// SPDX-License-Identifier: MIT
pragma solidity >=0.8.0;
/// @notice Signed 18 decimal fixed point (wad) arithmetic library.
/// @author Solmate (https://github.com/transmissions11/solmate/blob/main/src/utils/SignedWadMath.sol)
/// @author Modified from Remco Bloemen (https://xn--2-umb.com/22/exp-ln/index.html)
/// @dev Will not revert on overflow, only use where overflow is not possible.
function toWadUnsafe(uint256 x) pure returns (int256 r) {
/// @solidity memory-safe-assembly
assembly {
// Multiply x by 1e18.
r := mul(x, 1000000000000000000)
}
}
/// @dev Takes an integer amount of seconds and converts it to a wad amount of days.
/// @dev Will not revert on overflow, only use where overflow is not possible.
/// @dev Not meant for negative second amounts, it assumes x is positive.
function toDaysWadUnsafe(uint256 x) pure returns (int256 r) {
/// @solidity memory-safe-assembly
assembly {
// Multiply x by 1e18 and then divide it by 86400.
r := div(mul(x, 1000000000000000000), 86400)
}
}
/// @dev Takes a wad amount of days and converts it to an integer amount of seconds.
/// @dev Will not revert on overflow, only use where overflow is not possible.
/// @dev Not meant for negative day amounts, it assumes x is positive.
function fromDaysWadUnsafe(int256 x) pure returns (uint256 r) {
/// @solidity memory-safe-assembly
assembly {
// Multiply x by 86400 and then divide it by 1e18.
r := div(mul(x, 86400), 1000000000000000000)
}
}
/// @dev Will not revert on overflow, only use where overflow is not possible.
function unsafeWadMul(int256 x, int256 y) pure returns (int256 r) {
/// @solidity memory-safe-assembly
assembly {
// Multiply x by y and divide by 1e18.
r := sdiv(mul(x, y), 1000000000000000000)
}
}
/// @dev Will return 0 instead of reverting if y is zero and will
/// not revert on overflow, only use where overflow is not possible.
function unsafeWadDiv(int256 x, int256 y) pure returns (int256 r) {
/// @solidity memory-safe-assembly
assembly {
// Multiply x by 1e18 and divide it by y.
r := sdiv(mul(x, 1000000000000000000), y)
}
}
function wadMul(int256 x, int256 y) pure returns (int256 r) {
/// @solidity memory-safe-assembly
assembly {
// Store x * y in r for now.
r := mul(x, y)
// Equivalent to require(x == 0 || (x * y) / x == y)
if iszero(or(iszero(x), eq(sdiv(r, x), y))) {
revert(0, 0)
}
// Scale the result down by 1e18.
r := sdiv(r, 1000000000000000000)
}
}
function wadDiv(int256 x, int256 y) pure returns (int256 r) {
/// @solidity memory-safe-assembly
assembly {
// Store x * 1e18 in r for now.
r := mul(x, 1000000000000000000)
// Equivalent to require(y != 0 && ((x * 1e18) / 1e18 == x))
if iszero(and(iszero(iszero(y)), eq(sdiv(r, 1000000000000000000), x))) {
revert(0, 0)
}
// Divide r by y.
r := sdiv(r, y)
}
}
/// @dev Will not work with negative bases, only use when x is positive.
function wadPow(int256 x, int256 y) pure returns (int256) {
// Equivalent to x to the power of y because x ** y = (e ** ln(x)) ** y = e ** (ln(x) * y)
return wadExp((wadLn(x) * y) / 1e18); // Using ln(x) means x must be greater than 0.
}
function wadExp(int256 x) pure returns (int256 r) {
unchecked {
// When the result is < 0.5 we return zero. This happens when
// x <= floor(log(0.5e18) * 1e18) ~ -42e18
if (x <= -42139678854452767551) return 0;
// When the result is > (2**255 - 1) / 1e18 we can not represent it as an
// int. This happens when x >= floor(log((2**255 - 1) / 1e18) * 1e18) ~ 135.
if (x >= 135305999368893231589) revert("EXP_OVERFLOW");
// x is now in the range (-42, 136) * 1e18. Convert to (-42, 136) * 2**96
// for more intermediate precision and a binary basis. This base conversion
// is a multiplication by 1e18 / 2**96 = 5**18 / 2**78.
x = (x << 78) / 5**18;
// Reduce range of x to (-½ ln 2, ½ ln 2) * 2**96 by factoring out powers
// of two such that exp(x) = exp(x') * 2**k, where k is an integer.
// Solving this gives k = round(x / log(2)) and x' = x - k * log(2).
int256 k = ((x << 96) / 54916777467707473351141471128 + 2**95) >> 96;
x = x - k * 54916777467707473351141471128;
// k is in the range [-61, 195].
// Evaluate using a (6, 7)-term rational approximation.
// p is made monic, we'll multiply by a scale factor later.
int256 y = x + 1346386616545796478920950773328;
y = ((y * x) >> 96) + 57155421227552351082224309758442;
int256 p = y + x - 94201549194550492254356042504812;
p = ((p * y) >> 96) + 28719021644029726153956944680412240;
p = p * x + (4385272521454847904659076985693276 << 96);
// We leave p in 2**192 basis so we don't need to scale it back up for the division.
int256 q = x - 2855989394907223263936484059900;
q = ((q * x) >> 96) + 50020603652535783019961831881945;
q = ((q * x) >> 96) - 533845033583426703283633433725380;
q = ((q * x) >> 96) + 3604857256930695427073651918091429;
q = ((q * x) >> 96) - 14423608567350463180887372962807573;
q = ((q * x) >> 96) + 26449188498355588339934803723976023;
/// @solidity memory-safe-assembly
assembly {
// Div in assembly because solidity adds a zero check despite the unchecked.
// The q polynomial won't have zeros in the domain as all its roots are complex.
// No scaling is necessary because p is already 2**96 too large.
r := sdiv(p, q)
}
// r should be in the range (0.09, 0.25) * 2**96.
// We now need to multiply r by:
// * the scale factor s = ~6.031367120.
// * the 2**k factor from the range reduction.
// * the 1e18 / 2**96 factor for base conversion.
// We do this all at once, with an intermediate result in 2**213
// basis, so the final right shift is always by a positive amount.
r = int256((uint256(r) * 3822833074963236453042738258902158003155416615667) >> uint256(195 - k));
}
}
function wadLn(int256 x) pure returns (int256 r) {
unchecked {
require(x > 0, "UNDEFINED");
// We want to convert x from 10**18 fixed point to 2**96 fixed point.
// We do this by multiplying by 2**96 / 10**18. But since
// ln(x * C) = ln(x) + ln(C), we can simply do nothing here
// and add ln(2**96 / 10**18) at the end.
/// @solidity memory-safe-assembly
assembly {
r := shl(7, lt(0xffffffffffffffffffffffffffffffff, x))
r := or(r, shl(6, lt(0xffffffffffffffff, shr(r, x))))
r := or(r, shl(5, lt(0xffffffff, shr(r, x))))
r := or(r, shl(4, lt(0xffff, shr(r, x))))
r := or(r, shl(3, lt(0xff, shr(r, x))))
r := or(r, shl(2, lt(0xf, shr(r, x))))
r := or(r, shl(1, lt(0x3, shr(r, x))))
r := or(r, lt(0x1, shr(r, x)))
}
// Reduce range of x to (1, 2) * 2**96
// ln(2^k * x) = k * ln(2) + ln(x)
int256 k = r - 96;
x <<= uint256(159 - k);
x = int256(uint256(x) >> 159);
// Evaluate using a (8, 8)-term rational approximation.
// p is made monic, we will multiply by a scale factor later.
int256 p = x + 3273285459638523848632254066296;
p = ((p * x) >> 96) + 24828157081833163892658089445524;
p = ((p * x) >> 96) + 43456485725739037958740375743393;
p = ((p * x) >> 96) - 11111509109440967052023855526967;
p = ((p * x) >> 96) - 45023709667254063763336534515857;
p = ((p * x) >> 96) - 14706773417378608786704636184526;
p = p * x - (795164235651350426258249787498 << 96);
// We leave p in 2**192 basis so we don't need to scale it back up for the division.
// q is monic by convention.
int256 q = x + 5573035233440673466300451813936;
q = ((q * x) >> 96) + 71694874799317883764090561454958;
q = ((q * x) >> 96) + 283447036172924575727196451306956;
q = ((q * x) >> 96) + 401686690394027663651624208769553;
q = ((q * x) >> 96) + 204048457590392012362485061816622;
q = ((q * x) >> 96) + 31853899698501571402653359427138;
q = ((q * x) >> 96) + 909429971244387300277376558375;
/// @solidity memory-safe-assembly
assembly {
// Div in assembly because solidity adds a zero check despite the unchecked.
// The q polynomial is known not to have zeros in the domain.
// No scaling required because p is already 2**96 too large.
r := sdiv(p, q)
}
// r is in the range (0, 0.125) * 2**96
// Finalization, we need to:
// * multiply by the scale factor s = 5.549…
// * add ln(2**96 / 10**18)
// * add k * ln(2)
// * multiply by 10**18 / 2**96 = 5**18 >> 78
// mul s * 5e18 * 2**96, base is now 5**18 * 2**192
r *= 1677202110996718588342820967067443963516166;
// add ln(2) * k * 5e18 * 2**192
r += 16597577552685614221487285958193947469193820559219878177908093499208371 * k;
// add ln(2**96 / 10**18) * 5e18 * 2**192
r += 600920179829731861736702779321621459595472258049074101567377883020018308;
// base conversion: mul 2**18 / 2**192
r >>= 174;
}
}
/// @dev Will return 0 instead of reverting if y is zero.
function unsafeDiv(int256 x, int256 y) pure returns (int256 r) {
/// @solidity memory-safe-assembly
assembly {
// Divide x by y.
r := sdiv(x, y)
}
}