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Import prb-math for testing

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test/libsolidity/semanticTests/externalContracts/_prbmath/LICENSE.md Normal file
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# WTFPL
by Paul Razvan Berg (@PaulRBerg)
TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
0. You just DO WHAT THE FUCK YOU WANT TO.

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// SPDX-License-Identifier: WTFPL
pragma solidity >=0.8.0;
/// @dev Common mathematical functions used in both PRBMathSD59x18 and PRBMathUD60x18. Note that this shared library
/// does not always assume the signed 59.18-decimal fixed-point or the unsigned 60.18-decimal fixed-point
// representation. When it does not, it is annonated in the function's NatSpec documentation.
library PRBMathCommon {
/// @dev How many trailing decimals can be represented.
uint256 internal constant SCALE = 1e18;
/// @dev Largest power of two divisor of SCALE.
uint256 internal constant SCALE_LPOTD = 262144;
/// @dev SCALE inverted mod 2^256.
uint256 internal constant SCALE_INVERSE = 78156646155174841979727994598816262306175212592076161876661508869554232690281;
/// @notice Calculates the binary exponent of x using the binary fraction method.
/// @dev Uses 128.128-bit fixed-point numbers, which is the most efficient way.
/// See https://ethereum.stackexchange.com/a/96594/24693.
/// @param x The exponent as an unsigned 128.128-bit fixed-point number.
/// @return result The result as an unsigned 60x18 decimal fixed-point number.
function exp2(uint256 x) internal pure returns (uint256 result) {
unchecked {
// Start from 0.5 in the 128.128-bit fixed-point format.
result = 0x80000000000000000000000000000000;
// Multiply the result by root(2, 2^-i) when the bit at position i is 1. None of the intermediary results overflows
// because the initial result is 2^127 and all magic factors are less than 2^129.
if (x & 0x80000000000000000000000000000000 > 0) result = (result * 0x16A09E667F3BCC908B2FB1366EA957D3E) >> 128;
if (x & 0x40000000000000000000000000000000 > 0) result = (result * 0x1306FE0A31B7152DE8D5A46305C85EDED) >> 128;
if (x & 0x20000000000000000000000000000000 > 0) result = (result * 0x1172B83C7D517ADCDF7C8C50EB14A7920) >> 128;
if (x & 0x10000000000000000000000000000000 > 0) result = (result * 0x10B5586CF9890F6298B92B71842A98364) >> 128;
if (x & 0x8000000000000000000000000000000 > 0) result = (result * 0x1059B0D31585743AE7C548EB68CA417FE) >> 128;
if (x & 0x4000000000000000000000000000000 > 0) result = (result * 0x102C9A3E778060EE6F7CACA4F7A29BDE9) >> 128;
if (x & 0x2000000000000000000000000000000 > 0) result = (result * 0x10163DA9FB33356D84A66AE336DCDFA40) >> 128;
if (x & 0x1000000000000000000000000000000 > 0) result = (result * 0x100B1AFA5ABCBED6129AB13EC11DC9544) >> 128;
if (x & 0x800000000000000000000000000000 > 0) result = (result * 0x10058C86DA1C09EA1FF19D294CF2F679C) >> 128;
if (x & 0x400000000000000000000000000000 > 0) result = (result * 0x1002C605E2E8CEC506D21BFC89A23A011) >> 128;
if (x & 0x200000000000000000000000000000 > 0) result = (result * 0x100162F3904051FA128BCA9C55C31E5E0) >> 128;
if (x & 0x100000000000000000000000000000 > 0) result = (result * 0x1000B175EFFDC76BA38E31671CA939726) >> 128;
if (x & 0x80000000000000000000000000000 > 0) result = (result * 0x100058BA01FB9F96D6CACD4B180917C3E) >> 128;
if (x & 0x40000000000000000000000000000 > 0) result = (result * 0x10002C5CC37DA9491D0985C348C68E7B4) >> 128;
if (x & 0x20000000000000000000000000000 > 0) result = (result * 0x1000162E525EE054754457D5995292027) >> 128;
if (x & 0x10000000000000000000000000000 > 0) result = (result * 0x10000B17255775C040618BF4A4ADE83FD) >> 128;
if (x & 0x8000000000000000000000000000 > 0) result = (result * 0x1000058B91B5BC9AE2EED81E9B7D4CFAC) >> 128;
if (x & 0x4000000000000000000000000000 > 0) result = (result * 0x100002C5C89D5EC6CA4D7C8ACC017B7CA) >> 128;
if (x & 0x2000000000000000000000000000 > 0) result = (result * 0x10000162E43F4F831060E02D839A9D16D) >> 128;
if (x & 0x1000000000000000000000000000 > 0) result = (result * 0x100000B1721BCFC99D9F890EA06911763) >> 128;
if (x & 0x800000000000000000000000000 > 0) result = (result * 0x10000058B90CF1E6D97F9CA14DBCC1629) >> 128;
if (x & 0x400000000000000000000000000 > 0) result = (result * 0x1000002C5C863B73F016468F6BAC5CA2C) >> 128;
if (x & 0x200000000000000000000000000 > 0) result = (result * 0x100000162E430E5A18F6119E3C02282A6) >> 128;
if (x & 0x100000000000000000000000000 > 0) result = (result * 0x1000000B1721835514B86E6D96EFD1BFF) >> 128;
if (x & 0x80000000000000000000000000 > 0) result = (result * 0x100000058B90C0B48C6BE5DF846C5B2F0) >> 128;
if (x & 0x40000000000000000000000000 > 0) result = (result * 0x10000002C5C8601CC6B9E94213C72737B) >> 128;
if (x & 0x20000000000000000000000000 > 0) result = (result * 0x1000000162E42FFF037DF38AA2B219F07) >> 128;
if (x & 0x10000000000000000000000000 > 0) result = (result * 0x10000000B17217FBA9C739AA5819F44FA) >> 128;
if (x & 0x8000000000000000000000000 > 0) result = (result * 0x1000000058B90BFCDEE5ACD3C1CEDC824) >> 128;
if (x & 0x4000000000000000000000000 > 0) result = (result * 0x100000002C5C85FE31F35A6A30DA1BE51) >> 128;
if (x & 0x2000000000000000000000000 > 0) result = (result * 0x10000000162E42FF0999CE3541B9FFFD0) >> 128;
if (x & 0x1000000000000000000000000 > 0) result = (result * 0x100000000B17217F80F4EF5AADDA45554) >> 128;
if (x & 0x800000000000000000000000 > 0) result = (result * 0x10000000058B90BFBF8479BD5A81B51AE) >> 128;
if (x & 0x400000000000000000000000 > 0) result = (result * 0x1000000002C5C85FDF84BD62AE30A74CD) >> 128;
if (x & 0x200000000000000000000000 > 0) result = (result * 0x100000000162E42FEFB2FED257559BDAA) >> 128;
if (x & 0x100000000000000000000000 > 0) result = (result * 0x1000000000B17217F7D5A7716BBA4A9AF) >> 128;
if (x & 0x80000000000000000000000 > 0) result = (result * 0x100000000058B90BFBE9DDBAC5E109CCF) >> 128;
if (x & 0x40000000000000000000000 > 0) result = (result * 0x10000000002C5C85FDF4B15DE6F17EB0E) >> 128;
if (x & 0x20000000000000000000000 > 0) result = (result * 0x1000000000162E42FEFA494F1478FDE05) >> 128;
if (x & 0x10000000000000000000000 > 0) result = (result * 0x10000000000B17217F7D20CF927C8E94D) >> 128;
if (x & 0x8000000000000000000000 > 0) result = (result * 0x1000000000058B90BFBE8F71CB4E4B33E) >> 128;
if (x & 0x4000000000000000000000 > 0) result = (result * 0x100000000002C5C85FDF477B662B26946) >> 128;
if (x & 0x2000000000000000000000 > 0) result = (result * 0x10000000000162E42FEFA3AE53369388D) >> 128;
if (x & 0x1000000000000000000000 > 0) result = (result * 0x100000000000B17217F7D1D351A389D41) >> 128;
if (x & 0x800000000000000000000 > 0) result = (result * 0x10000000000058B90BFBE8E8B2D3D4EDF) >> 128;
if (x & 0x400000000000000000000 > 0) result = (result * 0x1000000000002C5C85FDF4741BEA6E77F) >> 128;
if (x & 0x200000000000000000000 > 0) result = (result * 0x100000000000162E42FEFA39FE95583C3) >> 128;
if (x & 0x100000000000000000000 > 0) result = (result * 0x1000000000000B17217F7D1CFB72B45E3) >> 128;
if (x & 0x80000000000000000000 > 0) result = (result * 0x100000000000058B90BFBE8E7CC35C3F2) >> 128;
if (x & 0x40000000000000000000 > 0) result = (result * 0x10000000000002C5C85FDF473E242EA39) >> 128;
if (x & 0x20000000000000000000 > 0) result = (result * 0x1000000000000162E42FEFA39F02B772C) >> 128;
if (x & 0x10000000000000000000 > 0) result = (result * 0x10000000000000B17217F7D1CF7D83C1A) >> 128;
if (x & 0x8000000000000000000 > 0) result = (result * 0x1000000000000058B90BFBE8E7BDCBE2E) >> 128;
if (x & 0x4000000000000000000 > 0) result = (result * 0x100000000000002C5C85FDF473DEA871F) >> 128;
if (x & 0x2000000000000000000 > 0) result = (result * 0x10000000000000162E42FEFA39EF44D92) >> 128;
if (x & 0x1000000000000000000 > 0) result = (result * 0x100000000000000B17217F7D1CF79E949) >> 128;
if (x & 0x800000000000000000 > 0) result = (result * 0x10000000000000058B90BFBE8E7BCE545) >> 128;
if (x & 0x400000000000000000 > 0) result = (result * 0x1000000000000002C5C85FDF473DE6ECA) >> 128;
if (x & 0x200000000000000000 > 0) result = (result * 0x100000000000000162E42FEFA39EF366F) >> 128;
if (x & 0x100000000000000000 > 0) result = (result * 0x1000000000000000B17217F7D1CF79AFA) >> 128;
if (x & 0x80000000000000000 > 0) result = (result * 0x100000000000000058B90BFBE8E7BCD6E) >> 128;
if (x & 0x40000000000000000 > 0) result = (result * 0x10000000000000002C5C85FDF473DE6B3) >> 128;
if (x & 0x20000000000000000 > 0) result = (result * 0x1000000000000000162E42FEFA39EF359) >> 128;
if (x & 0x10000000000000000 > 0) result = (result * 0x10000000000000000B17217F7D1CF79AC) >> 128;
// We do two things at the same time below:
//
// 1. Multiply the result by 2^n + 1, where 2^n is the integer part and 1 is an extra bit to account
// for the fact that we initially set the result to 0.5 We implement this by subtracting from 127
// instead of 128.
// 2. Convert the result to the unsigned 60.18-decimal fixed-point format.
//
// This works because result * SCALE * 2^ip / 2^127 = result * SCALE / 2^(127 - ip), where ip is the integer
// part and SCALE / 2^128 is what converts the result to our unsigned fixed-point format.
result *= SCALE;
result >>= (127 - (x >> 128));
}
}
/// @notice Finds the zero-based index of the first one in the binary representation of x.
/// @dev See the note on msb in the "Find First Set" Wikipedia article https://en.wikipedia.org/wiki/Find_first_set
/// @param x The uint256 number for which to find the index of the most significant bit.
/// @return msb The index of the most significant bit as an uint256.
function mostSignificantBit(uint256 x) internal pure returns (uint256 msb) {
if (x >= 2**128) {
x >>= 128;
msb += 128;
}
if (x >= 2**64) {
x >>= 64;
msb += 64;
}
if (x >= 2**32) {
x >>= 32;
msb += 32;
}
if (x >= 2**16) {
x >>= 16;
msb += 16;
}
if (x >= 2**8) {
x >>= 8;
msb += 8;
}
if (x >= 2**4) {
x >>= 4;
msb += 4;
}
if (x >= 2**2) {
x >>= 2;
msb += 2;
}
if (x >= 2**1) {
// No need to shift x any more.
msb += 1;
}
}
/// @notice Calculates floor(x*y÷denominator) with full precision.
///
/// @dev Credit to Remco Bloemen under MIT license https://xn--2-umb.com/21/muldiv.
///
/// Requirements:
/// - The denominator cannot be zero.
/// - The result must fit within uint256.
///
/// Caveats:
/// - This function does not work with fixed-point numbers.
///
/// @param x The multiplicand as an uint256.
/// @param y The multiplier as an uint256.
/// @param denominator The divisor as an uint256.
/// @return result The result as an uint256.
function mulDiv(
uint256 x,
uint256 y,
uint256 denominator
) internal pure returns (uint256 result) {
// 512-bit multiply [prod1 prod0] = x * y. Compute the product mod 2**256 and mod 2**256 - 1, then use
// use the Chinese Remainder Theorem to reconstruct the 512 bit result. The result is stored in two 256
// variables such that product = prod1 * 2**256 + prod0.
uint256 prod0; // Least significant 256 bits of the product
uint256 prod1; // Most significant 256 bits of the product
assembly {
let mm := mulmod(x, y, not(0))
prod0 := mul(x, y)
prod1 := sub(sub(mm, prod0), lt(mm, prod0))
}
// Handle non-overflow cases, 256 by 256 division
if (prod1 == 0) {
require(denominator > 0);
assembly {
result := div(prod0, denominator)
}
return result;
}
// Make sure the result is less than 2**256. Also prevents denominator == 0.
require(denominator > prod1);
///////////////////////////////////////////////
// 512 by 256 division.
///////////////////////////////////////////////
// Make division exact by subtracting the remainder from [prod1 prod0].
uint256 remainder;
assembly {
// Compute remainder using mulmod.
remainder := mulmod(x, y, denominator)
// Subtract 256 bit number from 512 bit number
prod1 := sub(prod1, gt(remainder, prod0))
prod0 := sub(prod0, remainder)
}
// Factor powers of two out of denominator and compute largest power of two divisor of denominator. Always >= 1.
// See https://cs.stackexchange.com/q/138556/92363.
unchecked {
// Does not overflow because the denominator cannot be zero at this stage in the function.
uint256 lpotdod = denominator & (~denominator + 1);
assembly {
// Divide denominator by lpotdod.
denominator := div(denominator, lpotdod)
// Divide [prod1 prod0] by lpotdod.
prod0 := div(prod0, lpotdod)
// Flip lpotdod such that it is 2**256 / lpotdod. If lpotdod is zero, then it becomes one.
lpotdod := add(div(sub(0, lpotdod), lpotdod), 1)
}
// Shift in bits from prod1 into prod0.
prod0 |= prod1 * lpotdod;
// Invert denominator mod 2**256. Now that denominator is an odd number, it has an inverse modulo 2**256 such
// that denominator * inv = 1 mod 2**256. Compute the inverse by starting with a seed that is correct for
// four bits. That is, denominator * inv = 1 mod 2**4
uint256 inverse = (3 * denominator) ^ 2;
// 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.
inverse *= 2 - denominator * inverse; // inverse mod 2**8
inverse *= 2 - denominator * inverse; // inverse mod 2**16
inverse *= 2 - denominator * inverse; // inverse mod 2**32
inverse *= 2 - denominator * inverse; // inverse mod 2**64
inverse *= 2 - denominator * inverse; // inverse mod 2**128
inverse *= 2 - denominator * inverse; // inverse mod 2**256
// Because the division is now exact we can divide by multiplying with the modular inverse of denominator.
// This will give us the correct result modulo 2**256. Since the precoditions guarantee that the outcome is
// less than 2**256, this is the final result. We don't need to compute the high bits of the result and prod1
// is no longer required.
result = prod0 * inverse;
return result;
}
}
/// @notice Calculates floor(x*y÷1e18) with full precision.
///
/// @dev Variant of "mulDiv" with constant folding, i.e. in which the denominator is always 1e18. Before returning the
/// final result, we add 1 if (x * y) % SCALE >= HALF_SCALE. Without this, 6.6e-19 would be truncated to 0 instead of
/// being rounded to 1e-18. See "Listing 6" and text above it at https://accu.org/index.php/journals/1717.
///
/// Requirements:
/// - The result must fit within uint256.
///
/// Caveats:
/// - The body is purposely left uncommented; see the NatSpec comments in "PRBMathCommon.mulDiv" to understand how this works.
/// - It is assumed that the result can never be type(uint256).max when x and y solve the following two queations:
/// 1. x * y = type(uint256).max * SCALE
/// 2. (x * y) % SCALE >= SCALE / 2
///
/// @param x The multiplicand as an unsigned 60.18-decimal fixed-point number.
/// @param y The multiplier as an unsigned 60.18-decimal fixed-point number.
/// @return result The result as an unsigned 60.18-decimal fixed-point number.
function mulDivFixedPoint(uint256 x, uint256 y) internal pure returns (uint256 result) {
uint256 prod0;
uint256 prod1;
assembly {
let mm := mulmod(x, y, not(0))
prod0 := mul(x, y)
prod1 := sub(sub(mm, prod0), lt(mm, prod0))
}
uint256 remainder;
uint256 roundUpUnit;
assembly {
remainder := mulmod(x, y, SCALE)
roundUpUnit := gt(remainder, 499999999999999999)
}
if (prod1 == 0) {
unchecked {
result = (prod0 / SCALE) + roundUpUnit;
return result;
}
}
require(SCALE > prod1);
assembly {
result := add(
mul(
or(
div(sub(prod0, remainder), SCALE_LPOTD),
mul(sub(prod1, gt(remainder, prod0)), add(div(sub(0, SCALE_LPOTD), SCALE_LPOTD), 1))
),
SCALE_INVERSE
),
roundUpUnit
)
}
}
/// @notice Calculates the square root of x, rounding down.
/// @dev Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method.
///
/// Caveats:
/// - This function does not work with fixed-point numbers.
///
/// @param x The uint256 number for which to calculate the square root.
/// @return result The result as an uint256.
function sqrt(uint256 x) internal pure returns (uint256 result) {
if (x == 0) {
return 0;
}
// Calculate the square root of the perfect square of a power of two that is the closest to x.
uint256 xAux = uint256(x);
result = 1;
if (xAux >= 0x100000000000000000000000000000000) {
xAux >>= 128;
result <<= 64;
}
if (xAux >= 0x10000000000000000) {
xAux >>= 64;
result <<= 32;
}
if (xAux >= 0x100000000) {
xAux >>= 32;
result <<= 16;
}
if (xAux >= 0x10000) {
xAux >>= 16;
result <<= 8;
}
if (xAux >= 0x100) {
xAux >>= 8;
result <<= 4;
}
if (xAux >= 0x10) {
xAux >>= 4;
result <<= 2;
}
if (xAux >= 0x8) {
result <<= 1;
}
// The operations can never overflow because the result is max 2^127 when it enters this block.
unchecked {
result = (result + x / result) >> 1;
result = (result + x / result) >> 1;
result = (result + x / result) >> 1;
result = (result + x / result) >> 1;
result = (result + x / result) >> 1;
result = (result + x / result) >> 1;
result = (result + x / result) >> 1; // Seven iterations should be enough
uint256 roundedDownResult = x / result;
return result >= roundedDownResult ? roundedDownResult : result;
}
}
}

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// SPDX-License-Identifier: WTFPL
pragma solidity >=0.8.0;
import "./PRBMathCommon.sol";
/// @title PRBMathSD59x18
/// @author Paul Razvan Berg
/// @notice Smart contract library for advanced fixed-point math. It works with int256 numbers considered to have 18
/// trailing decimals. We call this number representation signed 59.18-decimal fixed-point, since the numbers can have
/// a sign and there can be up to 59 digits in the integer part and up to 18 decimals in the fractional part. The numbers
/// are bound by the minimum and the maximum values permitted by the Solidity type int256.
library PRBMathSD59x18 {
/// @dev log2(e) as a signed 59.18-decimal fixed-point number.
int256 internal constant LOG2_E = 1442695040888963407;
/// @dev Half the SCALE number.
int256 internal constant HALF_SCALE = 5e17;
/// @dev The maximum value a signed 59.18-decimal fixed-point number can have.
int256 internal constant MAX_SD59x18 = 57896044618658097711785492504343953926634992332820282019728792003956564819967;
/// @dev The maximum whole value a signed 59.18-decimal fixed-point number can have.
int256 internal constant MAX_WHOLE_SD59x18 = 57896044618658097711785492504343953926634992332820282019728000000000000000000;
/// @dev The minimum value a signed 59.18-decimal fixed-point number can have.
int256 internal constant MIN_SD59x18 = -57896044618658097711785492504343953926634992332820282019728792003956564819968;
/// @dev The minimum whole value a signed 59.18-decimal fixed-point number can have.
int256 internal constant MIN_WHOLE_SD59x18 = -57896044618658097711785492504343953926634992332820282019728000000000000000000;
/// @dev How many trailing decimals can be represented.
int256 internal constant SCALE = 1e18;
/// INTERNAL FUNCTIONS ///
/// @notice Calculate the absolute value of x.
///
/// @dev Requirements:
/// - x must be greater than MIN_SD59x18.
///
/// @param x The number to calculate the absolute value for.
/// @param result The absolute value of x.
function abs(int256 x) internal pure returns (int256 result) {
unchecked {
require(x > MIN_SD59x18);
result = x < 0 ? -x : x;
}
}
/// @notice Calculates arithmetic average of x and y, rounding down.
/// @param x The first operand as a signed 59.18-decimal fixed-point number.
/// @param y The second operand as a signed 59.18-decimal fixed-point number.
/// @return result The arithmetic average as a signed 59.18-decimal fixed-point number.
function avg(int256 x, int256 y) internal pure returns (int256 result) {
// The operations can never overflow.
unchecked {
// The last operand checks if both x and y are odd and if that is the case, we add 1 to the result. We need
// to do this because if both numbers are odd, the 0.5 remainder gets truncated twice.
result = (x >> 1) + (y >> 1) + (x & y & 1);
}
}
/// @notice Yields the least greatest signed 59.18 decimal fixed-point number greater than or equal to x.
///
/// @dev Optimised for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts.
/// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions.
///
/// Requirements:
/// - x must be less than or equal to MAX_WHOLE_SD59x18.
///
/// @param x The signed 59.18-decimal fixed-point number to ceil.
/// @param result The least integer greater than or equal to x, as a signed 58.18-decimal fixed-point number.
function ceil(int256 x) internal pure returns (int256 result) {
require(x <= MAX_WHOLE_SD59x18);
unchecked {
int256 remainder = x % SCALE;
if (remainder == 0) {
result = x;
} else {
// Solidity uses C fmod style, which returns a modulus with the same sign as x.
result = x - remainder;
if (x > 0) {
result += SCALE;
}
}
}
}
/// @notice Divides two signed 59.18-decimal fixed-point numbers, returning a new signed 59.18-decimal fixed-point number.
///
/// @dev Variant of "mulDiv" that works with signed numbers. Works by computing the signs and the absolute values separately.
///
/// Requirements:
/// - All from "PRBMathCommon.mulDiv".
/// - None of the inputs can be type(int256).min.
/// - y cannot be zero.
/// - The result must fit within int256.
///
/// Caveats:
/// - All from "PRBMathCommon.mulDiv".
///
/// @param x The numerator as a signed 59.18-decimal fixed-point number.
/// @param y The denominator as a signed 59.18-decimal fixed-point number.
/// @param result The quotient as a signed 59.18-decimal fixed-point number.
function div(int256 x, int256 y) internal pure returns (int256 result) {
require(x > type(int256).min);
require(y > type(int256).min);
// Get hold of the absolute values of x and y.
uint256 ax;
uint256 ay;
unchecked {
ax = x < 0 ? uint256(-x) : uint256(x);
ay = y < 0 ? uint256(-y) : uint256(y);
}
// Compute the absolute value of (x*SCALE)÷y. The result must fit within int256.
uint256 resultUnsigned = PRBMathCommon.mulDiv(ax, uint256(SCALE), ay);
require(resultUnsigned <= uint256(type(int256).max));
// Get the signs of x and y.
uint256 sx;
uint256 sy;
assembly {
sx := sgt(x, sub(0, 1))
sy := sgt(y, sub(0, 1))
}
// XOR over sx and sy. This is basically checking whether the inputs have the same sign. If yes, the result
// should be positive. Otherwise, it should be negative.
result = sx ^ sy == 1 ? -int256(resultUnsigned) : int256(resultUnsigned);
}
/// @notice Returns Euler's number as a signed 59.18-decimal fixed-point number.
/// @dev See https://en.wikipedia.org/wiki/E_(mathematical_constant).
function e() internal pure returns (int256 result) {
result = 2718281828459045235;
}
/// @notice Calculates the natural exponent of x.
///
/// @dev Based on the insight that e^x = 2^(x * log2(e)).
///
/// Requirements:
/// - All from "log2".
/// - x must be less than 88722839111672999628.
///
/// @param x The exponent as a signed 59.18-decimal fixed-point number.
/// @return result The result as a signed 59.18-decimal fixed-point number.
function exp(int256 x) internal pure returns (int256 result) {
// Without this check, the value passed to "exp2" would be less than -59794705707972522261.
if (x < -41446531673892822322) {
return 0;
}
// Without this check, the value passed to "exp2" would be greater than 128e18.
require(x < 88722839111672999628);
// Do the fixed-point multiplication inline to save gas.
unchecked {
int256 doubleScaleProduct = x * LOG2_E;
result = exp2((doubleScaleProduct + HALF_SCALE) / SCALE);
}
}
/// @notice Calculates the binary exponent of x using the binary fraction method.
///
/// @dev See https://ethereum.stackexchange.com/q/79903/24693.
///
/// Requirements:
/// - x must be 128e18 or less.
/// - The result must fit within MAX_SD59x18.
///
/// Caveats:
/// - For any x less than -59794705707972522261, the result is zero.
///
/// @param x The exponent as a signed 59.18-decimal fixed-point number.
/// @return result The result as a signed 59.18-decimal fixed-point number.
function exp2(int256 x) internal pure returns (int256 result) {
// This works because 2^(-x) = 1/2^x.
if (x < 0) {
// 2**59.794705707972522262 is the maximum number whose inverse does not turn into zero.
if (x < -59794705707972522261) {
return 0;
}
// Do the fixed-point inversion inline to save gas. The numerator is SCALE * SCALE.
unchecked { result = 1e36 / exp2(-x); }
} else {
// 2**128 doesn't fit within the 128.128-bit fixed-point representation.
require(x < 128e18);
unchecked {
// Convert x to the 128.128-bit fixed-point format.
uint256 x128x128 = (uint256(x) << 128) / uint256(SCALE);
// Safe to convert the result to int256 directly because the maximum input allowed is 128e18.
result = int256(PRBMathCommon.exp2(x128x128));
}
}
}
/// @notice Yields the greatest signed 59.18 decimal fixed-point number less than or equal to x.
///
/// @dev Optimised for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts.
/// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions.
///
/// Requirements:
/// - x must be greater than or equal to MIN_WHOLE_SD59x18.
///
/// @param x The signed 59.18-decimal fixed-point number to floor.
/// @param result The greatest integer less than or equal to x, as a signed 58.18-decimal fixed-point number.
function floor(int256 x) internal pure returns (int256 result) {
require(x >= MIN_WHOLE_SD59x18);
unchecked {
int256 remainder = x % SCALE;
if (remainder == 0) {
result = x;
} else {
// Solidity uses C fmod style, which returns a modulus with the same sign as x.
result = x - remainder;
if (x < 0) {
result -= SCALE;
}
}
}
}
/// @notice Yields the excess beyond the floor of x for positive numbers and the part of the number to the right
/// of the radix point for negative numbers.
/// @dev Based on the odd function definition. https://en.wikipedia.org/wiki/Fractional_part
/// @param x The signed 59.18-decimal fixed-point number to get the fractional part of.
/// @param result The fractional part of x as a signed 59.18-decimal fixed-point number.
function frac(int256 x) internal pure returns (int256 result) {
unchecked { result = x % SCALE; }
}
/// @notice Calculates geometric mean of x and y, i.e. sqrt(x * y), rounding down.
///
/// @dev Requirements:
/// - x * y must fit within MAX_SD59x18, lest it overflows.
/// - x * y cannot be negative.
///
/// @param x The first operand as a signed 59.18-decimal fixed-point number.
/// @param y The second operand as a signed 59.18-decimal fixed-point number.
/// @return result The result as a signed 59.18-decimal fixed-point number.
function gm(int256 x, int256 y) internal pure returns (int256 result) {
if (x == 0) {
return 0;
}
unchecked {
// Checking for overflow this way is faster than letting Solidity do it.
int256 xy = x * y;
require(xy / x == y);
// The product cannot be negative.
require(xy >= 0);
// We don't need to multiply by the SCALE here because the x*y product had already picked up a factor of SCALE
// during multiplication. See the comments within the "sqrt" function.
result = int256(PRBMathCommon.sqrt(uint256(xy)));
}
}
/// @notice Calculates 1 / x, rounding towards zero.
///
/// @dev Requirements:
/// - x cannot be zero.
///
/// @param x The signed 59.18-decimal fixed-point number for which to calculate the inverse.
/// @return result The inverse as a signed 59.18-decimal fixed-point number.
function inv(int256 x) internal pure returns (int256 result) {
unchecked {
// 1e36 is SCALE * SCALE.
result = 1e36 / x;
}
}
/// @notice Calculates the natural logarithm of x.
///
/// @dev Based on the insight that ln(x) = log2(x) / log2(e).
///
/// Requirements:
/// - All from "log2".
///
/// Caveats:
/// - All from "log2".
/// - This doesn't return exactly 1 for 2718281828459045235, for that we would need more fine-grained precision.
///
/// @param x The signed 59.18-decimal fixed-point number for which to calculate the natural logarithm.
/// @return result The natural logarithm as a signed 59.18-decimal fixed-point number.
function ln(int256 x) internal pure returns (int256 result) {
// Do the fixed-point multiplication inline to save gas. This is overflow-safe because the maximum value that log2(x)
// can return is 195205294292027477728.
unchecked { result = (log2(x) * SCALE) / LOG2_E; }
}
/// @notice Calculates the common logarithm of x.
///
/// @dev First checks if x is an exact power of ten and it stops if yes. If it's not, calculates the common
/// logarithm based on the insight that log10(x) = log2(x) / log2(10).
///
/// Requirements:
/// - All from "log2".
///
/// Caveats:
/// - All from "log2".
///
/// @param x The signed 59.18-decimal fixed-point number for which to calculate the common logarithm.
/// @return result The common logarithm as a signed 59.18-decimal fixed-point number.
function log10(int256 x) internal pure returns (int256 result) {
require(x > 0);
// Note that the "mul" in this block is the assembly mul operation, not the "mul" function defined in this contract.
// prettier-ignore
assembly {
switch x
case 1 { result := mul(SCALE, sub(0, 18)) }
case 10 { result := mul(SCALE, sub(1, 18)) }
case 100 { result := mul(SCALE, sub(2, 18)) }
case 1000 { result := mul(SCALE, sub(3, 18)) }
case 10000 { result := mul(SCALE, sub(4, 18)) }
case 100000 { result := mul(SCALE, sub(5, 18)) }
case 1000000 { result := mul(SCALE, sub(6, 18)) }
case 10000000 { result := mul(SCALE, sub(7, 18)) }
case 100000000 { result := mul(SCALE, sub(8, 18)) }
case 1000000000 { result := mul(SCALE, sub(9, 18)) }
case 10000000000 { result := mul(SCALE, sub(10, 18)) }
case 100000000000 { result := mul(SCALE, sub(11, 18)) }
case 1000000000000 { result := mul(SCALE, sub(12, 18)) }
case 10000000000000 { result := mul(SCALE, sub(13, 18)) }
case 100000000000000 { result := mul(SCALE, sub(14, 18)) }
case 1000000000000000 { result := mul(SCALE, sub(15, 18)) }
case 10000000000000000 { result := mul(SCALE, sub(16, 18)) }
case 100000000000000000 { result := mul(SCALE, sub(17, 18)) }
case 1000000000000000000 { result := 0 }
case 10000000000000000000 { result := SCALE }
case 100000000000000000000 { result := mul(SCALE, 2) }
case 1000000000000000000000 { result := mul(SCALE, 3) }
case 10000000000000000000000 { result := mul(SCALE, 4) }
case 100000000000000000000000 { result := mul(SCALE, 5) }
case 1000000000000000000000000 { result := mul(SCALE, 6) }
case 10000000000000000000000000 { result := mul(SCALE, 7) }
case 100000000000000000000000000 { result := mul(SCALE, 8) }
case 1000000000000000000000000000 { result := mul(SCALE, 9) }
case 10000000000000000000000000000 { result := mul(SCALE, 10) }
case 100000000000000000000000000000 { result := mul(SCALE, 11) }
case 1000000000000000000000000000000 { result := mul(SCALE, 12) }
case 10000000000000000000000000000000 { result := mul(SCALE, 13) }
case 100000000000000000000000000000000 { result := mul(SCALE, 14) }
case 1000000000000000000000000000000000 { result := mul(SCALE, 15) }
case 10000000000000000000000000000000000 { result := mul(SCALE, 16) }
case 100000000000000000000000000000000000 { result := mul(SCALE, 17) }
case 1000000000000000000000000000000000000 { result := mul(SCALE, 18) }
case 10000000000000000000000000000000000000 { result := mul(SCALE, 19) }
case 100000000000000000000000000000000000000 { result := mul(SCALE, 20) }
case 1000000000000000000000000000000000000000 { result := mul(SCALE, 21) }
case 10000000000000000000000000000000000000000 { result := mul(SCALE, 22) }
case 100000000000000000000000000000000000000000 { result := mul(SCALE, 23) }
case 1000000000000000000000000000000000000000000 { result := mul(SCALE, 24) }
case 10000000000000000000000000000000000000000000 { result := mul(SCALE, 25) }
case 100000000000000000000000000000000000000000000 { result := mul(SCALE, 26) }
case 1000000000000000000000000000000000000000000000 { result := mul(SCALE, 27) }
case 10000000000000000000000000000000000000000000000 { result := mul(SCALE, 28) }
case 100000000000000000000000000000000000000000000000 { result := mul(SCALE, 29) }
case 1000000000000000000000000000000000000000000000000 { result := mul(SCALE, 30) }
case 10000000000000000000000000000000000000000000000000 { result := mul(SCALE, 31) }
case 100000000000000000000000000000000000000000000000000 { result := mul(SCALE, 32) }
case 1000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 33) }
case 10000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 34) }
case 100000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 35) }
case 1000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 36) }
case 10000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 37) }
case 100000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 38) }
case 1000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 39) }
case 10000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 40) }
case 100000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 41) }
case 1000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 42) }
case 10000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 43) }
case 100000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 44) }
case 1000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 45) }
case 10000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 46) }
case 100000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 47) }
case 1000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 48) }
case 10000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 49) }
case 100000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 50) }
case 1000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 51) }
case 10000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 52) }
case 100000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 53) }
case 1000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 54) }
case 10000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 55) }
case 100000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 56) }
case 1000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 57) }
case 10000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 58) }
default {
result := MAX_SD59x18
}
}
if (result == MAX_SD59x18) {
// Do the fixed-point division inline to save gas. The denominator is log2(10).
unchecked { result = (log2(x) * SCALE) / 332192809488736234; }
}
}
/// @notice Calculates the binary logarithm of x.
///
/// @dev Based on the iterative approximation algorithm.
/// https://en.wikipedia.org/wiki/Binary_logarithm#Iterative_approximation
///
/// Requirements:
/// - x must be greater than zero.
///
/// Caveats:
/// - The results are nor perfectly accurate to the last digit, due to the lossy precision of the iterative approximation.
///
/// @param x The signed 59.18-decimal fixed-point number for which to calculate the binary logarithm.
/// @return result The binary logarithm as a signed 59.18-decimal fixed-point number.
function log2(int256 x) internal pure returns (int256 result) {
require(x > 0);
unchecked {
// This works because log2(x) = -log2(1/x).
int256 sign;
if (x >= SCALE) {
sign = 1;
} else {
sign = -1;
// Do the fixed-point inversion inline to save gas. The numerator is SCALE * SCALE.
assembly {
x := div(1000000000000000000000000000000000000, x)
}
}
// Calculate the integer part of the logarithm and add it to the result and finally calculate y = x * 2^(-n).
uint256 n = PRBMathCommon.mostSignificantBit(uint256(x / SCALE));
// The integer part of the logarithm as a signed 59.18-decimal fixed-point number. The operation can't overflow
// because n is maximum 255, SCALE is 1e18 and sign is either 1 or -1.
result = int256(n) * SCALE;
// This is y = x * 2^(-n).
int256 y = x >> n;
// If y = 1, the fractional part is zero.
if (y == SCALE) {
return result * sign;
}
// Calculate the fractional part via the iterative approximation.
// The "delta >>= 1" part is equivalent to "delta /= 2", but shifting bits is faster.
for (int256 delta = int256(HALF_SCALE); delta > 0; delta >>= 1) {
y = (y * y) / SCALE;
// Is y^2 > 2 and so in the range [2,4)?
if (y >= 2 * SCALE) {
// Add the 2^(-m) factor to the logarithm.
result += delta;
// Corresponds to z/2 on Wikipedia.
y >>= 1;
}
}
result *= sign;
}
}
/// @notice Multiplies two signed 59.18-decimal fixed-point numbers together, returning a new signed 59.18-decimal
/// fixed-point number.
///
/// @dev Variant of "mulDiv" that works with signed numbers and employs constant folding, i.e. the denominator is
/// alawys 1e18.
///
/// Requirements:
/// - All from "PRBMathCommon.mulDivFixedPoint".
/// - The result must fit within MAX_SD59x18.
///
/// Caveats:
/// - The body is purposely left uncommented; see the NatSpec comments in "PRBMathCommon.mulDiv" to understand how this works.
///
/// @param x The multiplicand as a signed 59.18-decimal fixed-point number.
/// @param y The multiplier as a signed 59.18-decimal fixed-point number.
/// @return result The result as a signed 59.18-decimal fixed-point number.
function mul(int256 x, int256 y) internal pure returns (int256 result) {
require(x > MIN_SD59x18);
require(y > MIN_SD59x18);
unchecked {
uint256 ax;
uint256 ay;
ax = x < 0 ? uint256(-x) : uint256(x);
ay = y < 0 ? uint256(-y) : uint256(y);
uint256 resultUnsigned = PRBMathCommon.mulDivFixedPoint(ax, ay);
require(resultUnsigned <= uint256(MAX_SD59x18));
uint256 sx;
uint256 sy;
assembly {
sx := sgt(x, sub(0, 1))
sy := sgt(y, sub(0, 1))
}
result = sx ^ sy == 1 ? -int256(resultUnsigned) : int256(resultUnsigned);
}
}
/// @notice Retrieves PI as a signed 59.18-decimal fixed-point number.
function pi() internal pure returns (int256 result) {
result = 3141592653589793238;
}
/// @notice Raises x (signed 59.18-decimal fixed-point number) to the power of y (basic unsigned integer) using the
/// famous algorithm "exponentiation by squaring".
///
/// @dev See https://en.wikipedia.org/wiki/Exponentiation_by_squaring
///
/// Requirements:
/// - All from "abs" and "PRBMathCommon.mulDivFixedPoint".
/// - The result must fit within MAX_SD59x18.
///
/// Caveats:
/// - All from "PRBMathCommon.mulDivFixedPoint".
/// - Assumes 0^0 is 1.
///
/// @param x The base as a signed 59.18-decimal fixed-point number.
/// @param y The exponent as an uint256.
/// @return result The result as a signed 59.18-decimal fixed-point number.
function pow(int256 x, uint256 y) internal pure returns (int256 result) {
uint256 absX = uint256(abs(x));
// Calculate the first iteration of the loop in advance.
uint256 absResult = y & 1 > 0 ? absX : uint256(SCALE);
// Equivalent to "for(y /= 2; y > 0; y /= 2)" but faster.
for (y >>= 1; y > 0; y >>= 1) {
absX = PRBMathCommon.mulDivFixedPoint(absX, absX);
// Equivalent to "y % 2 == 1" but faster.
if (y & 1 > 0) {
absResult = PRBMathCommon.mulDivFixedPoint(absResult, absX);
}
}
// The result must fit within the 59.18-decimal fixed-point representation.
require(absResult <= uint256(MAX_SD59x18));
// Is the base negative and the exponent an odd number?
bool isNegative = x < 0 && y & 1 == 1;
result = isNegative ? -int256(absResult) : int256(absResult);
}
/// @notice Returns 1 as a signed 59.18-decimal fixed-point number.
function scale() internal pure returns (int256 result) {
result = SCALE;
}
/// @notice Calculates the square root of x, rounding down.
/// @dev Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method.
///
/// Requirements:
/// - x cannot be negative.
/// - x must be less than MAX_SD59x18 / SCALE.
///
/// Caveats:
/// - The maximum fixed-point number permitted is 57896044618658097711785492504343953926634.992332820282019729.
///
/// @param x The signed 59.18-decimal fixed-point number for which to calculate the square root.
/// @return result The result as a signed 59.18-decimal fixed-point .
function sqrt(int256 x) internal pure returns (int256 result) {
require(x >= 0);
require(x < 57896044618658097711785492504343953926634992332820282019729);
unchecked {
// Multiply x by the SCALE to account for the factor of SCALE that is picked up when multiplying two signed
// 59.18-decimal fixed-point numbers together (in this case, those two numbers are both the square root).
result = int256(PRBMathCommon.sqrt(uint256(x * SCALE)));
}
}
}

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test/libsolidity/semanticTests/externalContracts/_prbmath/PRBMathUD60x18.sol Normal file
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// SPDX-License-Identifier: WTFPL
pragma solidity >=0.8.0;
import "./PRBMathCommon.sol";
/// @title PRBMathUD60x18
/// @author Paul Razvan Berg
/// @notice Smart contract library for advanced fixed-point math. It works with uint256 numbers considered to have 18
/// trailing decimals. We call this number representation unsigned 60.18-decimal fixed-point, since there can be up to 60
/// digits in the integer part and up to 18 decimals in the fractional part. The numbers are bound by the minimum and the
/// maximum values permitted by the Solidity type uint256.
library PRBMathUD60x18 {
/// @dev Half the SCALE number.
uint256 internal constant HALF_SCALE = 5e17;
/// @dev log2(e) as an unsigned 60.18-decimal fixed-point number.
uint256 internal constant LOG2_E = 1442695040888963407;
/// @dev The maximum value an unsigned 60.18-decimal fixed-point number can have.
uint256 internal constant MAX_UD60x18 = 115792089237316195423570985008687907853269984665640564039457584007913129639935;
/// @dev The maximum whole value an unsigned 60.18-decimal fixed-point number can have.
uint256 internal constant MAX_WHOLE_UD60x18 = 115792089237316195423570985008687907853269984665640564039457000000000000000000;
/// @dev How many trailing decimals can be represented.
uint256 internal constant SCALE = 1e18;
/// @notice Calculates arithmetic average of x and y, rounding down.
/// @param x The first operand as an unsigned 60.18-decimal fixed-point number.
/// @param y The second operand as an unsigned 60.18-decimal fixed-point number.
/// @return result The arithmetic average as an usigned 60.18-decimal fixed-point number.
function avg(uint256 x, uint256 y) internal pure returns (uint256 result) {
// The operations can never overflow.
unchecked {
// The last operand checks if both x and y are odd and if that is the case, we add 1 to the result. We need
// to do this because if both numbers are odd, the 0.5 remainder gets truncated twice.
result = (x >> 1) + (y >> 1) + (x & y & 1);
}
}
/// @notice Yields the least unsigned 60.18 decimal fixed-point number greater than or equal to x.
///
/// @dev Optimised for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts.
/// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions.
///
/// Requirements:
/// - x must be less than or equal to MAX_WHOLE_UD60x18.
///
/// @param x The unsigned 60.18-decimal fixed-point number to ceil.
/// @param result The least integer greater than or equal to x, as an unsigned 60.18-decimal fixed-point number.
function ceil(uint256 x) internal pure returns (uint256 result) {
require(x <= MAX_WHOLE_UD60x18);
assembly {
// Equivalent to "x % SCALE" but faster.
let remainder := mod(x, SCALE)
// Equivalent to "SCALE - remainder" but faster.
let delta := sub(SCALE, remainder)
// Equivalent to "x + delta * (remainder > 0 ? 1 : 0)" but faster.
result := add(x, mul(delta, gt(remainder, 0)))
}
}
/// @notice Divides two unsigned 60.18-decimal fixed-point numbers, returning a new unsigned 60.18-decimal fixed-point number.
///
/// @dev Uses mulDiv to enable overflow-safe multiplication and division.
///
/// Requirements:
/// - y cannot be zero.
///
/// @param x The numerator as an unsigned 60.18-decimal fixed-point number.
/// @param y The denominator as an unsigned 60.18-decimal fixed-point number.
/// @param result The quotient as an unsigned 60.18-decimal fixed-point number.
function div(uint256 x, uint256 y) internal pure returns (uint256 result) {
result = PRBMathCommon.mulDiv(x, SCALE, y);
}
/// @notice Returns Euler's number as an unsigned 60.18-decimal fixed-point number.
/// @dev See https://en.wikipedia.org/wiki/E_(mathematical_constant).
function e() internal pure returns (uint256 result) {
result = 2718281828459045235;
}
/// @notice Calculates the natural exponent of x.
///
/// @dev Based on the insight that e^x = 2^(x * log2(e)).
///
/// Requirements:
/// - All from "log2".
/// - x must be less than 88722839111672999628.
///
/// @param x The exponent as an unsigned 60.18-decimal fixed-point number.
/// @return result The result as an unsigned 60.18-decimal fixed-point number.
function exp(uint256 x) internal pure returns (uint256 result) {
// Without this check, the value passed to "exp2" would be greater than 128e18.
require(x < 88722839111672999628);
// Do the fixed-point multiplication inline to save gas.
unchecked {
uint256 doubleScaleProduct = x * LOG2_E;
result = exp2((doubleScaleProduct + HALF_SCALE) / SCALE);
}
}
/// @notice Calculates the binary exponent of x using the binary fraction method.
///
/// @dev See https://ethereum.stackexchange.com/q/79903/24693.
///
/// Requirements:
/// - x must be 128e18 or less.
/// - The result must fit within MAX_UD60x18.
///
/// @param x The exponent as an unsigned 60.18-decimal fixed-point number.
/// @return result The result as an unsigned 60.18-decimal fixed-point number.
function exp2(uint256 x) internal pure returns (uint256 result) {
// 2**128 doesn't fit within the 128.128-bit format used internally in this function.
require(x < 128e18);
unchecked {
// Convert x to the 128.128-bit fixed-point format.
uint256 x128x128 = (x << 128) / SCALE;
// Pass x to the PRBMathCommon.exp2 function, which uses the 128.128-bit fixed-point number representation.
result = PRBMathCommon.exp2(x128x128);
}
}
/// @notice Yields the greatest unsigned 60.18 decimal fixed-point number less than or equal to x.
/// @dev Optimised for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts.
/// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions.
/// @param x The unsigned 60.18-decimal fixed-point number to floor.
/// @param result The greatest integer less than or equal to x, as an unsigned 60.18-decimal fixed-point number.
function floor(uint256 x) internal pure returns (uint256 result) {
assembly {
// Equivalent to "x % SCALE" but faster.
let remainder := mod(x, SCALE)
// Equivalent to "x - remainder * (remainder > 0 ? 1 : 0)" but faster.
result := sub(x, mul(remainder, gt(remainder, 0)))
}
}
/// @notice Yields the excess beyond the floor of x.
/// @dev Based on the odd function definition https://en.wikipedia.org/wiki/Fractional_part.
/// @param x The unsigned 60.18-decimal fixed-point number to get the fractional part of.
/// @param result The fractional part of x as an unsigned 60.18-decimal fixed-point number.
function frac(uint256 x) internal pure returns (uint256 result) {
assembly {
result := mod(x, SCALE)
}
}
/// @notice Calculates geometric mean of x and y, i.e. sqrt(x * y), rounding down.
///
/// @dev Requirements:
/// - x * y must fit within MAX_UD60x18, lest it overflows.
///
/// @param x The first operand as an unsigned 60.18-decimal fixed-point number.
/// @param y The second operand as an unsigned 60.18-decimal fixed-point number.
/// @return result The result as an unsigned 60.18-decimal fixed-point number.
function gm(uint256 x, uint256 y) internal pure returns (uint256 result) {
if (x == 0) {
return 0;
}
unchecked {
// Checking for overflow this way is faster than letting Solidity do it.
uint256 xy = x * y;
require(xy / x == y);
// We don't need to multiply by the SCALE here because the x*y product had already picked up a factor of SCALE
// during multiplication. See the comments within the "sqrt" function.
result = PRBMathCommon.sqrt(xy);
}
}
/// @notice Calculates 1 / x, rounding towards zero.
///
/// @dev Requirements:
/// - x cannot be zero.
///
/// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the inverse.
/// @return result The inverse as an unsigned 60.18-decimal fixed-point number.
function inv(uint256 x) internal pure returns (uint256 result) {
unchecked {
// 1e36 is SCALE * SCALE.
result = 1e36 / x;
}
}
/// @notice Calculates the natural logarithm of x.
///
/// @dev Based on the insight that ln(x) = log2(x) / log2(e).
///
/// Requirements:
/// - All from "log2".
///
/// Caveats:
/// - All from "log2".
/// - This doesn't return exactly 1 for 2718281828459045235, for that we would need more fine-grained precision.
///
/// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the natural logarithm.
/// @return result The natural logarithm as an unsigned 60.18-decimal fixed-point number.
function ln(uint256 x) internal pure returns (uint256 result) {
// Do the fixed-point multiplication inline to save gas. This is overflow-safe because the maximum value that log2(x)
// can return is 196205294292027477728.
unchecked { result = (log2(x) * SCALE) / LOG2_E; }
}
/// @notice Calculates the common logarithm of x.
///
/// @dev First checks if x is an exact power of ten and it stops if yes. If it's not, calculates the common
/// logarithm based on the insight that log10(x) = log2(x) / log2(10).
///
/// Requirements:
/// - All from "log2".
///
/// Caveats:
/// - All from "log2".
///
/// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the common logarithm.
/// @return result The common logarithm as an unsigned 60.18-decimal fixed-point number.
function log10(uint256 x) internal pure returns (uint256 result) {
require(x >= SCALE);
// Note that the "mul" in this block is the assembly mul operation, not the "mul" function defined in this contract.
// prettier-ignore
assembly {
switch x
case 1 { result := mul(SCALE, sub(0, 18)) }
case 10 { result := mul(SCALE, sub(1, 18)) }
case 100 { result := mul(SCALE, sub(2, 18)) }
case 1000 { result := mul(SCALE, sub(3, 18)) }
case 10000 { result := mul(SCALE, sub(4, 18)) }
case 100000 { result := mul(SCALE, sub(5, 18)) }
case 1000000 { result := mul(SCALE, sub(6, 18)) }
case 10000000 { result := mul(SCALE, sub(7, 18)) }
case 100000000 { result := mul(SCALE, sub(8, 18)) }
case 1000000000 { result := mul(SCALE, sub(9, 18)) }
case 10000000000 { result := mul(SCALE, sub(10, 18)) }
case 100000000000 { result := mul(SCALE, sub(11, 18)) }
case 1000000000000 { result := mul(SCALE, sub(12, 18)) }
case 10000000000000 { result := mul(SCALE, sub(13, 18)) }
case 100000000000000 { result := mul(SCALE, sub(14, 18)) }
case 1000000000000000 { result := mul(SCALE, sub(15, 18)) }
case 10000000000000000 { result := mul(SCALE, sub(16, 18)) }
case 100000000000000000 { result := mul(SCALE, sub(17, 18)) }
case 1000000000000000000 { result := 0 }
case 10000000000000000000 { result := SCALE }
case 100000000000000000000 { result := mul(SCALE, 2) }
case 1000000000000000000000 { result := mul(SCALE, 3) }
case 10000000000000000000000 { result := mul(SCALE, 4) }
case 100000000000000000000000 { result := mul(SCALE, 5) }
case 1000000000000000000000000 { result := mul(SCALE, 6) }
case 10000000000000000000000000 { result := mul(SCALE, 7) }
case 100000000000000000000000000 { result := mul(SCALE, 8) }
case 1000000000000000000000000000 { result := mul(SCALE, 9) }
case 10000000000000000000000000000 { result := mul(SCALE, 10) }
case 100000000000000000000000000000 { result := mul(SCALE, 11) }
case 1000000000000000000000000000000 { result := mul(SCALE, 12) }
case 10000000000000000000000000000000 { result := mul(SCALE, 13) }
case 100000000000000000000000000000000 { result := mul(SCALE, 14) }
case 1000000000000000000000000000000000 { result := mul(SCALE, 15) }
case 10000000000000000000000000000000000 { result := mul(SCALE, 16) }
case 100000000000000000000000000000000000 { result := mul(SCALE, 17) }
case 1000000000000000000000000000000000000 { result := mul(SCALE, 18) }
case 10000000000000000000000000000000000000 { result := mul(SCALE, 19) }
case 100000000000000000000000000000000000000 { result := mul(SCALE, 20) }
case 1000000000000000000000000000000000000000 { result := mul(SCALE, 21) }
case 10000000000000000000000000000000000000000 { result := mul(SCALE, 22) }
case 100000000000000000000000000000000000000000 { result := mul(SCALE, 23) }
case 1000000000000000000000000000000000000000000 { result := mul(SCALE, 24) }
case 10000000000000000000000000000000000000000000 { result := mul(SCALE, 25) }
case 100000000000000000000000000000000000000000000 { result := mul(SCALE, 26) }
case 1000000000000000000000000000000000000000000000 { result := mul(SCALE, 27) }
case 10000000000000000000000000000000000000000000000 { result := mul(SCALE, 28) }
case 100000000000000000000000000000000000000000000000 { result := mul(SCALE, 29) }
case 1000000000000000000000000000000000000000000000000 { result := mul(SCALE, 30) }
case 10000000000000000000000000000000000000000000000000 { result := mul(SCALE, 31) }
case 100000000000000000000000000000000000000000000000000 { result := mul(SCALE, 32) }
case 1000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 33) }
case 10000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 34) }
case 100000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 35) }
case 1000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 36) }
case 10000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 37) }
case 100000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 38) }
case 1000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 39) }
case 10000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 40) }
case 100000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 41) }
case 1000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 42) }
case 10000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 43) }
case 100000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 44) }
case 1000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 45) }
case 10000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 46) }
case 100000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 47) }
case 1000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 48) }
case 10000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 49) }
case 100000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 50) }
case 1000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 51) }
case 10000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 52) }
case 100000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 53) }
case 1000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 54) }
case 10000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 55) }
case 100000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 56) }
case 1000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 57) }
case 10000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 58) }
case 100000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 59) }
default {
result := MAX_UD60x18
}
}
if (result == MAX_UD60x18) {
// Do the fixed-point division inline to save gas. The denominator is log2(10).
unchecked { result = (log2(x) * SCALE) / 332192809488736234; }
}
}
/// @notice Calculates the binary logarithm of x.
///
/// @dev Based on the iterative approximation algorithm.
/// https://en.wikipedia.org/wiki/Binary_logarithm#Iterative_approximation
///
/// Requirements:
/// - x must be greater than or equal to SCALE, otherwise the result would be negative.
///
/// Caveats:
/// - The results are nor perfectly accurate to the last digit, due to the lossy precision of the iterative approximation.
///
/// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the binary logarithm.
/// @return result The binary logarithm as an unsigned 60.18-decimal fixed-point number.
function log2(uint256 x) internal pure returns (uint256 result) {
require(x >= SCALE);
unchecked {
// Calculate the integer part of the logarithm and add it to the result and finally calculate y = x * 2^(-n).
uint256 n = PRBMathCommon.mostSignificantBit(x / SCALE);
// The integer part of the logarithm as an unsigned 60.18-decimal fixed-point number. The operation can't overflow
// because n is maximum 255 and SCALE is 1e18.
result = n * SCALE;
// This is y = x * 2^(-n).
uint256 y = x >> n;
// If y = 1, the fractional part is zero.
if (y == SCALE) {
return result;
}
// Calculate the fractional part via the iterative approximation.
// The "delta >>= 1" part is equivalent to "delta /= 2", but shifting bits is faster.
for (uint256 delta = HALF_SCALE; delta > 0; delta >>= 1) {
y = (y * y) / SCALE;
// Is y^2 > 2 and so in the range [2,4)?
if (y >= 2 * SCALE) {
// Add the 2^(-m) factor to the logarithm.
result += delta;
// Corresponds to z/2 on Wikipedia.
y >>= 1;
}
}
}
}
/// @notice Multiplies two unsigned 60.18-decimal fixed-point numbers together, returning a new unsigned 60.18-decimal
/// fixed-point number.
/// @dev See the documentation for the "PRBMathCommon.mulDivFixedPoint" function.
/// @param x The multiplicand as an unsigned 60.18-decimal fixed-point number.
/// @param y The multiplier as an unsigned 60.18-decimal fixed-point number.
/// @return result The result as an unsigned 60.18-decimal fixed-point number.
function mul(uint256 x, uint256 y) internal pure returns (uint256 result) {
result = PRBMathCommon.mulDivFixedPoint(x, y);
}
/// @notice Retrieves PI as an unsigned 60.18-decimal fixed-point number.
function pi() internal pure returns (uint256 result) {
result = 3141592653589793238;
}
/// @notice Raises x (unsigned 60.18-decimal fixed-point number) to the power of y (basic unsigned integer) using the
/// famous algorithm "exponentiation by squaring".
///
/// @dev See https://en.wikipedia.org/wiki/Exponentiation_by_squaring
///
/// Requirements:
/// - The result must fit within MAX_UD60x18.
///
/// Caveats:
/// - All from "mul".
/// - Assumes 0^0 is 1.
///
/// @param x The base as an unsigned 60.18-decimal fixed-point number.
/// @param y The exponent as an uint256.
/// @return result The result as an unsigned 60.18-decimal fixed-point number.
function pow(uint256 x, uint256 y) internal pure returns (uint256 result) {
// Calculate the first iteration of the loop in advance.
result = y & 1 > 0 ? x : SCALE;
// Equivalent to "for(y /= 2; y > 0; y /= 2)" but faster.
for (y >>= 1; y > 0; y >>= 1) {
x = PRBMathCommon.mulDivFixedPoint(x, x);
// Equivalent to "y % 2 == 1" but faster.
if (y & 1 > 0) {
result = PRBMathCommon.mulDivFixedPoint(result, x);
}
}
}
/// @notice Returns 1 as an unsigned 60.18-decimal fixed-point number.
function scale() internal pure returns (uint256 result) {
result = SCALE;
}
/// @notice Calculates the square root of x, rounding down.
/// @dev Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method.
///
/// Requirements:
/// - x must be less than MAX_UD60x18 / SCALE.
///
/// Caveats:
/// - The maximum fixed-point number permitted is 115792089237316195423570985008687907853269.984665640564039458.
///
/// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the square root.
/// @return result The result as an unsigned 60.18-decimal fixed-point .
function sqrt(uint256 x) internal pure returns (uint256 result) {
require(x < 115792089237316195423570985008687907853269984665640564039458);
unchecked {
// Multiply x by the SCALE to account for the factor of SCALE that is picked up when multiplying two unsigned
// 60.18-decimal fixed-point numbers together (in this case, those two numbers are both the square root).
result = PRBMathCommon.sqrt(x * SCALE);
}
}
}

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Imported from https://github.com/hifi-finance/prb-math/commit/62021c1abc3413f20d0bdc8f941cf9f21d5a7d2d