489 lines
26 KiB
Solidity
489 lines
26 KiB
Solidity
pragma solidity ^0.5.2;
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import "./SafeMath.sol";
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contract BancorFormula {
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using SafeMath for uint256;
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uint256 private constant ONE = 1;
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uint8 private constant MIN_PRECISION = 32;
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uint8 private constant MAX_PRECISION = 127;
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/**
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Auto-generated via 'PrintIntScalingFactors.py'
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*/
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uint256 private constant FIXED_1 = 0x080000000000000000000000000000000;
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uint256 private constant FIXED_2 = 0x100000000000000000000000000000000;
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uint256 private constant MAX_NUM = 0x200000000000000000000000000000000;
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/**
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Auto-generated via 'PrintLn2ScalingFactors.py'
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*/
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uint256 private constant LN2_NUMERATOR = 0x3f80fe03f80fe03f80fe03f80fe03f8;
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uint256 private constant LN2_DENOMINATOR = 0x5b9de1d10bf4103d647b0955897ba80;
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/**
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Auto-generated via 'PrintFunctionOptimalLog.py' and 'PrintFunctionOptimalExp.py'
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*/
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uint256 private constant OPT_LOG_MAX_VAL = 0x15bf0a8b1457695355fb8ac404e7a79e3;
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uint256 private constant OPT_EXP_MAX_VAL = 0x800000000000000000000000000000000;
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/**
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Auto-generated via 'PrintFunctionConstructor.py'
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*/
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uint256[128] private maxExpArray;
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constructor() public {
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// maxExpArray[0] = 0x6bffffffffffffffffffffffffffffffff;
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// maxExpArray[1] = 0x67ffffffffffffffffffffffffffffffff;
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// maxExpArray[2] = 0x637fffffffffffffffffffffffffffffff;
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// maxExpArray[3] = 0x5f6fffffffffffffffffffffffffffffff;
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// maxExpArray[4] = 0x5b77ffffffffffffffffffffffffffffff;
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// maxExpArray[5] = 0x57b3ffffffffffffffffffffffffffffff;
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// maxExpArray[6] = 0x5419ffffffffffffffffffffffffffffff;
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// maxExpArray[7] = 0x50a2ffffffffffffffffffffffffffffff;
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// maxExpArray[8] = 0x4d517fffffffffffffffffffffffffffff;
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// maxExpArray[9] = 0x4a233fffffffffffffffffffffffffffff;
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// maxExpArray[10] = 0x47165fffffffffffffffffffffffffffff;
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// maxExpArray[11] = 0x4429afffffffffffffffffffffffffffff;
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// maxExpArray[12] = 0x415bc7ffffffffffffffffffffffffffff;
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// maxExpArray[13] = 0x3eab73ffffffffffffffffffffffffffff;
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// maxExpArray[14] = 0x3c1771ffffffffffffffffffffffffffff;
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// maxExpArray[15] = 0x399e96ffffffffffffffffffffffffffff;
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// maxExpArray[16] = 0x373fc47fffffffffffffffffffffffffff;
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// maxExpArray[17] = 0x34f9e8ffffffffffffffffffffffffffff;
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// maxExpArray[18] = 0x32cbfd5fffffffffffffffffffffffffff;
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// maxExpArray[19] = 0x30b5057fffffffffffffffffffffffffff;
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// maxExpArray[20] = 0x2eb40f9fffffffffffffffffffffffffff;
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// maxExpArray[21] = 0x2cc8340fffffffffffffffffffffffffff;
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// maxExpArray[22] = 0x2af09481ffffffffffffffffffffffffff;
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// maxExpArray[23] = 0x292c5bddffffffffffffffffffffffffff;
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// maxExpArray[24] = 0x277abdcdffffffffffffffffffffffffff;
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// maxExpArray[25] = 0x25daf6657fffffffffffffffffffffffff;
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// maxExpArray[26] = 0x244c49c65fffffffffffffffffffffffff;
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// maxExpArray[27] = 0x22ce03cd5fffffffffffffffffffffffff;
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// maxExpArray[28] = 0x215f77c047ffffffffffffffffffffffff;
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// maxExpArray[29] = 0x1fffffffffffffffffffffffffffffffff;
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// maxExpArray[30] = 0x1eaefdbdabffffffffffffffffffffffff;
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// maxExpArray[31] = 0x1d6bd8b2ebffffffffffffffffffffffff;
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maxExpArray[32] = 0x1c35fedd14ffffffffffffffffffffffff;
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maxExpArray[33] = 0x1b0ce43b323fffffffffffffffffffffff;
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maxExpArray[34] = 0x19f0028ec1ffffffffffffffffffffffff;
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maxExpArray[35] = 0x18ded91f0e7fffffffffffffffffffffff;
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maxExpArray[36] = 0x17d8ec7f0417ffffffffffffffffffffff;
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maxExpArray[37] = 0x16ddc6556cdbffffffffffffffffffffff;
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maxExpArray[38] = 0x15ecf52776a1ffffffffffffffffffffff;
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maxExpArray[39] = 0x15060c256cb2ffffffffffffffffffffff;
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maxExpArray[40] = 0x1428a2f98d72ffffffffffffffffffffff;
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maxExpArray[41] = 0x13545598e5c23fffffffffffffffffffff;
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maxExpArray[42] = 0x1288c4161ce1dfffffffffffffffffffff;
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maxExpArray[43] = 0x11c592761c666fffffffffffffffffffff;
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maxExpArray[44] = 0x110a688680a757ffffffffffffffffffff;
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maxExpArray[45] = 0x1056f1b5bedf77ffffffffffffffffffff;
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maxExpArray[46] = 0x0faadceceeff8bffffffffffffffffffff;
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maxExpArray[47] = 0x0f05dc6b27edadffffffffffffffffffff;
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maxExpArray[48] = 0x0e67a5a25da4107fffffffffffffffffff;
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maxExpArray[49] = 0x0dcff115b14eedffffffffffffffffffff;
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maxExpArray[50] = 0x0d3e7a392431239fffffffffffffffffff;
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maxExpArray[51] = 0x0cb2ff529eb71e4fffffffffffffffffff;
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maxExpArray[52] = 0x0c2d415c3db974afffffffffffffffffff;
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maxExpArray[53] = 0x0bad03e7d883f69bffffffffffffffffff;
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maxExpArray[54] = 0x0b320d03b2c343d5ffffffffffffffffff;
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maxExpArray[55] = 0x0abc25204e02828dffffffffffffffffff;
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maxExpArray[56] = 0x0a4b16f74ee4bb207fffffffffffffffff;
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maxExpArray[57] = 0x09deaf736ac1f569ffffffffffffffffff;
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maxExpArray[58] = 0x0976bd9952c7aa957fffffffffffffffff;
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maxExpArray[59] = 0x09131271922eaa606fffffffffffffffff;
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maxExpArray[60] = 0x08b380f3558668c46fffffffffffffffff;
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maxExpArray[61] = 0x0857ddf0117efa215bffffffffffffffff;
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maxExpArray[62] = 0x07ffffffffffffffffffffffffffffffff;
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maxExpArray[63] = 0x07abbf6f6abb9d087fffffffffffffffff;
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maxExpArray[64] = 0x075af62cbac95f7dfa7fffffffffffffff;
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maxExpArray[65] = 0x070d7fb7452e187ac13fffffffffffffff;
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maxExpArray[66] = 0x06c3390ecc8af379295fffffffffffffff;
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maxExpArray[67] = 0x067c00a3b07ffc01fd6fffffffffffffff;
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maxExpArray[68] = 0x0637b647c39cbb9d3d27ffffffffffffff;
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maxExpArray[69] = 0x05f63b1fc104dbd39587ffffffffffffff;
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maxExpArray[70] = 0x05b771955b36e12f7235ffffffffffffff;
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maxExpArray[71] = 0x057b3d49dda84556d6f6ffffffffffffff;
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maxExpArray[72] = 0x054183095b2c8ececf30ffffffffffffff;
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maxExpArray[73] = 0x050a28be635ca2b888f77fffffffffffff;
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maxExpArray[74] = 0x04d5156639708c9db33c3fffffffffffff;
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maxExpArray[75] = 0x04a23105873875bd52dfdfffffffffffff;
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maxExpArray[76] = 0x0471649d87199aa990756fffffffffffff;
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maxExpArray[77] = 0x04429a21a029d4c1457cfbffffffffffff;
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maxExpArray[78] = 0x0415bc6d6fb7dd71af2cb3ffffffffffff;
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maxExpArray[79] = 0x03eab73b3bbfe282243ce1ffffffffffff;
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maxExpArray[80] = 0x03c1771ac9fb6b4c18e229ffffffffffff;
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maxExpArray[81] = 0x0399e96897690418f785257fffffffffff;
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maxExpArray[82] = 0x0373fc456c53bb779bf0ea9fffffffffff;
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maxExpArray[83] = 0x034f9e8e490c48e67e6ab8bfffffffffff;
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maxExpArray[84] = 0x032cbfd4a7adc790560b3337ffffffffff;
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maxExpArray[85] = 0x030b50570f6e5d2acca94613ffffffffff;
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maxExpArray[86] = 0x02eb40f9f620fda6b56c2861ffffffffff;
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maxExpArray[87] = 0x02cc8340ecb0d0f520a6af58ffffffffff;
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maxExpArray[88] = 0x02af09481380a0a35cf1ba02ffffffffff;
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maxExpArray[89] = 0x0292c5bdd3b92ec810287b1b3fffffffff;
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maxExpArray[90] = 0x0277abdcdab07d5a77ac6d6b9fffffffff;
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maxExpArray[91] = 0x025daf6654b1eaa55fd64df5efffffffff;
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maxExpArray[92] = 0x0244c49c648baa98192dce88b7ffffffff;
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maxExpArray[93] = 0x022ce03cd5619a311b2471268bffffffff;
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maxExpArray[94] = 0x0215f77c045fbe885654a44a0fffffffff;
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maxExpArray[95] = 0x01ffffffffffffffffffffffffffffffff;
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maxExpArray[96] = 0x01eaefdbdaaee7421fc4d3ede5ffffffff;
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maxExpArray[97] = 0x01d6bd8b2eb257df7e8ca57b09bfffffff;
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maxExpArray[98] = 0x01c35fedd14b861eb0443f7f133fffffff;
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maxExpArray[99] = 0x01b0ce43b322bcde4a56e8ada5afffffff;
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maxExpArray[100] = 0x019f0028ec1fff007f5a195a39dfffffff;
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maxExpArray[101] = 0x018ded91f0e72ee74f49b15ba527ffffff;
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maxExpArray[102] = 0x017d8ec7f04136f4e5615fd41a63ffffff;
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maxExpArray[103] = 0x016ddc6556cdb84bdc8d12d22e6fffffff;
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maxExpArray[104] = 0x015ecf52776a1155b5bd8395814f7fffff;
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maxExpArray[105] = 0x015060c256cb23b3b3cc3754cf40ffffff;
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maxExpArray[106] = 0x01428a2f98d728ae223ddab715be3fffff;
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maxExpArray[107] = 0x013545598e5c23276ccf0ede68034fffff;
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maxExpArray[108] = 0x01288c4161ce1d6f54b7f61081194fffff;
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maxExpArray[109] = 0x011c592761c666aa641d5a01a40f17ffff;
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maxExpArray[110] = 0x0110a688680a7530515f3e6e6cfdcdffff;
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maxExpArray[111] = 0x01056f1b5bedf75c6bcb2ce8aed428ffff;
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maxExpArray[112] = 0x00faadceceeff8a0890f3875f008277fff;
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maxExpArray[113] = 0x00f05dc6b27edad306388a600f6ba0bfff;
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maxExpArray[114] = 0x00e67a5a25da41063de1495d5b18cdbfff;
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maxExpArray[115] = 0x00dcff115b14eedde6fc3aa5353f2e4fff;
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maxExpArray[116] = 0x00d3e7a3924312399f9aae2e0f868f8fff;
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maxExpArray[117] = 0x00cb2ff529eb71e41582cccd5a1ee26fff;
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maxExpArray[118] = 0x00c2d415c3db974ab32a51840c0b67edff;
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maxExpArray[119] = 0x00bad03e7d883f69ad5b0a186184e06bff;
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maxExpArray[120] = 0x00b320d03b2c343d4829abd6075f0cc5ff;
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maxExpArray[121] = 0x00abc25204e02828d73c6e80bcdb1a95bf;
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maxExpArray[122] = 0x00a4b16f74ee4bb2040a1ec6c15fbbf2df;
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maxExpArray[123] = 0x009deaf736ac1f569deb1b5ae3f36c130f;
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maxExpArray[124] = 0x00976bd9952c7aa957f5937d790ef65037;
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maxExpArray[125] = 0x009131271922eaa6064b73a22d0bd4f2bf;
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maxExpArray[126] = 0x008b380f3558668c46c91c49a2f8e967b9;
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maxExpArray[127] = 0x00857ddf0117efa215952912839f6473e6;
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}
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/**
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General Description:
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Determine a value of precision.
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Calculate an integer approximation of (_baseN / _baseD) ^ (_expN / _expD) * 2 ^ precision.
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Return the result along with the precision used.
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Detailed Description:
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Instead of calculating "base ^ exp", we calculate "e ^ (log(base) * exp)".
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The value of "log(base)" is represented with an integer slightly smaller than "log(base) * 2 ^ precision".
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The larger "precision" is, the more accurately this value represents the real value.
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However, the larger "precision" is, the more bits are required in order to store this value.
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And the exponentiation function, which takes "x" and calculates "e ^ x", is limited to a maximum exponent (maximum value of "x").
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This maximum exponent depends on the "precision" used, and it is given by "maxExpArray[precision] >> (MAX_PRECISION - precision)".
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Hence we need to determine the highest precision which can be used for the given input, before calling the exponentiation function.
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This allows us to compute "base ^ exp" with maximum accuracy and without exceeding 256 bits in any of the intermediate computations.
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This functions assumes that "_expN < 2 ^ 256 / log(MAX_NUM - 1)", otherwise the multiplication should be replaced with a "safeMul".
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*/
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function power(
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uint256 _baseN,
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uint256 _baseD,
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uint32 _expN,
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uint32 _expD) internal view returns (uint256, uint8)
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{
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require(_baseN < MAX_NUM, "SNT available is invalid");
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uint256 baseLog;
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uint256 base = _baseN * FIXED_1 / _baseD;
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if (base < OPT_LOG_MAX_VAL) {
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baseLog = optimalLog(base);
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} else {
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baseLog = generalLog(base);
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}
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uint256 baseLogTimesExp = baseLog * _expN / _expD;
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if (baseLogTimesExp < OPT_EXP_MAX_VAL) {
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return (optimalExp(baseLogTimesExp), MAX_PRECISION);
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} else {
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uint8 precision = findPositionInMaxExpArray(baseLogTimesExp);
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return (generalExp(baseLogTimesExp >> (MAX_PRECISION - precision), precision), precision);
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}
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}
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/**
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Compute log(x / FIXED_1) * FIXED_1.
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This functions assumes that "x >= FIXED_1", because the output would be negative otherwise.
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*/
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function generalLog(uint256 x) internal pure returns (uint256) {
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uint256 res = 0;
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// If x >= 2, then we compute the integer part of log2(x), which is larger than 0.
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if (x >= FIXED_2) {
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uint8 count = floorLog2(x / FIXED_1);
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x >>= count; // now x < 2
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res = count * FIXED_1;
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}
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// If x > 1, then we compute the fraction part of log2(x), which is larger than 0.
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if (x > FIXED_1) {
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for (uint8 i = MAX_PRECISION; i > 0; --i) {
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x = (x * x) / FIXED_1; // now 1 < x < 4
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if (x >= FIXED_2) {
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x >>= 1; // now 1 < x < 2
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res += ONE << (i - 1);
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}
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}
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}
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return res * LN2_NUMERATOR / LN2_DENOMINATOR;
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}
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/**
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Compute the largest integer smaller than or equal to the binary logarithm of the input.
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*/
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function floorLog2(uint256 _n) internal pure returns (uint8) {
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uint8 res = 0;
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if (_n < 256) {
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// At most 8 iterations
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while (_n > 1) {
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_n >>= 1;
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res += 1;
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}
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} else {
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// Exactly 8 iterations
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for (uint8 s = 128; s > 0; s >>= 1) {
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if (_n >= (ONE << s)) {
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_n >>= s;
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res |= s;
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}
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}
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}
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return res;
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}
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/**
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The global "maxExpArray" is sorted in descending order, and therefore the following statements are equivalent:
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- This function finds the position of [the smallest value in "maxExpArray" larger than or equal to "x"]
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- This function finds the highest position of [a value in "maxExpArray" larger than or equal to "x"]
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*/
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function findPositionInMaxExpArray(uint256 _x) internal view returns (uint8) {
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uint8 lo = MIN_PRECISION;
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uint8 hi = MAX_PRECISION;
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while (lo + 1 < hi) {
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uint8 mid = (lo + hi) / 2;
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if (maxExpArray[mid] >= _x)
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lo = mid;
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else
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hi = mid;
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}
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if (maxExpArray[hi] >= _x)
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return hi;
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if (maxExpArray[lo] >= _x)
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return lo;
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require(false, "Could not find a suitable position");
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return 0;
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}
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/**
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This function can be auto-generated by the script 'PrintFunctionGeneralExp.py'.
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It approximates "e ^ x" via maclaurin summation: "(x^0)/0! + (x^1)/1! + ... + (x^n)/n!".
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It returns "e ^ (x / 2 ^ precision) * 2 ^ precision", that is, the result is upshifted for accuracy.
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The global "maxExpArray" maps each "precision" to "((maximumExponent + 1) << (MAX_PRECISION - precision)) - 1".
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The maximum permitted value for "x" is therefore given by "maxExpArray[precision] >> (MAX_PRECISION - precision)".
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*/
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function generalExp(uint256 _x, uint8 _precision) internal pure returns (uint256) {
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uint256 xi = _x;
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uint256 res = 0;
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xi = (xi * _x) >> _precision;
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res += xi * 0x3442c4e6074a82f1797f72ac0000000; // add x^02 * (33! / 02!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x116b96f757c380fb287fd0e40000000; // add x^03 * (33! / 03!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x045ae5bdd5f0e03eca1ff4390000000; // add x^04 * (33! / 04!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x00defabf91302cd95b9ffda50000000; // add x^05 * (33! / 05!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x002529ca9832b22439efff9b8000000; // add x^06 * (33! / 06!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x00054f1cf12bd04e516b6da88000000; // add x^07 * (33! / 07!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x0000a9e39e257a09ca2d6db51000000; // add x^08 * (33! / 08!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x000012e066e7b839fa050c309000000; // add x^09 * (33! / 09!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x000001e33d7d926c329a1ad1a800000; // add x^10 * (33! / 10!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x0000002bee513bdb4a6b19b5f800000; // add x^11 * (33! / 11!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x00000003a9316fa79b88eccf2a00000; // add x^12 * (33! / 12!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x0000000048177ebe1fa812375200000; // add x^13 * (33! / 13!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x0000000005263fe90242dcbacf00000; // add x^14 * (33! / 14!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x000000000057e22099c030d94100000; // add x^15 * (33! / 15!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x0000000000057e22099c030d9410000; // add x^16 * (33! / 16!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x00000000000052b6b54569976310000; // add x^17 * (33! / 17!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x00000000000004985f67696bf748000; // add x^18 * (33! / 18!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x000000000000003dea12ea99e498000; // add x^19 * (33! / 19!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x00000000000000031880f2214b6e000; // add x^20 * (33! / 20!)
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xi = (xi * _x) >> _precision;
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res += xi * 0x000000000000000025bcff56eb36000; // add x^21 * (33! / 21!)
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xi = (xi * _x) >> _precision;
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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'
|
|
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;
|
|
}
|
|
|
|
/**
|
|
Return 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;
|
|
}
|
|
} |