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FixedPointMathLib.sol
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FixedPointMathLib.sol
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// SPDX-License-Identifier: MIT
pragma solidity ^0.8.4;
/// @notice Arithmetic library with operations for fixed-point numbers.
/// @author Solady (https://github.com/vectorized/solady/blob/main/src/utils/FixedPointMathLib.sol)
/// @author Modified from Solmate (https://github.com/transmissions11/solmate/blob/main/src/utils/FixedPointMathLib.sol)
library FixedPointMathLib {
/*´:°•.°+.*•´.*:˚.°*.˚•´.°:°•.°•.*•´.*:˚.°*.˚•´.°:°•.°+.*•´.*:*/
/* CUSTOM ERRORS */
/*.•°:°.´+˚.*°.˚:*.´•*.+°.•°:´*.´•*.•°.•°:°.´:•˚°.*°.˚:*.´+°.•*/
/// @dev The operation failed, as the output exceeds the maximum value of uint256.
error ExpOverflow();
/// @dev The operation failed, as the output exceeds the maximum value of uint256.
error FactorialOverflow();
/// @dev The operation failed, due to an overflow.
error RPowOverflow();
/// @dev The mantissa is too big to fit.
error MantissaOverflow();
/// @dev The operation failed, due to an multiplication overflow.
error MulWadFailed();
/// @dev The operation failed, due to an multiplication overflow.
error SMulWadFailed();
/// @dev The operation failed, either due to a multiplication overflow, or a division by a zero.
error DivWadFailed();
/// @dev The operation failed, either due to a multiplication overflow, or a division by a zero.
error SDivWadFailed();
/// @dev The operation failed, either due to a multiplication overflow, or a division by a zero.
error MulDivFailed();
/// @dev The division failed, as the denominator is zero.
error DivFailed();
/// @dev The full precision multiply-divide operation failed, either due
/// to the result being larger than 256 bits, or a division by a zero.
error FullMulDivFailed();
/// @dev The output is undefined, as the input is less-than-or-equal to zero.
error LnWadUndefined();
/// @dev The input outside the acceptable domain.
error OutOfDomain();
/*´:°•.°+.*•´.*:˚.°*.˚•´.°:°•.°•.*•´.*:˚.°*.˚•´.°:°•.°+.*•´.*:*/
/* CONSTANTS */
/*.•°:°.´+˚.*°.˚:*.´•*.+°.•°:´*.´•*.•°.•°:°.´:•˚°.*°.˚:*.´+°.•*/
/// @dev The scalar of ETH and most ERC20s.
uint256 internal constant WAD = 1e18;
/*´:°•.°+.*•´.*:˚.°*.˚•´.°:°•.°•.*•´.*:˚.°*.˚•´.°:°•.°+.*•´.*:*/
/* SIMPLIFIED FIXED POINT OPERATIONS */
/*.•°:°.´+˚.*°.˚:*.´•*.+°.•°:´*.´•*.•°.•°:°.´:•˚°.*°.˚:*.´+°.•*/
/// @dev Equivalent to `(x * y) / WAD` rounded down.
function mulWad(uint256 x, uint256 y) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
// Equivalent to `require(y == 0 || x <= type(uint256).max / y)`.
if gt(x, div(not(0), y)) {
if y {
mstore(0x00, 0xbac65e5b) // `MulWadFailed()`.
revert(0x1c, 0x04)
}
}
z := div(mul(x, y), WAD)
}
}
/// @dev Equivalent to `(x * y) / WAD` rounded down.
function sMulWad(int256 x, int256 y) internal pure returns (int256 z) {
/// @solidity memory-safe-assembly
assembly {
z := mul(x, y)
// Equivalent to `require((x == 0 || z / x == y) && !(x == -1 && y == type(int256).min))`.
if iszero(gt(or(iszero(x), eq(sdiv(z, x), y)), lt(not(x), eq(y, shl(255, 1))))) {
mstore(0x00, 0xedcd4dd4) // `SMulWadFailed()`.
revert(0x1c, 0x04)
}
z := sdiv(z, WAD)
}
}
/// @dev Equivalent to `(x * y) / WAD` rounded down, but without overflow checks.
function rawMulWad(uint256 x, uint256 y) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
z := div(mul(x, y), WAD)
}
}
/// @dev Equivalent to `(x * y) / WAD` rounded down, but without overflow checks.
function rawSMulWad(int256 x, int256 y) internal pure returns (int256 z) {
/// @solidity memory-safe-assembly
assembly {
z := sdiv(mul(x, y), WAD)
}
}
/// @dev Equivalent to `(x * y) / WAD` rounded up.
function mulWadUp(uint256 x, uint256 y) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
z := mul(x, y)
// Equivalent to `require(y == 0 || x <= type(uint256).max / y)`.
if iszero(eq(div(z, y), x)) {
if y {
mstore(0x00, 0xbac65e5b) // `MulWadFailed()`.
revert(0x1c, 0x04)
}
}
z := add(iszero(iszero(mod(z, WAD))), div(z, WAD))
}
}
/// @dev Equivalent to `(x * y) / WAD` rounded up, but without overflow checks.
function rawMulWadUp(uint256 x, uint256 y) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
z := add(iszero(iszero(mod(mul(x, y), WAD))), div(mul(x, y), WAD))
}
}
/// @dev Equivalent to `(x * WAD) / y` rounded down.
function divWad(uint256 x, uint256 y) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
// Equivalent to `require(y != 0 && x <= type(uint256).max / WAD)`.
if iszero(mul(y, lt(x, add(1, div(not(0), WAD))))) {
mstore(0x00, 0x7c5f487d) // `DivWadFailed()`.
revert(0x1c, 0x04)
}
z := div(mul(x, WAD), y)
}
}
/// @dev Equivalent to `(x * WAD) / y` rounded down.
function sDivWad(int256 x, int256 y) internal pure returns (int256 z) {
/// @solidity memory-safe-assembly
assembly {
z := mul(x, WAD)
// Equivalent to `require(y != 0 && ((x * WAD) / WAD == x))`.
if iszero(mul(y, eq(sdiv(z, WAD), x))) {
mstore(0x00, 0x5c43740d) // `SDivWadFailed()`.
revert(0x1c, 0x04)
}
z := sdiv(z, y)
}
}
/// @dev Equivalent to `(x * WAD) / y` rounded down, but without overflow and divide by zero checks.
function rawDivWad(uint256 x, uint256 y) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
z := div(mul(x, WAD), y)
}
}
/// @dev Equivalent to `(x * WAD) / y` rounded down, but without overflow and divide by zero checks.
function rawSDivWad(int256 x, int256 y) internal pure returns (int256 z) {
/// @solidity memory-safe-assembly
assembly {
z := sdiv(mul(x, WAD), y)
}
}
/// @dev Equivalent to `(x * WAD) / y` rounded up.
function divWadUp(uint256 x, uint256 y) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
// Equivalent to `require(y != 0 && x <= type(uint256).max / WAD)`.
if iszero(mul(y, lt(x, add(1, div(not(0), WAD))))) {
mstore(0x00, 0x7c5f487d) // `DivWadFailed()`.
revert(0x1c, 0x04)
}
z := add(iszero(iszero(mod(mul(x, WAD), y))), div(mul(x, WAD), y))
}
}
/// @dev Equivalent to `(x * WAD) / y` rounded up, but without overflow and divide by zero checks.
function rawDivWadUp(uint256 x, uint256 y) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
z := add(iszero(iszero(mod(mul(x, WAD), y))), div(mul(x, WAD), y))
}
}
/// @dev Equivalent to `x` to the power of `y`.
/// because `x ** y = (e ** ln(x)) ** y = e ** (ln(x) * y)`.
/// Note: This function is an approximation.
function powWad(int256 x, int256 y) internal pure returns (int256) {
// Using `ln(x)` means `x` must be greater than 0.
return expWad((lnWad(x) * y) / int256(WAD));
}
/// @dev Returns `exp(x)`, denominated in `WAD`.
/// Credit to Remco Bloemen under MIT license: https://2π.com/22/exp-ln
/// Note: This function is an approximation. Monotonically increasing.
function expWad(int256 x) internal pure returns (int256 r) {
unchecked {
// When the result is less than 0.5 we return zero.
// This happens when `x <= (log(1e-18) * 1e18) ~ -4.15e19`.
if (x <= -41446531673892822313) return r;
/// @solidity memory-safe-assembly
assembly {
// When the result is greater than `(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 iszero(slt(x, 135305999368893231589)) {
mstore(0x00, 0xa37bfec9) // `ExpOverflow()`.
revert(0x1c, 0x04)
}
}
// `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)
);
}
}
/// @dev Returns `ln(x)`, denominated in `WAD`.
/// Credit to Remco Bloemen under MIT license: https://2π.com/22/exp-ln
/// Note: This function is an approximation. Monotonically increasing.
function lnWad(int256 x) internal pure returns (int256 r) {
/// @solidity memory-safe-assembly
assembly {
// 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.
// Compute `k = log2(x) - 96`, `r = 159 - k = 255 - log2(x) = 255 ^ log2(x)`.
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))))
// We place the check here for more optimal stack operations.
if iszero(sgt(x, 0)) {
mstore(0x00, 0x1615e638) // `LnWadUndefined()`.
revert(0x1c, 0x04)
}
// forgefmt: disable-next-item
r := xor(r, byte(and(0x1f, shr(shr(r, x), 0x8421084210842108cc6318c6db6d54be)),
0xf8f9f9faf9fdfafbf9fdfcfdfafbfcfef9fafdfafcfcfbfefafafcfbffffffff))
// Reduce range of x to (1, 2) * 2**96
// ln(2^k * x) = k * ln(2) + ln(x)
x := shr(159, shl(r, x))
// Evaluate using a (8, 8)-term rational approximation.
// `p` is made monic, we will multiply by a scale factor later.
// forgefmt: disable-next-item
let p := sub( // This heavily nested expression is to avoid stack-too-deep for via-ir.
sar(96, mul(add(43456485725739037958740375743393,
sar(96, mul(add(24828157081833163892658089445524,
sar(96, mul(add(3273285459638523848632254066296,
x), x))), x))), x)), 11111509109440967052023855526967)
p := sub(sar(96, mul(p, x)), 45023709667254063763336534515857)
p := sub(sar(96, mul(p, x)), 14706773417378608786704636184526)
p := sub(mul(p, x), shl(96, 795164235651350426258249787498))
// 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.
let q := add(5573035233440673466300451813936, x)
q := add(71694874799317883764090561454958, sar(96, mul(x, q)))
q := add(283447036172924575727196451306956, sar(96, mul(x, q)))
q := add(401686690394027663651624208769553, sar(96, mul(x, q)))
q := add(204048457590392012362485061816622, sar(96, mul(x, q)))
q := add(31853899698501571402653359427138, sar(96, mul(x, q)))
q := add(909429971244387300277376558375, sar(96, mul(x, q)))
// `p / q` 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`.
// The q polynomial is known not to have zeros in the domain.
// No scaling required because p is already `2**96` too large.
p := sdiv(p, q)
// Multiply by the scaling factor: `s * 5**18 * 2**96`, base is now `5**18 * 2**192`.
p := mul(1677202110996718588342820967067443963516166, p)
// Add `ln(2) * k * 5**18 * 2**192`.
// forgefmt: disable-next-item
p := add(mul(16597577552685614221487285958193947469193820559219878177908093499208371, sub(159, r)), p)
// Add `ln(2**96 / 10**18) * 5**18 * 2**192`.
p := add(600920179829731861736702779321621459595472258049074101567377883020018308, p)
// Base conversion: mul `2**18 / 2**192`.
r := sar(174, p)
}
}
/// @dev Returns `W_0(x)`, denominated in `WAD`.
/// See: https://en.wikipedia.org/wiki/Lambert_W_function
/// a.k.a. Product log function. This is an approximation of the principal branch.
/// Note: This function is an approximation. Monotonically increasing.
function lambertW0Wad(int256 x) internal pure returns (int256 w) {
// forgefmt: disable-next-item
unchecked {
if ((w = x) <= -367879441171442322) revert OutOfDomain(); // `x` less than `-1/e`.
(int256 wad, int256 p) = (int256(WAD), x);
uint256 c; // Whether we need to avoid catastrophic cancellation.
uint256 i = 4; // Number of iterations.
if (w <= 0x1ffffffffffff) {
if (-0x4000000000000 <= w) {
i = 1; // Inputs near zero only take one step to converge.
} else if (w <= -0x3ffffffffffffff) {
i = 32; // Inputs near `-1/e` take very long to converge.
}
} else if (uint256(w >> 63) == uint256(0)) {
/// @solidity memory-safe-assembly
assembly {
// Inline log2 for more performance, since the range is small.
let v := shr(49, w)
let l := shl(3, lt(0xff, v))
l := add(or(l, byte(and(0x1f, shr(shr(l, v), 0x8421084210842108cc6318c6db6d54be)),
0x0706060506020504060203020504030106050205030304010505030400000000)), 49)
w := sdiv(shl(l, 7), byte(sub(l, 31), 0x0303030303030303040506080c13))
c := gt(l, 60)
i := add(2, add(gt(l, 53), c))
}
} else {
int256 ll = lnWad(w = lnWad(w));
/// @solidity memory-safe-assembly
assembly {
// `w = ln(x) - ln(ln(x)) + b * ln(ln(x)) / ln(x)`.
w := add(sdiv(mul(ll, 1023715080943847266), w), sub(w, ll))
i := add(3, iszero(shr(68, x)))
c := iszero(shr(143, x))
}
if (c == uint256(0)) {
do { // If `x` is big, use Newton's so that intermediate values won't overflow.
int256 e = expWad(w);
/// @solidity memory-safe-assembly
assembly {
let t := mul(w, div(e, wad))
w := sub(w, sdiv(sub(t, x), div(add(e, t), wad)))
}
if (p <= w) break;
p = w;
} while (--i != uint256(0));
/// @solidity memory-safe-assembly
assembly {
w := sub(w, sgt(w, 2))
}
return w;
}
}
do { // Otherwise, use Halley's for faster convergence.
int256 e = expWad(w);
/// @solidity memory-safe-assembly
assembly {
let t := add(w, wad)
let s := sub(mul(w, e), mul(x, wad))
w := sub(w, sdiv(mul(s, wad), sub(mul(e, t), sdiv(mul(add(t, wad), s), add(t, t)))))
}
if (p <= w) break;
p = w;
} while (--i != c);
/// @solidity memory-safe-assembly
assembly {
w := sub(w, sgt(w, 2))
}
// For certain ranges of `x`, we'll use the quadratic-rate recursive formula of
// R. Iacono and J.P. Boyd for the last iteration, to avoid catastrophic cancellation.
if (c == uint256(0)) return w;
int256 t = w | 1;
/// @solidity memory-safe-assembly
assembly {
x := sdiv(mul(x, wad), t)
}
x = (t * (wad + lnWad(x)));
/// @solidity memory-safe-assembly
assembly {
w := sdiv(x, add(wad, t))
}
}
}
/*´:°•.°+.*•´.*:˚.°*.˚•´.°:°•.°•.*•´.*:˚.°*.˚•´.°:°•.°+.*•´.*:*/
/* GENERAL NUMBER UTILITIES */
/*.•°:°.´+˚.*°.˚:*.´•*.+°.•°:´*.´•*.•°.•°:°.´:•˚°.*°.˚:*.´+°.•*/
/// @dev Calculates `floor(x * y / d)` with full precision.
/// Throws if result overflows a uint256 or when `d` is zero.
/// Credit to Remco Bloemen under MIT license: https://2π.com/21/muldiv
function fullMulDiv(uint256 x, uint256 y, uint256 d) internal pure returns (uint256 result) {
/// @solidity memory-safe-assembly
assembly {
// 512-bit multiply `[p1 p0] = x * y`.
// Compute the product mod `2**256` and mod `2**256 - 1`
// then use the Chinese Remainder Theorem to reconstruct
// the 512 bit result. The result is stored in two 256
// variables such that `product = p1 * 2**256 + p0`.
// Temporarily use `result` as `p0` to save gas.
result := mul(x, y) // Lower 256 bits of `x * y`.
for {} 1 {} {
// If overflows.
if iszero(mul(or(iszero(x), eq(div(result, x), y)), d)) {
let mm := mulmod(x, y, not(0))
let p1 := sub(mm, add(result, lt(mm, result))) // Upper 256 bits of `x * y`.
/*------------------- 512 by 256 division --------------------*/
// Make division exact by subtracting the remainder from `[p1 p0]`.
let r := mulmod(x, y, d) // Compute remainder using mulmod.
let t := and(d, sub(0, d)) // The least significant bit of `d`. `t >= 1`.
// Make sure the result is less than `2**256`. Also prevents `d == 0`.
// Placing the check here seems to give more optimal stack operations.
if iszero(gt(d, p1)) {
mstore(0x00, 0xae47f702) // `FullMulDivFailed()`.
revert(0x1c, 0x04)
}
d := div(d, t) // Divide `d` by `t`, which is a power of two.
// Invert `d mod 2**256`
// Now that `d` is an odd number, it has an inverse
// modulo `2**256` such that `d * inv = 1 mod 2**256`.
// Compute the inverse by starting with a seed that is correct
// correct for four bits. That is, `d * inv = 1 mod 2**4`.
let inv := xor(2, mul(3, d))
// Now use Newton-Raphson iteration to improve the precision.
// Thanks to Hensel's lifting lemma, this also works in modular
// arithmetic, doubling the correct bits in each step.
inv := mul(inv, sub(2, mul(d, inv))) // inverse mod 2**8
inv := mul(inv, sub(2, mul(d, inv))) // inverse mod 2**16
inv := mul(inv, sub(2, mul(d, inv))) // inverse mod 2**32
inv := mul(inv, sub(2, mul(d, inv))) // inverse mod 2**64
inv := mul(inv, sub(2, mul(d, inv))) // inverse mod 2**128
result :=
mul(
// Divide [p1 p0] by the factors of two.
// Shift in bits from `p1` into `p0`. For this we need
// to flip `t` such that it is `2**256 / t`.
or(
mul(sub(p1, gt(r, result)), add(div(sub(0, t), t), 1)),
div(sub(result, r), t)
),
mul(sub(2, mul(d, inv)), inv) // inverse mod 2**256
)
break
}
result := div(result, d)
break
}
}
}
/// @dev Calculates `floor(x * y / d)` with full precision.
/// Behavior is undefined if `d` is zero or the final result cannot fit in 256 bits.
/// Performs the full 512 bit calculation regardless.
function fullMulDivUnchecked(uint256 x, uint256 y, uint256 d)
internal
pure
returns (uint256 result)
{
/// @solidity memory-safe-assembly
assembly {
result := mul(x, y)
let mm := mulmod(x, y, not(0))
let p1 := sub(mm, add(result, lt(mm, result)))
let t := and(d, sub(0, d))
let r := mulmod(x, y, d)
d := div(d, t)
let inv := xor(2, mul(3, d))
inv := mul(inv, sub(2, mul(d, inv)))
inv := mul(inv, sub(2, mul(d, inv)))
inv := mul(inv, sub(2, mul(d, inv)))
inv := mul(inv, sub(2, mul(d, inv)))
inv := mul(inv, sub(2, mul(d, inv)))
result :=
mul(
or(mul(sub(p1, gt(r, result)), add(div(sub(0, t), t), 1)), div(sub(result, r), t)),
mul(sub(2, mul(d, inv)), inv)
)
}
}
/// @dev Calculates `floor(x * y / d)` with full precision, rounded up.
/// Throws if result overflows a uint256 or when `d` is zero.
/// Credit to Uniswap-v3-core under MIT license:
/// https://github.com/Uniswap/v3-core/blob/main/contracts/libraries/FullMath.sol
function fullMulDivUp(uint256 x, uint256 y, uint256 d) internal pure returns (uint256 result) {
result = fullMulDiv(x, y, d);
/// @solidity memory-safe-assembly
assembly {
if mulmod(x, y, d) {
result := add(result, 1)
if iszero(result) {
mstore(0x00, 0xae47f702) // `FullMulDivFailed()`.
revert(0x1c, 0x04)
}
}
}
}
/// @dev Returns `floor(x * y / d)`.
/// Reverts if `x * y` overflows, or `d` is zero.
function mulDiv(uint256 x, uint256 y, uint256 d) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
z := mul(x, y)
// Equivalent to `require(d != 0 && (y == 0 || x <= type(uint256).max / y))`.
if iszero(mul(or(iszero(x), eq(div(z, x), y)), d)) {
mstore(0x00, 0xad251c27) // `MulDivFailed()`.
revert(0x1c, 0x04)
}
z := div(z, d)
}
}
/// @dev Returns `ceil(x * y / d)`.
/// Reverts if `x * y` overflows, or `d` is zero.
function mulDivUp(uint256 x, uint256 y, uint256 d) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
z := mul(x, y)
// Equivalent to `require(d != 0 && (y == 0 || x <= type(uint256).max / y))`.
if iszero(mul(or(iszero(x), eq(div(z, x), y)), d)) {
mstore(0x00, 0xad251c27) // `MulDivFailed()`.
revert(0x1c, 0x04)
}
z := add(iszero(iszero(mod(z, d))), div(z, d))
}
}
/// @dev Returns `ceil(x / d)`.
/// Reverts if `d` is zero.
function divUp(uint256 x, uint256 d) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
if iszero(d) {
mstore(0x00, 0x65244e4e) // `DivFailed()`.
revert(0x1c, 0x04)
}
z := add(iszero(iszero(mod(x, d))), div(x, d))
}
}
/// @dev Returns `max(0, x - y)`.
function zeroFloorSub(uint256 x, uint256 y) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
z := mul(gt(x, y), sub(x, y))
}
}
/// @dev Returns `condition ? x : y`, without branching.
function ternary(bool condition, uint256 x, uint256 y) internal pure returns (uint256 result) {
/// @solidity memory-safe-assembly
assembly {
result := xor(x, mul(xor(x, y), iszero(condition)))
}
}
/// @dev Exponentiate `x` to `y` by squaring, denominated in base `b`.
/// Reverts if the computation overflows.
function rpow(uint256 x, uint256 y, uint256 b) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
z := mul(b, iszero(y)) // `0 ** 0 = 1`. Otherwise, `0 ** n = 0`.
if x {
z := xor(b, mul(xor(b, x), and(y, 1))) // `z = isEven(y) ? scale : x`
let half := shr(1, b) // Divide `b` by 2.
// Divide `y` by 2 every iteration.
for { y := shr(1, y) } y { y := shr(1, y) } {
let xx := mul(x, x) // Store x squared.
let xxRound := add(xx, half) // Round to the nearest number.
// Revert if `xx + half` overflowed, or if `x ** 2` overflows.
if or(lt(xxRound, xx), shr(128, x)) {
mstore(0x00, 0x49f7642b) // `RPowOverflow()`.
revert(0x1c, 0x04)
}
x := div(xxRound, b) // Set `x` to scaled `xxRound`.
// If `y` is odd:
if and(y, 1) {
let zx := mul(z, x) // Compute `z * x`.
let zxRound := add(zx, half) // Round to the nearest number.
// If `z * x` overflowed or `zx + half` overflowed:
if or(xor(div(zx, x), z), lt(zxRound, zx)) {
// Revert if `x` is non-zero.
if x {
mstore(0x00, 0x49f7642b) // `RPowOverflow()`.
revert(0x1c, 0x04)
}
}
z := div(zxRound, b) // Return properly scaled `zxRound`.
}
}
}
}
}
/// @dev Returns the square root of `x`, rounded down.
function sqrt(uint256 x) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
// `floor(sqrt(2**15)) = 181`. `sqrt(2**15) - 181 = 2.84`.
z := 181 // The "correct" value is 1, but this saves a multiplication later.
// This segment is to get a reasonable initial estimate for the Babylonian method. With a bad
// start, the correct # of bits increases ~linearly each iteration instead of ~quadratically.
// Let `y = x / 2**r`. We check `y >= 2**(k + 8)`
// but shift right by `k` bits to ensure that if `x >= 256`, then `y >= 256`.
let r := shl(7, lt(0xffffffffffffffffffffffffffffffffff, x))
r := or(r, shl(6, lt(0xffffffffffffffffff, shr(r, x))))
r := or(r, shl(5, lt(0xffffffffff, shr(r, x))))
r := or(r, shl(4, lt(0xffffff, shr(r, x))))
z := shl(shr(1, r), z)
// Goal was to get `z*z*y` within a small factor of `x`. More iterations could
// get y in a tighter range. Currently, we will have y in `[256, 256*(2**16))`.
// We ensured `y >= 256` so that the relative difference between `y` and `y+1` is small.
// That's not possible if `x < 256` but we can just verify those cases exhaustively.
// Now, `z*z*y <= x < z*z*(y+1)`, and `y <= 2**(16+8)`, and either `y >= 256`, or `x < 256`.
// Correctness can be checked exhaustively for `x < 256`, so we assume `y >= 256`.
// Then `z*sqrt(y)` is within `sqrt(257)/sqrt(256)` of `sqrt(x)`, or about 20bps.
// For `s` in the range `[1/256, 256]`, the estimate `f(s) = (181/1024) * (s+1)`
// is in the range `(1/2.84 * sqrt(s), 2.84 * sqrt(s))`,
// with largest error when `s = 1` and when `s = 256` or `1/256`.
// Since `y` is in `[256, 256*(2**16))`, let `a = y/65536`, so that `a` is in `[1/256, 256)`.
// Then we can estimate `sqrt(y)` using
// `sqrt(65536) * 181/1024 * (a + 1) = 181/4 * (y + 65536)/65536 = 181 * (y + 65536)/2**18`.
// There is no overflow risk here since `y < 2**136` after the first branch above.
z := shr(18, mul(z, add(shr(r, x), 65536))) // A `mul()` is saved from starting `z` at 181.
// Given the worst case multiplicative error of 2.84 above, 7 iterations should be enough.
z := shr(1, add(z, div(x, z)))
z := shr(1, add(z, div(x, z)))
z := shr(1, add(z, div(x, z)))
z := shr(1, add(z, div(x, z)))
z := shr(1, add(z, div(x, z)))
z := shr(1, add(z, div(x, z)))
z := shr(1, add(z, div(x, z)))
// If `x+1` is a perfect square, the Babylonian method cycles between
// `floor(sqrt(x))` and `ceil(sqrt(x))`. This statement ensures we return floor.
// See: https://en.wikipedia.org/wiki/Integer_square_root#Using_only_integer_division
z := sub(z, lt(div(x, z), z))
}
}
/// @dev Returns the cube root of `x`, rounded down.
/// Credit to bout3fiddy and pcaversaccio under AGPLv3 license:
/// https://github.com/pcaversaccio/snekmate/blob/main/src/utils/Math.vy
/// Formally verified by xuwinnie:
/// https://github.com/vectorized/solady/blob/main/audits/xuwinnie-solady-cbrt-proof.pdf
function cbrt(uint256 x) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
let 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))))
// Makeshift lookup table to nudge the approximate log2 result.
z := div(shl(div(r, 3), shl(lt(0xf, shr(r, x)), 0xf)), xor(7, mod(r, 3)))
// Newton-Raphson's.
z := div(add(add(div(x, mul(z, z)), z), z), 3)
z := div(add(add(div(x, mul(z, z)), z), z), 3)
z := div(add(add(div(x, mul(z, z)), z), z), 3)
z := div(add(add(div(x, mul(z, z)), z), z), 3)
z := div(add(add(div(x, mul(z, z)), z), z), 3)
z := div(add(add(div(x, mul(z, z)), z), z), 3)
z := div(add(add(div(x, mul(z, z)), z), z), 3)
// Round down.
z := sub(z, lt(div(x, mul(z, z)), z))
}
}
/// @dev Returns the square root of `x`, denominated in `WAD`, rounded down.
function sqrtWad(uint256 x) internal pure returns (uint256 z) {
unchecked {
if (x <= type(uint256).max / 10 ** 18) return sqrt(x * 10 ** 18);
z = (1 + sqrt(x)) * 10 ** 9;
z = (fullMulDivUnchecked(x, 10 ** 18, z) + z) >> 1;
}
/// @solidity memory-safe-assembly
assembly {
z := sub(z, gt(999999999999999999, sub(mulmod(z, z, x), 1))) // Round down.
}
}
/// @dev Returns the cube root of `x`, denominated in `WAD`, rounded down.
/// Formally verified by xuwinnie:
/// https://github.com/vectorized/solady/blob/main/audits/xuwinnie-solady-cbrt-proof.pdf
function cbrtWad(uint256 x) internal pure returns (uint256 z) {
unchecked {
if (x <= type(uint256).max / 10 ** 36) return cbrt(x * 10 ** 36);
z = (1 + cbrt(x)) * 10 ** 12;
z = (fullMulDivUnchecked(x, 10 ** 36, z * z) + z + z) / 3;
}
/// @solidity memory-safe-assembly
assembly {
let p := x
for {} 1 {} {
if iszero(shr(229, p)) {
if iszero(shr(199, p)) {
p := mul(p, 100000000000000000) // 10 ** 17.
break
}
p := mul(p, 100000000) // 10 ** 8.
break
}
if iszero(shr(249, p)) { p := mul(p, 100) }
break
}
let t := mulmod(mul(z, z), z, p)
z := sub(z, gt(lt(t, shr(1, p)), iszero(t))) // Round down.
}
}
/// @dev Returns the factorial of `x`.
function factorial(uint256 x) internal pure returns (uint256 result) {
/// @solidity memory-safe-assembly
assembly {
result := 1
if iszero(lt(x, 58)) {
mstore(0x00, 0xaba0f2a2) // `FactorialOverflow()`.
revert(0x1c, 0x04)
}
for {} x { x := sub(x, 1) } { result := mul(result, x) }
}
}
/// @dev Returns the log2 of `x`.
/// Equivalent to computing the index of the most significant bit (MSB) of `x`.
/// Returns 0 if `x` is zero.
function log2(uint256 x) internal pure returns (uint256 r) {
/// @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))))
// forgefmt: disable-next-item
r := or(r, byte(and(0x1f, shr(shr(r, x), 0x8421084210842108cc6318c6db6d54be)),
0x0706060506020504060203020504030106050205030304010505030400000000))
}
}
/// @dev Returns the log2 of `x`, rounded up.
/// Returns 0 if `x` is zero.
function log2Up(uint256 x) internal pure returns (uint256 r) {
r = log2(x);
/// @solidity memory-safe-assembly
assembly {
r := add(r, lt(shl(r, 1), x))
}
}
/// @dev Returns the log10 of `x`.
/// Returns 0 if `x` is zero.
function log10(uint256 x) internal pure returns (uint256 r) {
/// @solidity memory-safe-assembly
assembly {
if iszero(lt(x, 100000000000000000000000000000000000000)) {
x := div(x, 100000000000000000000000000000000000000)
r := 38
}
if iszero(lt(x, 100000000000000000000)) {
x := div(x, 100000000000000000000)
r := add(r, 20)
}
if iszero(lt(x, 10000000000)) {
x := div(x, 10000000000)
r := add(r, 10)
}
if iszero(lt(x, 100000)) {
x := div(x, 100000)
r := add(r, 5)
}
r := add(r, add(gt(x, 9), add(gt(x, 99), add(gt(x, 999), gt(x, 9999)))))
}
}
/// @dev Returns the log10 of `x`, rounded up.
/// Returns 0 if `x` is zero.
function log10Up(uint256 x) internal pure returns (uint256 r) {
r = log10(x);
/// @solidity memory-safe-assembly
assembly {
r := add(r, lt(exp(10, r), x))
}
}
/// @dev Returns the log256 of `x`.
/// Returns 0 if `x` is zero.
function log256(uint256 x) internal pure returns (uint256 r) {
/// @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(shr(3, r), lt(0xff, shr(r, x)))
}
}
/// @dev Returns the log256 of `x`, rounded up.
/// Returns 0 if `x` is zero.
function log256Up(uint256 x) internal pure returns (uint256 r) {
r = log256(x);
/// @solidity memory-safe-assembly
assembly {
r := add(r, lt(shl(shl(3, r), 1), x))
}
}
/// @dev Returns the scientific notation format `mantissa * 10 ** exponent` of `x`.
/// Useful for compressing prices (e.g. using 25 bit mantissa and 7 bit exponent).
function sci(uint256 x) internal pure returns (uint256 mantissa, uint256 exponent) {
/// @solidity memory-safe-assembly
assembly {
mantissa := x
if mantissa {
if iszero(mod(mantissa, 1000000000000000000000000000000000)) {
mantissa := div(mantissa, 1000000000000000000000000000000000)
exponent := 33
}
if iszero(mod(mantissa, 10000000000000000000)) {
mantissa := div(mantissa, 10000000000000000000)
exponent := add(exponent, 19)
}
if iszero(mod(mantissa, 1000000000000)) {
mantissa := div(mantissa, 1000000000000)
exponent := add(exponent, 12)
}
if iszero(mod(mantissa, 1000000)) {
mantissa := div(mantissa, 1000000)
exponent := add(exponent, 6)
}
if iszero(mod(mantissa, 10000)) {
mantissa := div(mantissa, 10000)
exponent := add(exponent, 4)
}
if iszero(mod(mantissa, 100)) {
mantissa := div(mantissa, 100)
exponent := add(exponent, 2)
}
if iszero(mod(mantissa, 10)) {
mantissa := div(mantissa, 10)
exponent := add(exponent, 1)
}
}
}
}
/// @dev Convenience function for packing `x` into a smaller number using `sci`.
/// The `mantissa` will be in bits [7..255] (the upper 249 bits).
/// The `exponent` will be in bits [0..6] (the lower 7 bits).
/// Use `SafeCastLib` to safely ensure that the `packed` number is small
/// enough to fit in the desired unsigned integer type:
/// ```
/// uint32 packed = SafeCastLib.toUint32(FixedPointMathLib.packSci(777 ether));
/// ```
function packSci(uint256 x) internal pure returns (uint256 packed) {
(x, packed) = sci(x); // Reuse for `mantissa` and `exponent`.
/// @solidity memory-safe-assembly
assembly {
if shr(249, x) {
mstore(0x00, 0xce30380c) // `MantissaOverflow()`.
revert(0x1c, 0x04)
}
packed := or(shl(7, x), packed)
}
}
/// @dev Convenience function for unpacking a packed number from `packSci`.
function unpackSci(uint256 packed) internal pure returns (uint256 unpacked) {
unchecked {
unpacked = (packed >> 7) * 10 ** (packed & 0x7f);
}
}
/// @dev Returns the average of `x` and `y`. Rounds towards zero.
function avg(uint256 x, uint256 y) internal pure returns (uint256 z) {
unchecked {
z = (x & y) + ((x ^ y) >> 1);
}
}
/// @dev Returns the average of `x` and `y`. Rounds towards negative infinity.
function avg(int256 x, int256 y) internal pure returns (int256 z) {
unchecked {
z = (x >> 1) + (y >> 1) + (x & y & 1);
}
}
/// @dev Returns the absolute value of `x`.
function abs(int256 x) internal pure returns (uint256 z) {
unchecked {
z = (uint256(x) + uint256(x >> 255)) ^ uint256(x >> 255);
}
}
/// @dev Returns the absolute distance between `x` and `y`.
function dist(uint256 x, uint256 y) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
z := add(xor(sub(0, gt(x, y)), sub(y, x)), gt(x, y))
}
}
/// @dev Returns the absolute distance between `x` and `y`.
function dist(int256 x, int256 y) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
z := add(xor(sub(0, sgt(x, y)), sub(y, x)), sgt(x, y))
}
}
/// @dev Returns the minimum of `x` and `y`.
function min(uint256 x, uint256 y) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
z := xor(x, mul(xor(x, y), lt(y, x)))
}
}
/// @dev Returns the minimum of `x` and `y`.
function min(int256 x, int256 y) internal pure returns (int256 z) {
/// @solidity memory-safe-assembly
assembly {
z := xor(x, mul(xor(x, y), slt(y, x)))
}
}
/// @dev Returns the maximum of `x` and `y`.
function max(uint256 x, uint256 y) internal pure returns (uint256 z) {
/// @solidity memory-safe-assembly
assembly {
z := xor(x, mul(xor(x, y), gt(y, x)))
}
}
/// @dev Returns the maximum of `x` and `y`.
function max(int256 x, int256 y) internal pure returns (int256 z) {