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feat: continuous compounding #254

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merged 6 commits into from
Aug 11, 2023
Merged

feat: continuous compounding #254

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acf23d1
feat: continuous compounding
pakim249CAL Aug 9, 2023
459d1ad
chore: Format
pakim249CAL Aug 9, 2023
7b896b6
test: cut down wadmath
pakim249CAL Aug 9, 2023
bce0fec
test: use bound
pakim249CAL Aug 9, 2023
43927fe
Merge remote-tracking branch 'origin/main' into refactor/compound
pakim249CAL Aug 9, 2023
bb78b2a
Merge remote-tracking branch 'origin/main' into refactor/compound
pakim249CAL Aug 10, 2023
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17 changes: 3 additions & 14 deletions src/libraries/FixedPointMathLib.sol
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Original file line number Diff line number Diff line change
Expand Up @@ -30,11 +30,11 @@ library FixedPointMathLib {
}

/// @dev The sum of the last three terms in a four term taylor series expansion
/// to approximate a compound interest rate: (1 + x)^n - 1.
/// to approximate a continuous compound interest rate: e^(nx) - 1.
function wTaylorCompounded(uint256 x, uint256 n) internal pure returns (uint256) {
uint256 firstTerm = x * n;
uint256 secondTerm = mulWadDown(firstTerm, x * zeroFloorSub(n, 1)) / 2;
uint256 thirdTerm = mulWadDown(secondTerm, x * zeroFloorSub(n, 2)) / 3;
uint256 secondTerm = mulWadDown(firstTerm, firstTerm) / 2;
uint256 thirdTerm = mulWadDown(secondTerm, firstTerm) / 3;

return firstTerm + secondTerm + thirdTerm;
}
Expand Down Expand Up @@ -65,15 +65,4 @@ library FixedPointMathLib {
z := add(gt(mod(mul(x, y), denominator), 0), div(mul(x, y), denominator))
}
}

/*//////////////////////////////////////////////////////////////
INTEGER OPERATIONS
//////////////////////////////////////////////////////////////*/

/// @dev Returns max(x - y, 0).
function zeroFloorSub(uint256 x, uint256 y) internal pure returns (uint256 z) {
assembly {
z := mul(gt(x, y), sub(x, y))
}
}
}
15 changes: 4 additions & 11 deletions test/forge/Math.t.sol
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Expand Up @@ -4,28 +4,21 @@ pragma solidity ^0.8.0;
import "forge-std/Test.sol";

import "src/libraries/FixedPointMathLib.sol";
import "test/forge/helpers/WadMath.sol";

contract MathTest is Test {
using FixedPointMathLib for uint256;

function testWTaylorCompounded(uint256 rate, uint256 timeElapsed) public {
// Assume rate is less than a ~500% APY. (~180% APR)
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vm.assume(rate < (FixedPointMathLib.WAD / 20_000_000) && timeElapsed < 365 days);
rate = bound(rate, 0, FixedPointMathLib.WAD / 20_000_000);
timeElapsed = bound(timeElapsed, 0, 365 days);
uint256 result = rate.wTaylorCompounded(timeElapsed) + FixedPointMathLib.WAD;
uint256 toCompare = wPow(FixedPointMathLib.WAD + rate, timeElapsed);
uint256 toCompare = WadMath.wadExpUp(rate * timeElapsed);
assertLe(result, toCompare, "rate should be less than the compounded rate");
assertGe(
result, FixedPointMathLib.WAD + timeElapsed * rate, "rate should be greater than the simple interest rate"
);
assertLe((toCompare - result) * 100_00 / toCompare, 8_00, "The error should be less than or equal to 8%");
}

// Exponentiation by squaring with rounding up.
function wPow(uint256 x, uint256 n) private pure returns (uint256 z) {
z = FixedPointMathLib.WAD;
for (; n != 0; n /= 2) {
z = n % 2 != 0 ? z.mulWadUp(x) : z;
x = x.mulWadUp(x);
}
}
}
166 changes: 166 additions & 0 deletions test/forge/helpers/WadMath.sol
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@@ -0,0 +1,166 @@
// SPDX-License-Identifier: GPL-3.0-or-later
pragma solidity ^0.8.0;

/**
* @title Fixed point arithmetic library
* @author Alberto Cuesta Cañada, Jacob Eliosoff, Alex Roan
* @notice This library is imported for testing purposes only. It is not used in production.
* Code is cut from https://raw.githubusercontent.com/usmfum/USM/master/contracts/WadMath.sol
*/
library WadMath {
uint256 public constant WAD = 1e18;
uint256 public constant FLOOR_LOG_2_WAD_SCALED = 158961593653514369813532673448321674075; // log2(1e18) * 2**121
uint256 public constant CEIL_LOG_2_E_SCALED_OVER_WAD = 3835341275459348170; // log2(e) * 2**121 / 1e18

function wadDivUp(uint256 x, uint256 y) internal pure returns (uint256 z) {
z = x * WAD + y; // 101 (1.01) / 1000 (10) -> (101 * 100 + 1000 - 1) / 1000 -> 11 (0.11 = 0.101 rounded up).
unchecked {
z -= 1;
} // Can do unchecked subtraction since division in next line will catch y = 0 case anyway
z /= y;
}

function wadExpUp(uint256 y) internal pure returns (uint256 z) {
uint256 exponent = FLOOR_LOG_2_WAD_SCALED - CEIL_LOG_2_E_SCALED_OVER_WAD * y;
require(exponent <= type(uint128).max, "exponent overflow");
uint256 wadOneOverExpY = pow_2(uint128(exponent));
z = wadDivUp(WAD, wadOneOverExpY);
}

/**
* Calculate 2 raised into given power.
*
* @param x power to raise 2 into, multiplied by 2^121
* @return z 2 raised into given power
*/
function pow_2(uint128 x) internal pure returns (uint128 z) {
unchecked {
uint256 r = 0x80000000000000000000000000000000;
if (x & 0x1000000000000000000000000000000 > 0) r = r * 0xb504f333f9de6484597d89b3754abe9f >> 127;
if (x & 0x800000000000000000000000000000 > 0) r = r * 0x9837f0518db8a96f46ad23182e42f6f6 >> 127;
if (x & 0x400000000000000000000000000000 > 0) r = r * 0x8b95c1e3ea8bd6e6fbe4628758a53c90 >> 127;
if (x & 0x200000000000000000000000000000 > 0) r = r * 0x85aac367cc487b14c5c95b8c2154c1b2 >> 127;
if (x & 0x100000000000000000000000000000 > 0) r = r * 0x82cd8698ac2ba1d73e2a475b46520bff >> 127;
if (x & 0x80000000000000000000000000000 > 0) r = r * 0x8164d1f3bc0307737be56527bd14def4 >> 127;
if (x & 0x40000000000000000000000000000 > 0) r = r * 0x80b1ed4fd999ab6c25335719b6e6fd20 >> 127;
if (x & 0x20000000000000000000000000000 > 0) r = r * 0x8058d7d2d5e5f6b094d589f608ee4aa2 >> 127;
if (x & 0x10000000000000000000000000000 > 0) r = r * 0x802c6436d0e04f50ff8ce94a6797b3ce >> 127;
if (x & 0x8000000000000000000000000000 > 0) r = r * 0x8016302f174676283690dfe44d11d008 >> 127;
if (x & 0x4000000000000000000000000000 > 0) r = r * 0x800b179c82028fd0945e54e2ae18f2f0 >> 127;
if (x & 0x2000000000000000000000000000 > 0) r = r * 0x80058baf7fee3b5d1c718b38e549cb93 >> 127;
if (x & 0x1000000000000000000000000000 > 0) r = r * 0x8002c5d00fdcfcb6b6566a58c048be1f >> 127;
if (x & 0x800000000000000000000000000 > 0) r = r * 0x800162e61bed4a48e84c2e1a463473d9 >> 127;
if (x & 0x400000000000000000000000000 > 0) r = r * 0x8000b17292f702a3aa22beacca949013 >> 127;
if (x & 0x200000000000000000000000000 > 0) r = r * 0x800058b92abbae02030c5fa5256f41fe >> 127;
if (x & 0x100000000000000000000000000 > 0) r = r * 0x80002c5c8dade4d71776c0f4dbea67d6 >> 127;
if (x & 0x80000000000000000000000000 > 0) r = r * 0x8000162e44eaf636526be456600bdbe4 >> 127;
if (x & 0x40000000000000000000000000 > 0) r = r * 0x80000b1721fa7c188307016c1cd4e8b6 >> 127;
if (x & 0x20000000000000000000000000 > 0) r = r * 0x8000058b90de7e4cecfc487503488bb1 >> 127;
if (x & 0x10000000000000000000000000 > 0) r = r * 0x800002c5c8678f36cbfce50a6de60b14 >> 127;
if (x & 0x8000000000000000000000000 > 0) r = r * 0x80000162e431db9f80b2347b5d62e516 >> 127;
if (x & 0x4000000000000000000000000 > 0) r = r * 0x800000b1721872d0c7b08cf1e0114152 >> 127;
if (x & 0x2000000000000000000000000 > 0) r = r * 0x80000058b90c1aa8a5c3736cb77e8dff >> 127;
if (x & 0x1000000000000000000000000 > 0) r = r * 0x8000002c5c8605a4635f2efc2362d978 >> 127;
if (x & 0x800000000000000000000000 > 0) r = r * 0x800000162e4300e635cf4a109e3939bd >> 127;
if (x & 0x400000000000000000000000 > 0) r = r * 0x8000000b17217ff81bef9c551590cf83 >> 127;
if (x & 0x200000000000000000000000 > 0) r = r * 0x800000058b90bfdd4e39cd52c0cfa27c >> 127;
if (x & 0x100000000000000000000000 > 0) r = r * 0x80000002c5c85fe6f72d669e0e76e411 >> 127;
if (x & 0x80000000000000000000000 > 0) r = r * 0x8000000162e42ff18f9ad35186d0df28 >> 127;
if (x & 0x40000000000000000000000 > 0) r = r * 0x80000000b17217f84cce71aa0dcfffe7 >> 127;
if (x & 0x20000000000000000000000 > 0) r = r * 0x8000000058b90bfc07a77ad56ed22aaa >> 127;
if (x & 0x10000000000000000000000 > 0) r = r * 0x800000002c5c85fdfc23cdead40da8d6 >> 127;
if (x & 0x8000000000000000000000 > 0) r = r * 0x80000000162e42fefc25eb1571853a66 >> 127;
if (x & 0x4000000000000000000000 > 0) r = r * 0x800000000b17217f7d97f692baacded5 >> 127;
if (x & 0x2000000000000000000000 > 0) r = r * 0x80000000058b90bfbead3b8b5dd254d7 >> 127;
if (x & 0x1000000000000000000000 > 0) r = r * 0x8000000002c5c85fdf4eedd62f084e67 >> 127;
if (x & 0x800000000000000000000 > 0) r = r * 0x800000000162e42fefa58aef378bf586 >> 127;
if (x & 0x400000000000000000000 > 0) r = r * 0x8000000000b17217f7d24a78a3c7ef02 >> 127;
if (x & 0x200000000000000000000 > 0) r = r * 0x800000000058b90bfbe9067c93e474a6 >> 127;
if (x & 0x100000000000000000000 > 0) r = r * 0x80000000002c5c85fdf47b8e5a72599f >> 127;
if (x & 0x80000000000000000000 > 0) r = r * 0x8000000000162e42fefa3bdb315934a2 >> 127;
if (x & 0x40000000000000000000 > 0) r = r * 0x80000000000b17217f7d1d7299b49c46 >> 127;
if (x & 0x20000000000000000000 > 0) r = r * 0x8000000000058b90bfbe8e9a8d1c4ea0 >> 127;
if (x & 0x10000000000000000000 > 0) r = r * 0x800000000002c5c85fdf4745969ea76f >> 127;
if (x & 0x8000000000000000000 > 0) r = r * 0x80000000000162e42fefa3a0df5373bf >> 127;
if (x & 0x4000000000000000000 > 0) r = r * 0x800000000000b17217f7d1cff4aac1e1 >> 127;
if (x & 0x2000000000000000000 > 0) r = r * 0x80000000000058b90bfbe8e7db95a2f1 >> 127;
if (x & 0x1000000000000000000 > 0) r = r * 0x8000000000002c5c85fdf473e61ae1f8 >> 127;
if (x & 0x800000000000000000 > 0) r = r * 0x800000000000162e42fefa39f121751c >> 127;
if (x & 0x400000000000000000 > 0) r = r * 0x8000000000000b17217f7d1cf815bb96 >> 127;
if (x & 0x200000000000000000 > 0) r = r * 0x800000000000058b90bfbe8e7bec1e0d >> 127;
if (x & 0x100000000000000000 > 0) r = r * 0x80000000000002c5c85fdf473dee5f17 >> 127;
if (x & 0x80000000000000000 > 0) r = r * 0x8000000000000162e42fefa39ef5438f >> 127;
if (x & 0x40000000000000000 > 0) r = r * 0x80000000000000b17217f7d1cf7a26c8 >> 127;
if (x & 0x20000000000000000 > 0) r = r * 0x8000000000000058b90bfbe8e7bcf4a4 >> 127;
if (x & 0x10000000000000000 > 0) r = r * 0x800000000000002c5c85fdf473de72a2 >> 127;
if (x & 0x8000000000000000 > 0) r = r * 0x80000000000000162e42fefa39ef3765 >> 127;
if (x & 0x4000000000000000 > 0) r = r * 0x800000000000000b17217f7d1cf79b37 >> 127;
if (x & 0x2000000000000000 > 0) r = r * 0x80000000000000058b90bfbe8e7bcd7d >> 127;
if (x & 0x1000000000000000 > 0) r = r * 0x8000000000000002c5c85fdf473de6b6 >> 127;
if (x & 0x800000000000000 > 0) r = r * 0x800000000000000162e42fefa39ef359 >> 127;
if (x & 0x400000000000000 > 0) r = r * 0x8000000000000000b17217f7d1cf79ac >> 127;
if (x & 0x200000000000000 > 0) r = r * 0x800000000000000058b90bfbe8e7bcd6 >> 127;
if (x & 0x100000000000000 > 0) r = r * 0x80000000000000002c5c85fdf473de6a >> 127;
if (x & 0x80000000000000 > 0) r = r * 0x8000000000000000162e42fefa39ef35 >> 127;
if (x & 0x40000000000000 > 0) r = r * 0x80000000000000000b17217f7d1cf79a >> 127;
if (x & 0x20000000000000 > 0) r = r * 0x8000000000000000058b90bfbe8e7bcd >> 127;
if (x & 0x10000000000000 > 0) r = r * 0x800000000000000002c5c85fdf473de6 >> 127;
if (x & 0x8000000000000 > 0) r = r * 0x80000000000000000162e42fefa39ef3 >> 127;
if (x & 0x4000000000000 > 0) r = r * 0x800000000000000000b17217f7d1cf79 >> 127;
if (x & 0x2000000000000 > 0) r = r * 0x80000000000000000058b90bfbe8e7bc >> 127;
if (x & 0x1000000000000 > 0) r = r * 0x8000000000000000002c5c85fdf473de >> 127;
if (x & 0x800000000000 > 0) r = r * 0x800000000000000000162e42fefa39ef >> 127;
if (x & 0x400000000000 > 0) r = r * 0x8000000000000000000b17217f7d1cf7 >> 127;
if (x & 0x200000000000 > 0) r = r * 0x800000000000000000058b90bfbe8e7b >> 127;
if (x & 0x100000000000 > 0) r = r * 0x80000000000000000002c5c85fdf473d >> 127;
if (x & 0x80000000000 > 0) r = r * 0x8000000000000000000162e42fefa39e >> 127;
if (x & 0x40000000000 > 0) r = r * 0x80000000000000000000b17217f7d1cf >> 127;
if (x & 0x20000000000 > 0) r = r * 0x8000000000000000000058b90bfbe8e7 >> 127;
if (x & 0x10000000000 > 0) r = r * 0x800000000000000000002c5c85fdf473 >> 127;
if (x & 0x8000000000 > 0) r = r * 0x80000000000000000000162e42fefa39 >> 127;
if (x & 0x4000000000 > 0) r = r * 0x800000000000000000000b17217f7d1c >> 127;
if (x & 0x2000000000 > 0) r = r * 0x80000000000000000000058b90bfbe8e >> 127;
if (x & 0x1000000000 > 0) r = r * 0x8000000000000000000002c5c85fdf47 >> 127;
if (x & 0x800000000 > 0) r = r * 0x800000000000000000000162e42fefa3 >> 127;
if (x & 0x400000000 > 0) r = r * 0x8000000000000000000000b17217f7d1 >> 127;
if (x & 0x200000000 > 0) r = r * 0x800000000000000000000058b90bfbe8 >> 127;
if (x & 0x100000000 > 0) r = r * 0x80000000000000000000002c5c85fdf4 >> 127;
if (x & 0x80000000 > 0) r = r * 0x8000000000000000000000162e42fefa >> 127;
if (x & 0x40000000 > 0) r = r * 0x80000000000000000000000b17217f7d >> 127;
if (x & 0x20000000 > 0) r = r * 0x8000000000000000000000058b90bfbe >> 127;
if (x & 0x10000000 > 0) r = r * 0x800000000000000000000002c5c85fdf >> 127;
if (x & 0x8000000 > 0) r = r * 0x80000000000000000000000162e42fef >> 127;
if (x & 0x4000000 > 0) r = r * 0x800000000000000000000000b17217f7 >> 127;
if (x & 0x2000000 > 0) r = r * 0x80000000000000000000000058b90bfb >> 127;
if (x & 0x1000000 > 0) r = r * 0x8000000000000000000000002c5c85fd >> 127;
if (x & 0x800000 > 0) r = r * 0x800000000000000000000000162e42fe >> 127;
if (x & 0x400000 > 0) r = r * 0x8000000000000000000000000b17217f >> 127;
if (x & 0x200000 > 0) r = r * 0x800000000000000000000000058b90bf >> 127;
if (x & 0x100000 > 0) r = r * 0x80000000000000000000000002c5c85f >> 127;
if (x & 0x80000 > 0) r = r * 0x8000000000000000000000000162e42f >> 127;
if (x & 0x40000 > 0) r = r * 0x80000000000000000000000000b17217 >> 127;
if (x & 0x20000 > 0) r = r * 0x8000000000000000000000000058b90b >> 127;
if (x & 0x10000 > 0) r = r * 0x800000000000000000000000002c5c85 >> 127;
if (x & 0x8000 > 0) r = r * 0x80000000000000000000000000162e42 >> 127;
if (x & 0x4000 > 0) r = r * 0x800000000000000000000000000b1721 >> 127;
if (x & 0x2000 > 0) r = r * 0x80000000000000000000000000058b90 >> 127;
if (x & 0x1000 > 0) r = r * 0x8000000000000000000000000002c5c8 >> 127;
if (x & 0x800 > 0) r = r * 0x800000000000000000000000000162e4 >> 127;
if (x & 0x400 > 0) r = r * 0x8000000000000000000000000000b172 >> 127;
if (x & 0x200 > 0) r = r * 0x800000000000000000000000000058b9 >> 127;
if (x & 0x100 > 0) r = r * 0x80000000000000000000000000002c5c >> 127;
if (x & 0x80 > 0) r = r * 0x8000000000000000000000000000162e >> 127;
if (x & 0x40 > 0) r = r * 0x80000000000000000000000000000b17 >> 127;
if (x & 0x20 > 0) r = r * 0x8000000000000000000000000000058b >> 127;
if (x & 0x10 > 0) r = r * 0x800000000000000000000000000002c5 >> 127;
if (x & 0x8 > 0) r = r * 0x80000000000000000000000000000162 >> 127;
if (x & 0x4 > 0) r = r * 0x800000000000000000000000000000b1 >> 127;
if (x & 0x2 > 0) r = r * 0x80000000000000000000000000000058 >> 127;
if (x & 0x1 > 0) r = r * 0x8000000000000000000000000000002c >> 127;

r >>= 127 - (x >> 121);

z = uint128(r);
}
}
}