (uint256) { // validate input require(_supply > 0 && _connectorBalance > 0 && _connectorWeight > 0 && _connectorWeight <= MAX_WEIGHT); // special case for 0 deposit amount if (_depositAmount == 0) return 0; // special case if the weight = 100% if (_connectorWeight == MAX_WEIGHT) return safeMul(_supply, _depositAmount) / _connectorBalance; uint256 result; uint8 precision; uint256 baseN = safeAdd(_depositAmount, _connectorBalance); (result, precision) = power(baseN, _connectorBalance, _connectorWeight, MAX_WEIGHT); uint256 temp = safeMul(_supply, result) >> precision; return temp - _supply; } /** @dev given a token supply, connector balance, weight and a sell amount (in the main token), calculates the return for a given conversion (in the connector token) Formula: Return = _connectorBalance * (1 - (1 - _sellAmount / _supply) ^ (1 / (_connectorWeight / 1000000))) @param _supply token total supply @param _connectorBalance total connector @param _connectorWeight constant connector Weight, represented in ppm, 1-1000000 @param _sellAmount sell amount, in the token itself @return sale return amount */ function calculateSaleReturn(uint256 _supply, uint256 _connectorBalance, uint32 _connectorWeight, uint256 _sellAmount) public view returns (uint256) { // validate input require(_supply > 0 && _connectorBalance > 0 && _connectorWeight > 0 && _connectorWeight <= MAX_WEIGHT && _sellAmount <= _supply); // special case for 0 sell amount if (_sellAmount == 0) return 0; // special case for selling the entire supply if (_sellAmount == _supply) return _connectorBalance; // special case if the weight = 100% if (_connectorWeight == MAX_WEIGHT) return safeMul(_connectorBalance, _sellAmount) / _supply; uint256 result; uint8 precision; uint256 baseD = _supply - _sellAmount; (result, precision) = power(_supply, baseD, MAX_WEIGHT, _connectorWeight); uint256 temp1 = safeMul(_connectorBalance, result); uint256 temp2 = _connectorBalance << precision; return (temp1 - temp2) / result; } /** @dev given two connector balances/weights and a sell amount (in the first connector token), calculates the return for a conversion from the first connector token to the second connector token (in the second connector token) Formula: Return = _toConnectorBalance * (1 - (_fromConnectorBalance / (_fromConnectorBalance + _amount)) ^ (_fromConnectorWeight / _toConnectorWeight)) @param _fromConnectorBalance input connector balance @param _fromConnectorWeight input connector weight, represented in ppm, 1-1000000 @param _toConnectorBalance output connector balance @param _toConnectorWeight output connector weight, represented in ppm, 1-1000000 @param _amount input connector amount @return second connector amount */ function calculateCrossConnectorReturn(uint256 _fromConnectorBalance, uint32 _fromConnectorWeight, uint256 _toConnectorBalance, uint32 _toConnectorWeight, uint256 _amount) public view returns (uint256) { // validate input require(_fromConnectorBalance > 0 && _fromConnectorWeight > 0 && _fromConnectorWeight <= MAX_WEIGHT && _toConnectorBalance > 0 && _toConnectorWeight > 0 && _toConnectorWeight <= MAX_WEIGHT); // special case for equal weights if (_fromConnectorWeight == _toConnectorWeight) return safeMul(_toConnectorBalance, _amount) / safeAdd(_fromConnectorBalance, _amount); uint256 result; uint8 precision; uint256 baseN = safeAdd(_fromConnectorBalance, _amount); (result, precision) = power(baseN, _fromConnectorBalance, _fromConnectorWeight, _toConnectorWeight); uint256 temp1 = safeMul(_toConnectorBalance, result); uint256 temp2 = _toConnectorBalance << precision; return (temp1 - temp2) / result; } /** 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". */ function power(uint256 _baseN, uint256 _baseD, uint32 _expN, uint32 _expD) internal view returns (uint256, uint8) { assert(_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); } } /** Compute 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; } /** Compute 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; } /** 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; assert(false); return 0; } /** 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! } /** Return log(x / FIXED_1) * FIXED_1 Input range: FIXED_1 <= x <= LOG_EXP_MAX_VAL - 1 Auto-generated via 'PrintFunctionOptimalLog.py' */ 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;} if (x >= 0xa45af1e1f40c333b3de1db4dd55f29a7) {res += 0x20000000000000000000000000000000; x = x * FIXED_1 / 0xa45af1e1f40c333b3de1db4dd55f29a7;} if (x >= 0x910b022db7ae67ce76b441c27035c6a1) {res += 0x10000000000000000000000000000000; x = x * FIXED_1 / 0x910b022db7ae67ce76b441c27035c6a1;} if (x >= 0x88415abbe9a76bead8d00cf112e4d4a8) {res += 0x08000000000000000000000000000000; x = x * FIXED_1 / 0x88415abbe9a76bead8d00cf112e4d4a8;} if (x >= 0x84102b00893f64c705e841d5d4064bd3) {res += 0x04000000000000000000000000000000; x = x * FIXED_1 / 0x84102b00893f64c705e841d5d4064bd3;} if (x >= 0x8204055aaef1c8bd5c3259f4822735a2) {res += 0x02000000000000000000000000000000; x = x * FIXED_1 / 0x8204055aaef1c8bd5c3259f4822735a2;} if (x >= 0x810100ab00222d861931c15e39b44e99) {res += 0x01000000000000000000000000000000; x = x * FIXED_1 / 0x810100ab00222d861931c15e39b44e99;} if (x >= 0x808040155aabbbe9451521693554f733) {res += 0x00800000000000000000000000000000; x = x * FIXED_1 / 0x808040155aabbbe9451521693554f733;} z = y = x - FIXED_1; w = y * y / FIXED_1; res += z * (0x100000000000000000000000000000000 - y) / 0x100000000000000000000000000000000; z = z * w / FIXED_1; res += z * (0x0aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa - y) / 0x200000000000000000000000000000000; z = z * w / FIXED_1; res += z * (0x099999999999999999999999999999999 - y) / 0x300000000000000000000000000000000; z = z * w / FIXED_1; res += z * (0x092492492492492492492492492492492 - y) / 0x400000000000000000000000000000000; z = z * w / FIXED_1; res += z * (0x08e38e38e38e38e38e38e38e38e38e38e - y) / 0x500000000000000000000000000000000; z = z * w / FIXED_1; res += z * (0x08ba2e8ba2e8ba2e8ba2e8ba2e8ba2e8b - y) / 0x600000000000000000000000000000000; z = z * w / FIXED_1; res += z * (0x089d89d89d89d89d89d89d89d89d89d89 - y) / 0x700000000000000000000000000000000; z = z * w / FIXED_1; res += z * (0x088888888888888888888888888888888 - y) / 0x800000000000000000000000000000000; return res; } /** Return e ^ (x / FIXED_1) * FIXED_1 Input range: 0 <= x <= OPT_EXP_MAX_VAL - 1 Auto-generated via 'PrintFunctionOptimalExp.py' */ function optimalExp(uint256 x) internal pure returns (uint256) { uint256 res = 0; uint256 y; uint256 z; z = y = x % 0x10000000000000000000000000000000; 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; if ((x & 0x020000000000000000000000000000000) != 0) res = res * 0x18ebef9eac820ae8682b9793ac6d1e778 / 0x1368b2fc6f9609fe7aceb46aa619baed4; if ((x & 0x040000000000000000000000000000000) != 0) res = res * 0x1368b2fc6f9609fe7aceb46aa619baed5 / 0x0bc5ab1b16779be3575bd8f0520a9f21f; if ((x & 0x080000000000000000000000000000000) != 0) res = res * 0x0bc5ab1b16779be3575bd8f0520a9f21e / 0x0454aaa8efe072e7f6ddbab84b40a55c9; if ((x & 0x100000000000000000000000000000000) != 0) res = res * 0x0454aaa8efe072e7f6ddbab84b40a55c5 / 0x00960aadc109e7a3bf4578099615711ea; if ((x & 0x200000000000000000000000000000000) != 0) res = res * 0x00960aadc109e7a3bf4578099615711d7 / 0x0002bf84208204f5977f9a8cf01fdce3d; if ((x & 0x400000000000000000000000000000000) != 0) res = res * 0x0002bf84208204f5977f9a8cf01fdc307 / 0x0000003c6ab775dd0b95b4cbee7e65d11; return res; } /** @dev given a token supply, connector balance, weight and and a sell amount (in the main token), calculates the return for a given conversion (in the connector token) Formula: Return = _connectorBalance * ((1 + _sellAmount / _supply) ^ (1/(_connectorWeight / 1000000)) - 1) @param _supply token total supply @param _connectorBalance total connector balance @param _connectorWeight connector weight, represented in ppm, 1-1000000 @param _buyAmount buy amount, in the main token @return purchase require amount */ function calculatePurchaseRequire(uint256 _connectorBalance, uint256 _supply, uint32 _connectorWeight, uint256 _buyAmount) public view returns (uint256) { // validate input require(_supply > 0 && _connectorBalance > 0 && _connectorWeight > 0 && _connectorWeight <= MAX_WEIGHT); // special case for 0 deposit amount if (_buyAmount == 0) return 0; // special case if the weight = 100% if (_connectorWeight == MAX_WEIGHT) return safeMul(_connectorBalance, _buyAmount) / _supply; uint256 result; uint8 precision; uint256 baseN = safeAdd(_buyAmount, _supply); (result, precision) = power(baseN, _supply, MAX_WEIGHT, _connectorWeight); uint256 temp = safeMul(_connectorBalance, result) >> precision; return temp - _connectorBalance; } /** @dev given a token supply, connector balance, weight and a sell amount (in the connector token), calculates the return for a given conversion (in the main token) Formula: Return = _supply * (1 - (1 - _sellAmount / _connectorBalance) ^ (_connectorWeight / 1000000)) @param _connectorBalance total connector @param _supply token total supply @param _connectorWeight constant connector Weight, represented in ppm, 1-1000000 @param _expectedSellReturn expected sell return, in the connector token @return sale return amount */ function calculateSaleRequire(uint256 _connectorBalance, uint256 _supply, uint32 _connectorWeight, uint256 _expectedSellReturn) public view returns (uint256) { // validate input require(_supply > 0 && _connectorBalance > 0 && _connectorWeight > 0 && _connectorWeight <= MAX_WEIGHT && _expectedSellReturn <= _connectorBalance); // special case for 0 sell amount if (_expectedSellReturn == 0) return 0; // special case for selling the entire supply if (_expectedSellReturn == _connectorBalance) return _supply; // special case if the weight = 100% if (_connectorWeight == MAX_WEIGHT) return safeMul(_supply, _expectedSellReturn) / _supply; uint256 result; uint8 precision; uint256 baseD = _connectorBalance - _expectedSellReturn; (result, precision) = power(_connectorBalance, baseD, _connectorWeight, MAX_WEIGHT); uint256 temp1 = safeMul(_supply, result); uint256 temp2 = _supply << precision; return (temp1 - temp2) / result; } }