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TickMath.sol
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TickMath.sol
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// SPDX-License-Identifier: GPL-2.0-or-later
pragma solidity ^0.8.20;
/// @title Math library for computing sqrt prices from ticks and vice versa
/// @notice Computes sqrt price for ticks of size 1.0001, i.e. sqrt(1.0001^tick) as fixed point Q64.96 numbers. Supports
/// prices between 2**-128 and 2**128
library TickMath {
/// @notice Thrown when the tick passed to #getSqrtRatioAtTick is not between MIN_TICK and MAX_TICK
error InvalidTick();
/// @notice Thrown when the ratio passed to #getTickAtSqrtRatio does not correspond to a price between MIN_TICK and MAX_TICK
error InvalidSqrtRatio();
/// @dev The minimum tick that may be passed to #getSqrtRatioAtTick computed from log base 1.0001 of 2**-128
int24 internal constant MIN_TICK = -887272;
/// @dev The maximum tick that may be passed to #getSqrtRatioAtTick computed from log base 1.0001 of 2**128
int24 internal constant MAX_TICK = -MIN_TICK;
/// @dev The minimum value that can be returned from #getSqrtRatioAtTick. Equivalent to getSqrtRatioAtTick(MIN_TICK)
uint160 internal constant MIN_SQRT_RATIO = 4295128739;
/// @dev The maximum value that can be returned from #getSqrtRatioAtTick. Equivalent to getSqrtRatioAtTick(MAX_TICK)
uint160 internal constant MAX_SQRT_RATIO = 1461446703485210103287273052203988822378723970342;
/// @notice Given a tickSpacing, compute the maximum usable tick
function maxUsableTick(int24 tickSpacing) internal pure returns (int24) {
unchecked {
return (MAX_TICK / tickSpacing) * tickSpacing;
}
}
/// @notice Given a tickSpacing, compute the minimum usable tick
function minUsableTick(int24 tickSpacing) internal pure returns (int24) {
unchecked {
return (MIN_TICK / tickSpacing) * tickSpacing;
}
}
/// @notice Calculates sqrt(1.0001^tick) * 2^96
/// @dev Throws if |tick| > max tick
/// @param tick The input tick for the above formula
/// @return sqrtPriceX96 A Fixed point Q64.96 number representing the sqrt of the ratio of the two assets (currency1/currency0)
/// at the given tick
function getSqrtRatioAtTick(int24 tick) internal pure returns (uint160 sqrtPriceX96) {
unchecked {
uint256 absTick = tick < 0 ? uint256(-int256(tick)) : uint256(int256(tick));
if (absTick > uint256(int256(MAX_TICK))) revert InvalidTick();
uint256 ratio =
absTick & 0x1 != 0 ? 0xfffcb933bd6fad37aa2d162d1a594001 : 0x100000000000000000000000000000000;
if (absTick & 0x2 != 0) ratio = (ratio * 0xfff97272373d413259a46990580e213a) >> 128;
if (absTick & 0x4 != 0) ratio = (ratio * 0xfff2e50f5f656932ef12357cf3c7fdcc) >> 128;
if (absTick & 0x8 != 0) ratio = (ratio * 0xffe5caca7e10e4e61c3624eaa0941cd0) >> 128;
if (absTick & 0x10 != 0) ratio = (ratio * 0xffcb9843d60f6159c9db58835c926644) >> 128;
if (absTick & 0x20 != 0) ratio = (ratio * 0xff973b41fa98c081472e6896dfb254c0) >> 128;
if (absTick & 0x40 != 0) ratio = (ratio * 0xff2ea16466c96a3843ec78b326b52861) >> 128;
if (absTick & 0x80 != 0) ratio = (ratio * 0xfe5dee046a99a2a811c461f1969c3053) >> 128;
if (absTick & 0x100 != 0) ratio = (ratio * 0xfcbe86c7900a88aedcffc83b479aa3a4) >> 128;
if (absTick & 0x200 != 0) ratio = (ratio * 0xf987a7253ac413176f2b074cf7815e54) >> 128;
if (absTick & 0x400 != 0) ratio = (ratio * 0xf3392b0822b70005940c7a398e4b70f3) >> 128;
if (absTick & 0x800 != 0) ratio = (ratio * 0xe7159475a2c29b7443b29c7fa6e889d9) >> 128;
if (absTick & 0x1000 != 0) ratio = (ratio * 0xd097f3bdfd2022b8845ad8f792aa5825) >> 128;
if (absTick & 0x2000 != 0) ratio = (ratio * 0xa9f746462d870fdf8a65dc1f90e061e5) >> 128;
if (absTick & 0x4000 != 0) ratio = (ratio * 0x70d869a156d2a1b890bb3df62baf32f7) >> 128;
if (absTick & 0x8000 != 0) ratio = (ratio * 0x31be135f97d08fd981231505542fcfa6) >> 128;
if (absTick & 0x10000 != 0) ratio = (ratio * 0x9aa508b5b7a84e1c677de54f3e99bc9) >> 128;
if (absTick & 0x20000 != 0) ratio = (ratio * 0x5d6af8dedb81196699c329225ee604) >> 128;
if (absTick & 0x40000 != 0) ratio = (ratio * 0x2216e584f5fa1ea926041bedfe98) >> 128;
if (absTick & 0x80000 != 0) ratio = (ratio * 0x48a170391f7dc42444e8fa2) >> 128;
if (tick > 0) ratio = type(uint256).max / ratio;
// this divides by 1<<32 rounding up to go from a Q128.128 to a Q128.96.
// we then downcast because we know the result always fits within 160 bits due to our tick input constraint
// we round up in the division so getTickAtSqrtRatio of the output price is always consistent
sqrtPriceX96 = uint160((ratio >> 32) + (ratio % (1 << 32) == 0 ? 0 : 1));
}
}
/// @notice Calculates the greatest tick value such that getRatioAtTick(tick) <= ratio
/// @dev Throws in case sqrtPriceX96 < MIN_SQRT_RATIO, as MIN_SQRT_RATIO is the lowest value getRatioAtTick may
/// ever return.
/// @param sqrtPriceX96 The sqrt ratio for which to compute the tick as a Q64.96
/// @return tick The greatest tick for which the ratio is less than or equal to the input ratio
function getTickAtSqrtRatio(uint160 sqrtPriceX96) internal pure returns (int24 tick) {
unchecked {
// second inequality must be < because the price can never reach the price at the max tick
if (sqrtPriceX96 < MIN_SQRT_RATIO || sqrtPriceX96 >= MAX_SQRT_RATIO) revert InvalidSqrtRatio();
uint256 ratio = uint256(sqrtPriceX96) << 32;
uint256 r = ratio;
uint256 msb = 0;
assembly {
let f := shl(7, gt(r, 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF))
msb := or(msb, f)
r := shr(f, r)
}
assembly {
let f := shl(6, gt(r, 0xFFFFFFFFFFFFFFFF))
msb := or(msb, f)
r := shr(f, r)
}
assembly {
let f := shl(5, gt(r, 0xFFFFFFFF))
msb := or(msb, f)
r := shr(f, r)
}
assembly {
let f := shl(4, gt(r, 0xFFFF))
msb := or(msb, f)
r := shr(f, r)
}
assembly {
let f := shl(3, gt(r, 0xFF))
msb := or(msb, f)
r := shr(f, r)
}
assembly {
let f := shl(2, gt(r, 0xF))
msb := or(msb, f)
r := shr(f, r)
}
assembly {
let f := shl(1, gt(r, 0x3))
msb := or(msb, f)
r := shr(f, r)
}
assembly {
let f := gt(r, 0x1)
msb := or(msb, f)
}
if (msb >= 128) r = ratio >> (msb - 127);
else r = ratio << (127 - msb);
int256 log_2 = (int256(msb) - 128) << 64;
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(63, f))
r := shr(f, r)
}
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(62, f))
r := shr(f, r)
}
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(61, f))
r := shr(f, r)
}
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(60, f))
r := shr(f, r)
}
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(59, f))
r := shr(f, r)
}
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(58, f))
r := shr(f, r)
}
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(57, f))
r := shr(f, r)
}
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(56, f))
r := shr(f, r)
}
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(55, f))
r := shr(f, r)
}
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(54, f))
r := shr(f, r)
}
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(53, f))
r := shr(f, r)
}
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(52, f))
r := shr(f, r)
}
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(51, f))
r := shr(f, r)
}
assembly {
r := shr(127, mul(r, r))
let f := shr(128, r)
log_2 := or(log_2, shl(50, f))
}
int256 log_sqrt10001 = log_2 * 255738958999603826347141; // 128.128 number
int24 tickLow = int24((log_sqrt10001 - 3402992956809132418596140100660247210) >> 128);
int24 tickHi = int24((log_sqrt10001 + 291339464771989622907027621153398088495) >> 128);
tick = tickLow == tickHi ? tickLow : getSqrtRatioAtTick(tickHi) <= sqrtPriceX96 ? tickHi : tickLow;
}
}
}