// SPDX-License-Identifier: BSD-4-Clause /* * ABDK Math 64.64 Smart Contract Library. Copyright © 2019 by ABDK Consulting. * Author: Mikhail Vladimirov */ pragma solidity ^0.8.13; /** * Smart contract library of mathematical functions operating with signed * 64.64-bit fixed point numbers. Signed 64.64-bit fixed point number is * basically a simple fraction whose numerator is signed 128-bit integer and * denominator is 2^64. As long as denominator is always the same, there is no * need to store it, thus in Solidity signed 64.64-bit fixed point numbers are * represented by int128 type holding only the numerator. */ library ABDKMath64x64 { /* * Minimum value signed 64.64-bit fixed point number may have. */ int128 public constant MIN_64x64 = -0x80000000000000000000000000000000; /* * Maximum value signed 64.64-bit fixed point number may have. */ int128 public constant MAX_64x64 = 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF; /** * Convert signed 256-bit integer number into signed 64.64-bit fixed point * number. Revert on overflow. * * @param x signed 256-bit integer number * @return signed 64.64-bit fixed point number */ function fromInt(int256 x) internal pure returns (int128) { unchecked { require(x >= -0x8000000000000000 && x <= 0x7FFFFFFFFFFFFFFF); return int128(x << 64); } } /** * Convert signed 64.64 fixed point number into signed 64-bit integer number * rounding down. * * @param x signed 64.64-bit fixed point number * @return signed 64-bit integer number */ function toInt(int128 x) internal pure returns (int64) { unchecked { return int64(x >> 64); } } /** * Convert unsigned 256-bit integer number into signed 64.64-bit fixed point * number. Revert on overflow. * * @param x unsigned 256-bit integer number * @return signed 64.64-bit fixed point number */ function fromUInt(uint256 x) internal pure returns (int128) { unchecked { require(x <= 0x7FFFFFFFFFFFFFFF); return int128(int256(x << 64)); } } /** * Convert signed 64.64 fixed point number into unsigned 64-bit integer * number rounding down. Revert on underflow. * * @param x signed 64.64-bit fixed point number * @return unsigned 64-bit integer number */ function toUInt(int128 x) internal pure returns (uint64) { unchecked { require(x >= 0); return uint64(uint128(x >> 64)); } } /** * Convert signed 128.128 fixed point number into signed 64.64-bit fixed point * number rounding down. Revert on overflow. * * @param x signed 128.128-bin fixed point number * @return signed 64.64-bit fixed point number */ function from128x128(int256 x) internal pure returns (int128) { unchecked { int256 result = x >> 64; require(result >= MIN_64x64 && result <= MAX_64x64); return int128(result); } } /** * Convert signed 64.64 fixed point number into signed 128.128 fixed point * number. * * @param x signed 64.64-bit fixed point number * @return signed 128.128 fixed point number */ function to128x128(int128 x) internal pure returns (int256) { unchecked { return int256(x) << 64; } } /** * Calculate x + y. Revert on overflow. * * @param x signed 64.64-bit fixed point number * @param y signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function add(int128 x, int128 y) internal pure returns (int128) { unchecked { int256 result = int256(x) + y; require(result >= MIN_64x64 && result <= MAX_64x64); return int128(result); } } /** * Calculate x - y. Revert on overflow. * * @param x signed 64.64-bit fixed point number * @param y signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function sub(int128 x, int128 y) internal pure returns (int128) { unchecked { int256 result = int256(x) - y; require(result >= MIN_64x64 && result <= MAX_64x64); return int128(result); } } /** * Calculate x * y rounding down. Revert on overflow. * * @param x signed 64.64-bit fixed point number * @param y signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function mul(int128 x, int128 y) internal pure returns (int128) { unchecked { int256 result = int256(x) * y >> 64; require(result >= MIN_64x64 && result <= MAX_64x64); return int128(result); } } /** * Calculate x * y rounding towards zero, where x is signed 64.64 fixed point * number and y is signed 256-bit integer number. Revert on overflow. * * @param x signed 64.64 fixed point number * @param y signed 256-bit integer number * @return signed 256-bit integer number */ function muli(int128 x, int256 y) internal pure returns (int256) { unchecked { if (x == MIN_64x64) { require( y >= -0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF && y <= 0x1000000000000000000000000000000000000000000000000 ); return -y << 63; } else { bool negativeResult = false; if (x < 0) { x = -x; negativeResult = true; } if (y < 0) { y = -y; // We rely on overflow behavior here negativeResult = !negativeResult; } uint256 absoluteResult = mulu(x, uint256(y)); if (negativeResult) { require(absoluteResult <= 0x8000000000000000000000000000000000000000000000000000000000000000); return -int256(absoluteResult); // We rely on overflow behavior here } else { require(absoluteResult <= 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF); return int256(absoluteResult); } } } } /** * Calculate x * y rounding down, where x is signed 64.64 fixed point number * and y is unsigned 256-bit integer number. Revert on overflow. * * @param x signed 64.64 fixed point number * @param y unsigned 256-bit integer number * @return unsigned 256-bit integer number */ function mulu(int128 x, uint256 y) internal pure returns (uint256) { unchecked { if (y == 0) return 0; require(x >= 0); uint256 lo = (uint256(int256(x)) * (y & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)) >> 64; uint256 hi = uint256(int256(x)) * (y >> 128); require(hi <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF); hi <<= 64; require(hi <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF - lo); return hi + lo; } } /** * Calculate x / y rounding towards zero. Revert on overflow or when y is * zero. * * @param x signed 64.64-bit fixed point number * @param y signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function div(int128 x, int128 y) internal pure returns (int128) { unchecked { require(y != 0); int256 result = (int256(x) << 64) / y; require(result >= MIN_64x64 && result <= MAX_64x64); return int128(result); } } /** * Calculate x / y rounding towards zero, where x and y are signed 256-bit * integer numbers. Revert on overflow or when y is zero. * * @param x signed 256-bit integer number * @param y signed 256-bit integer number * @return signed 64.64-bit fixed point number */ function divi(int256 x, int256 y) internal pure returns (int128) { unchecked { require(y != 0); bool negativeResult = false; if (x < 0) { x = -x; // We rely on overflow behavior here negativeResult = true; } if (y < 0) { y = -y; // We rely on overflow behavior here negativeResult = !negativeResult; } uint128 absoluteResult = divuu(uint256(x), uint256(y)); if (negativeResult) { require(absoluteResult <= 0x80000000000000000000000000000000); return -int128(absoluteResult); // We rely on overflow behavior here } else { require(absoluteResult <= 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF); return int128(absoluteResult); // We rely on overflow behavior here } } } /** * Calculate x / y rounding towards zero, where x and y are unsigned 256-bit * integer numbers. Revert on overflow or when y is zero. * * @param x unsigned 256-bit integer number * @param y unsigned 256-bit integer number * @return signed 64.64-bit fixed point number */ function divu(uint256 x, uint256 y) internal pure returns (int128) { unchecked { require(y != 0); uint128 result = divuu(x, y); require(result <= uint128(MAX_64x64)); return int128(result); } } /** * Calculate -x. Revert on overflow. * * @param x signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function neg(int128 x) internal pure returns (int128) { unchecked { require(x != MIN_64x64); return -x; } } /** * Calculate |x|. Revert on overflow. * * @param x signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function abs(int128 x) internal pure returns (int128) { unchecked { require(x != MIN_64x64); return x < 0 ? -x : x; } } /** * Calculate 1 / x rounding towards zero. Revert on overflow or when x is * zero. * * @param x signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function inv(int128 x) internal pure returns (int128) { unchecked { require(x != 0); int256 result = int256(0x100000000000000000000000000000000) / x; require(result >= MIN_64x64 && result <= MAX_64x64); return int128(result); } } /** * Calculate arithmetics average of x and y, i.e. (x + y) / 2 rounding down. * * @param x signed 64.64-bit fixed point number * @param y signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function avg(int128 x, int128 y) internal pure returns (int128) { unchecked { return int128((int256(x) + int256(y)) >> 1); } } /** * Calculate geometric average of x and y, i.e. sqrt (x * y) rounding down. * Revert on overflow or in case x * y is negative. * * @param x signed 64.64-bit fixed point number * @param y signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function gavg(int128 x, int128 y) internal pure returns (int128) { unchecked { int256 m = int256(x) * int256(y); require(m >= 0); require(m < 0x4000000000000000000000000000000000000000000000000000000000000000); return int128(sqrtu(uint256(m))); } } /** * Calculate x^y assuming 0^0 is 1, where x is signed 64.64 fixed point number * and y is unsigned 256-bit integer number. Revert on overflow. * * @param x signed 64.64-bit fixed point number * @param y uint256 value * @return signed 64.64-bit fixed point number */ function pow(int128 x, uint256 y) internal pure returns (int128) { unchecked { bool negative = x < 0 && y & 1 == 1; uint256 absX = uint128(x < 0 ? -x : x); uint256 absResult; absResult = 0x100000000000000000000000000000000; if (absX <= 0x10000000000000000) { absX <<= 63; while (y != 0) { if (y & 0x1 != 0) { absResult = absResult * absX >> 127; } absX = absX * absX >> 127; if (y & 0x2 != 0) { absResult = absResult * absX >> 127; } absX = absX * absX >> 127; if (y & 0x4 != 0) { absResult = absResult * absX >> 127; } absX = absX * absX >> 127; if (y & 0x8 != 0) { absResult = absResult * absX >> 127; } absX = absX * absX >> 127; y >>= 4; } absResult >>= 64; } else { uint256 absXShift = 63; if (absX < 0x1000000000000000000000000) { absX <<= 32; absXShift -= 32; } if (absX < 0x10000000000000000000000000000) { absX <<= 16; absXShift -= 16; } if (absX < 0x1000000000000000000000000000000) { absX <<= 8; absXShift -= 8; } if (absX < 0x10000000000000000000000000000000) { absX <<= 4; absXShift -= 4; } if (absX < 0x40000000000000000000000000000000) { absX <<= 2; absXShift -= 2; } if (absX < 0x80000000000000000000000000000000) { absX <<= 1; absXShift -= 1; } uint256 resultShift = 0; while (y != 0) { require(absXShift < 64); if (y & 0x1 != 0) { absResult = absResult * absX >> 127; resultShift += absXShift; if (absResult > 0x100000000000000000000000000000000) { absResult >>= 1; resultShift += 1; } } absX = absX * absX >> 127; absXShift <<= 1; if (absX >= 0x100000000000000000000000000000000) { absX >>= 1; absXShift += 1; } y >>= 1; } require(resultShift < 64); absResult >>= 64 - resultShift; } int256 result = negative ? -int256(absResult) : int256(absResult); require(result >= MIN_64x64 && result <= MAX_64x64); return int128(result); } } /** * Calculate sqrt (x) rounding down. Revert if x < 0. * * @param x signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function sqrt(int128 x) internal pure returns (int128) { unchecked { require(x >= 0); return int128(sqrtu(uint256(int256(x)) << 64)); } } /** * Calculate binary logarithm of x. Revert if x <= 0. * * @param x signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function log_2(int128 x) internal pure returns (int128) { unchecked { require(x > 0); int256 msb = 0; int256 xc = x; if (xc >= 0x10000000000000000) { xc >>= 64; msb += 64; } if (xc >= 0x100000000) { xc >>= 32; msb += 32; } if (xc >= 0x10000) { xc >>= 16; msb += 16; } if (xc >= 0x100) { xc >>= 8; msb += 8; } if (xc >= 0x10) { xc >>= 4; msb += 4; } if (xc >= 0x4) { xc >>= 2; msb += 2; } if (xc >= 0x2) msb += 1; // No need to shift xc anymore int256 result = msb - 64 << 64; uint256 ux = uint256(int256(x)) << uint256(127 - msb); for (int256 bit = 0x8000000000000000; bit > 0; bit >>= 1) { ux *= ux; uint256 b = ux >> 255; ux >>= 127 + b; result += bit * int256(b); } return int128(result); } } /** * Calculate natural logarithm of x. Revert if x <= 0. * * @param x signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function ln(int128 x) internal pure returns (int128) { unchecked { require(x > 0); return int128(int256(uint256(int256(log_2(x))) * 0xB17217F7D1CF79ABC9E3B39803F2F6AF >> 128)); } } /** * Calculate binary exponent of x. Revert on overflow. * * @param x signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function exp_2(int128 x) internal pure returns (int128) { unchecked { require(x < 0x400000000000000000); // Overflow if (x < -0x400000000000000000) return 0; // Underflow uint256 result = 0x80000000000000000000000000000000; if (x & 0x8000000000000000 > 0) { result = result * 0x16A09E667F3BCC908B2FB1366EA957D3E >> 128; } if (x & 0x4000000000000000 > 0) { result = result * 0x1306FE0A31B7152DE8D5A46305C85EDEC >> 128; } if (x & 0x2000000000000000 > 0) { result = result * 0x1172B83C7D517ADCDF7C8C50EB14A791F >> 128; } if (x & 0x1000000000000000 > 0) { result = result * 0x10B5586CF9890F6298B92B71842A98363 >> 128; } if (x & 0x800000000000000 > 0) { result = result * 0x1059B0D31585743AE7C548EB68CA417FD >> 128; } if (x & 0x400000000000000 > 0) { result = result * 0x102C9A3E778060EE6F7CACA4F7A29BDE8 >> 128; } if (x & 0x200000000000000 > 0) { result = result * 0x10163DA9FB33356D84A66AE336DCDFA3F >> 128; } if (x & 0x100000000000000 > 0) { result = result * 0x100B1AFA5ABCBED6129AB13EC11DC9543 >> 128; } if (x & 0x80000000000000 > 0) { result = result * 0x10058C86DA1C09EA1FF19D294CF2F679B >> 128; } if (x & 0x40000000000000 > 0) { result = result * 0x1002C605E2E8CEC506D21BFC89A23A00F >> 128; } if (x & 0x20000000000000 > 0) { result = result * 0x100162F3904051FA128BCA9C55C31E5DF >> 128; } if (x & 0x10000000000000 > 0) { result = result * 0x1000B175EFFDC76BA38E31671CA939725 >> 128; } if (x & 0x8000000000000 > 0) { result = result * 0x100058BA01FB9F96D6CACD4B180917C3D >> 128; } if (x & 0x4000000000000 > 0) { result = result * 0x10002C5CC37DA9491D0985C348C68E7B3 >> 128; } if (x & 0x2000000000000 > 0) { result = result * 0x1000162E525EE054754457D5995292026 >> 128; } if (x & 0x1000000000000 > 0) { result = result * 0x10000B17255775C040618BF4A4ADE83FC >> 128; } if (x & 0x800000000000 > 0) { result = result * 0x1000058B91B5BC9AE2EED81E9B7D4CFAB >> 128; } if (x & 0x400000000000 > 0) { result = result * 0x100002C5C89D5EC6CA4D7C8ACC017B7C9 >> 128; } if (x & 0x200000000000 > 0) { result = result * 0x10000162E43F4F831060E02D839A9D16D >> 128; } if (x & 0x100000000000 > 0) { result = result * 0x100000B1721BCFC99D9F890EA06911763 >> 128; } if (x & 0x80000000000 > 0) { result = result * 0x10000058B90CF1E6D97F9CA14DBCC1628 >> 128; } if (x & 0x40000000000 > 0) { result = result * 0x1000002C5C863B73F016468F6BAC5CA2B >> 128; } if (x & 0x20000000000 > 0) { result = result * 0x100000162E430E5A18F6119E3C02282A5 >> 128; } if (x & 0x10000000000 > 0) { result = result * 0x1000000B1721835514B86E6D96EFD1BFE >> 128; } if (x & 0x8000000000 > 0) { result = result * 0x100000058B90C0B48C6BE5DF846C5B2EF >> 128; } if (x & 0x4000000000 > 0) { result = result * 0x10000002C5C8601CC6B9E94213C72737A >> 128; } if (x & 0x2000000000 > 0) { result = result * 0x1000000162E42FFF037DF38AA2B219F06 >> 128; } if (x & 0x1000000000 > 0) { result = result * 0x10000000B17217FBA9C739AA5819F44F9 >> 128; } if (x & 0x800000000 > 0) { result = result * 0x1000000058B90BFCDEE5ACD3C1CEDC823 >> 128; } if (x & 0x400000000 > 0) { result = result * 0x100000002C5C85FE31F35A6A30DA1BE50 >> 128; } if (x & 0x200000000 > 0) { result = result * 0x10000000162E42FF0999CE3541B9FFFCF >> 128; } if (x & 0x100000000 > 0) { result = result * 0x100000000B17217F80F4EF5AADDA45554 >> 128; } if (x & 0x80000000 > 0) { result = result * 0x10000000058B90BFBF8479BD5A81B51AD >> 128; } if (x & 0x40000000 > 0) { result = result * 0x1000000002C5C85FDF84BD62AE30A74CC >> 128; } if (x & 0x20000000 > 0) { result = result * 0x100000000162E42FEFB2FED257559BDAA >> 128; } if (x & 0x10000000 > 0) { result = result * 0x1000000000B17217F7D5A7716BBA4A9AE >> 128; } if (x & 0x8000000 > 0) { result = result * 0x100000000058B90BFBE9DDBAC5E109CCE >> 128; } if (x & 0x4000000 > 0) { result = result * 0x10000000002C5C85FDF4B15DE6F17EB0D >> 128; } if (x & 0x2000000 > 0) { result = result * 0x1000000000162E42FEFA494F1478FDE05 >> 128; } if (x & 0x1000000 > 0) { result = result * 0x10000000000B17217F7D20CF927C8E94C >> 128; } if (x & 0x800000 > 0) { result = result * 0x1000000000058B90BFBE8F71CB4E4B33D >> 128; } if (x & 0x400000 > 0) { result = result * 0x100000000002C5C85FDF477B662B26945 >> 128; } if (x & 0x200000 > 0) { result = result * 0x10000000000162E42FEFA3AE53369388C >> 128; } if (x & 0x100000 > 0) { result = result * 0x100000000000B17217F7D1D351A389D40 >> 128; } if (x & 0x80000 > 0) { result = result * 0x10000000000058B90BFBE8E8B2D3D4EDE >> 128; } if (x & 0x40000 > 0) { result = result * 0x1000000000002C5C85FDF4741BEA6E77E >> 128; } if (x & 0x20000 > 0) { result = result * 0x100000000000162E42FEFA39FE95583C2 >> 128; } if (x & 0x10000 > 0) { result = result * 0x1000000000000B17217F7D1CFB72B45E1 >> 128; } if (x & 0x8000 > 0) { result = result * 0x100000000000058B90BFBE8E7CC35C3F0 >> 128; } if (x & 0x4000 > 0) { result = result * 0x10000000000002C5C85FDF473E242EA38 >> 128; } if (x & 0x2000 > 0) { result = result * 0x1000000000000162E42FEFA39F02B772C >> 128; } if (x & 0x1000 > 0) { result = result * 0x10000000000000B17217F7D1CF7D83C1A >> 128; } if (x & 0x800 > 0) { result = result * 0x1000000000000058B90BFBE8E7BDCBE2E >> 128; } if (x & 0x400 > 0) { result = result * 0x100000000000002C5C85FDF473DEA871F >> 128; } if (x & 0x200 > 0) { result = result * 0x10000000000000162E42FEFA39EF44D91 >> 128; } if (x & 0x100 > 0) { result = result * 0x100000000000000B17217F7D1CF79E949 >> 128; } if (x & 0x80 > 0) { result = result * 0x10000000000000058B90BFBE8E7BCE544 >> 128; } if (x & 0x40 > 0) { result = result * 0x1000000000000002C5C85FDF473DE6ECA >> 128; } if (x & 0x20 > 0) { result = result * 0x100000000000000162E42FEFA39EF366F >> 128; } if (x & 0x10 > 0) { result = result * 0x1000000000000000B17217F7D1CF79AFA >> 128; } if (x & 0x8 > 0) { result = result * 0x100000000000000058B90BFBE8E7BCD6D >> 128; } if (x & 0x4 > 0) { result = result * 0x10000000000000002C5C85FDF473DE6B2 >> 128; } if (x & 0x2 > 0) { result = result * 0x1000000000000000162E42FEFA39EF358 >> 128; } if (x & 0x1 > 0) { result = result * 0x10000000000000000B17217F7D1CF79AB >> 128; } result >>= uint256(int256(63 - (x >> 64))); require(result <= uint256(int256(MAX_64x64))); return int128(int256(result)); } } /** * Calculate natural exponent of x. Revert on overflow. * * @param x signed 64.64-bit fixed point number * @return signed 64.64-bit fixed point number */ function exp(int128 x) internal pure returns (int128) { unchecked { require(x < 0x400000000000000000); // Overflow if (x < -0x400000000000000000) return 0; // Underflow return exp_2(int128(int256(x) * 0x171547652B82FE1777D0FFDA0D23A7D12 >> 128)); } } /** * Calculate x / y rounding towards zero, where x and y are unsigned 256-bit * integer numbers. Revert on overflow or when y is zero. * * @param x unsigned 256-bit integer number * @param y unsigned 256-bit integer number * @return unsigned 64.64-bit fixed point number */ function divuu(uint256 x, uint256 y) private pure returns (uint128) { unchecked { require(y != 0); uint256 result; if (x <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF) { result = (x << 64) / y; } else { uint256 msb = 192; uint256 xc = x >> 192; if (xc >= 0x100000000) { xc >>= 32; msb += 32; } if (xc >= 0x10000) { xc >>= 16; msb += 16; } if (xc >= 0x100) { xc >>= 8; msb += 8; } if (xc >= 0x10) { xc >>= 4; msb += 4; } if (xc >= 0x4) { xc >>= 2; msb += 2; } if (xc >= 0x2) msb += 1; // No need to shift xc anymore result = (x << 255 - msb) / ((y - 1 >> msb - 191) + 1); require(result <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF); uint256 hi = result * (y >> 128); uint256 lo = result * (y & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF); uint256 xh = x >> 192; uint256 xl = x << 64; if (xl < lo) xh -= 1; xl -= lo; // We rely on overflow behavior here lo = hi << 128; if (xl < lo) xh -= 1; xl -= lo; // We rely on overflow behavior here assert(xh == hi >> 128); result += xl / y; } require(result <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF); return uint128(result); } } /** * Calculate sqrt (x) rounding down, where x is unsigned 256-bit integer * number. * * @param x unsigned 256-bit integer number * @return unsigned 128-bit integer number */ function sqrtu(uint256 x) private pure returns (uint128) { unchecked { if (x == 0) { return 0; } else { uint256 xx = x; uint256 r = 1; if (xx >= 0x100000000000000000000000000000000) { xx >>= 128; r <<= 64; } if (xx >= 0x10000000000000000) { xx >>= 64; r <<= 32; } if (xx >= 0x100000000) { xx >>= 32; r <<= 16; } if (xx >= 0x10000) { xx >>= 16; r <<= 8; } if (xx >= 0x100) { xx >>= 8; r <<= 4; } if (xx >= 0x10) { xx >>= 4; r <<= 2; } if (xx >= 0x8) r <<= 1; r = (r + x / r) >> 1; r = (r + x / r) >> 1; r = (r + x / r) >> 1; r = (r + x / r) >> 1; r = (r + x / r) >> 1; r = (r + x / r) >> 1; r = (r + x / r) >> 1; // Seven iterations should be enough uint256 r1 = x / r; return uint128(r < r1 ? r : r1); } } } }

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