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neuroswish/Power.sol

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Fractional Exponentiation in Solidity based on Bancor formula
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Power.sol
// SPDX-License-Identifier: GPL-3.0-or-later
pragma solidity ^0.8.4;
/**
* @title Power
* @author neuroswish
* @notice Functions to calculate fractional exponents
*/
contract Power {
uint256 private constant ONE = 1;
uint32 private constant MAX_WEIGHT = 1000000;
uint8 private constant MIN_PRECISION = 32;
uint8 private constant MAX_PRECISION = 127;
uint256 private constant FIXED_1 = 0x080000000000000000000000000000000;
uint256 private constant FIXED_2 = 0x100000000000000000000000000000000;
uint256 private constant MAX_NUM = 0x200000000000000000000000000000000;
uint256 private constant LN2_NUMERATOR = 0x3f80fe03f80fe03f80fe03f80fe03f8;
uint256 private constant LN2_DENOMINATOR =
0x5b9de1d10bf4103d647b0955897ba80;
uint256 private constant OPT_LOG_MAX_VAL =
0x15bf0a8b1457695355fb8ac404e7a79e3;
uint256 private constant OPT_EXP_MAX_VAL =
0x800000000000000000000000000000000;
uint256[128] private maxExpArray;
function initMaxExpArray() public {
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;
}
constructor() {
initMaxExpArray();
}
/**
* @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 result)
{
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) {
result = hi;
return result;
}
if (maxExpArray[lo] >= _x) {
result = lo;
return result;
}
require(false);
}
/**
* @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 <= OPT_LOG_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;
}
}
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