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gyro2Math.ts
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gyro2Math.ts
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import { BigNumber } from '@ethersproject/bignumber';
import { WeiPerEther as ONE } from '@ethersproject/constants';
import {
sqrt,
mulUp,
divUp,
mulDown,
divDown,
} from '../gyroHelpers/gyroSignedFixedPoint';
/////////
/// Virtual Parameter calculations
/////////
export function _findVirtualParams(
invariant: BigNumber,
sqrtAlpha: BigNumber,
sqrtBeta: BigNumber
): [BigNumber, BigNumber] {
return [divDown(invariant, sqrtBeta), mulDown(invariant, sqrtAlpha)];
}
/////////
/// Invariant Calculation
/////////
export function _calculateInvariant(
balances: BigNumber[], // balances
sqrtAlpha: BigNumber,
sqrtBeta: BigNumber
): BigNumber {
/**********************************************************************************************
// Calculate with quadratic formula
// 0 = (1-sqrt(alpha/beta)*L^2 - (y/sqrt(beta)+x*sqrt(alpha))*L - x*y)
// 0 = a*L^2 + b*L + c
// here a > 0, b < 0, and c < 0, which is a special case that works well w/o negative numbers
// taking mb = -b and mc = -c: (1/2)
// mb + (mb^2 + 4 * a * mc)^ //
// L = ------------------------------------------ //
// 2 * a //
// //
**********************************************************************************************/
const [a, mb, bSquare, mc] = _calculateQuadraticTerms(
balances,
sqrtAlpha,
sqrtBeta
);
const invariant = _calculateQuadratic(a, mb, bSquare, mc);
return invariant;
}
export function _calculateQuadraticTerms(
balances: BigNumber[],
sqrtAlpha: BigNumber,
sqrtBeta: BigNumber
): [BigNumber, BigNumber, BigNumber, BigNumber] {
const a = ONE.sub(divDown(sqrtAlpha, sqrtBeta));
const bterm0 = divDown(balances[1], sqrtBeta);
const bterm1 = mulDown(balances[0], sqrtAlpha);
const mb = bterm0.add(bterm1);
const mc = mulDown(balances[0], balances[1]);
// For better fixed point precision, calculate in expanded form w/ re-ordering of multiplications
// b^2 = x^2 * alpha + x*y*2*sqrt(alpha/beta) + y^2 / beta
let bSquare = mulDown(
mulDown(mulDown(balances[0], balances[0]), sqrtAlpha),
sqrtAlpha
);
const bSq2 = divDown(
mulDown(
mulDown(mulDown(balances[0], balances[1]), ONE.mul(2)),
sqrtAlpha
),
sqrtBeta
);
const bSq3 = divDown(
mulDown(balances[1], balances[1]),
mulUp(sqrtBeta, sqrtBeta)
);
bSquare = bSquare.add(bSq2).add(bSq3);
return [a, mb, bSquare, mc];
}
export function _calculateQuadratic(
a: BigNumber,
mb: BigNumber,
bSquare: BigNumber,
mc: BigNumber
): BigNumber {
const denominator = mulUp(a, ONE.mul(2));
// order multiplications for fixed point precision
const addTerm = mulDown(mulDown(mc, ONE.mul(4)), a);
// The minus sign in the radicand cancels out in this special case, so we add
const radicand = bSquare.add(addTerm);
const sqrResult = sqrt(radicand, BigNumber.from(5));
// The minus sign in the numerator cancels out in this special case
const numerator = mb.add(sqrResult);
const invariant = divDown(numerator, denominator);
return invariant;
}
/////////
/// Swap functions
/////////
// SwapType = 'swapExactIn'
export function _calcOutGivenIn(
balanceIn: BigNumber,
balanceOut: BigNumber,
amountIn: BigNumber,
virtualParamIn: BigNumber,
virtualParamOut: BigNumber
): BigNumber {
/**********************************************************************************************
// Described for X = `in' asset and Y = `out' asset, but equivalent for the other case //
// dX = incrX = amountIn > 0 //
// dY = incrY = amountOut < 0 //
// x = balanceIn x' = x + virtualParamX //
// y = balanceOut y' = y + virtualParamY //
// L = inv.Liq / L^2 \ //
// - dy = y' - | -------------------------- | //
// x' = virtIn \ ( x' + dX) / //
// y' = virtOut //
// Note that -dy > 0 is what the trader receives. //
// We exploit the fact that this formula is symmetric up to virtualParam{X,Y}. //
**********************************************************************************************/
// The factors in total lead to a multiplicative "safety margin" between the employed virtual offsets
// very slightly larger than 3e-18.
const virtInOver = balanceIn.add(mulUp(virtualParamIn, ONE.add(2)));
const virtOutUnder = balanceOut.add(mulDown(virtualParamOut, ONE.sub(1)));
const amountOut = divDown(
mulDown(virtOutUnder, amountIn),
virtInOver.add(amountIn)
);
if (amountOut.gt(balanceOut)) throw new Error('ASSET_BOUNDS_EXCEEDED');
return amountOut;
}
// SwapType = 'swapExactOut'
export function _calcInGivenOut(
balanceIn: BigNumber,
balanceOut: BigNumber,
amountOut: BigNumber,
virtualParamIn: BigNumber,
virtualParamOut: BigNumber
): BigNumber {
/**********************************************************************************************
// dX = incrX = amountIn > 0 //
// dY = incrY = amountOut < 0 //
// x = balanceIn x' = x + virtualParamX //
// y = balanceOut y' = y + virtualParamY //
// x = balanceIn //
// L = inv.Liq / L^2 \ //
// dx = | -------------------------- | - x' //
// x' = virtIn \ ( y' + dy) / //
// y' = virtOut //
// Note that dy < 0 < dx. //
**********************************************************************************************/
if (amountOut.gt(balanceOut)) throw new Error('ASSET_BOUNDS_EXCEEDED');
// The factors in total lead to a multiplicative "safety margin" between the employed virtual offsets
// very slightly larger than 3e-18.
const virtInOver = balanceIn.add(mulUp(virtualParamIn, ONE.add(2)));
const virtOutUnder = balanceOut.add(mulDown(virtualParamOut, ONE.sub(1)));
const amountIn = divUp(
mulUp(virtInOver, amountOut),
virtOutUnder.sub(amountOut)
);
return amountIn;
}
// /////////
// /// Spot price function
// /////////
export function _calculateNewSpotPrice(
balances: BigNumber[],
inAmount: BigNumber,
outAmount: BigNumber,
virtualParamIn: BigNumber,
virtualParamOut: BigNumber,
swapFee: BigNumber
): BigNumber {
/**********************************************************************************************
// dX = incrX = amountIn > 0 //
// dY = incrY = amountOut < 0 //
// x = balanceIn x' = x + virtualParamX //
// y = balanceOut y' = y + virtualParamY //
// s = swapFee //
// L = inv.Liq 1 / x' + (1 - s) * dx \ //
// p_y = --- | -------------------------- | //
// x' = virtIn 1-s \ y' + dy / //
// y' = virtOut //
// Note that dy < 0 < dx. //
**********************************************************************************************/
const afterFeeMultiplier = ONE.sub(swapFee); // 1 - s
const virtIn = balances[0].add(virtualParamIn); // x + virtualParamX = x'
const numerator = virtIn.add(mulDown(afterFeeMultiplier, inAmount)); // x' + (1 - s) * dx
const virtOut = balances[1].add(virtualParamOut); // y + virtualParamY = y'
const denominator = mulDown(afterFeeMultiplier, virtOut.sub(outAmount)); // (1 - s) * (y' + dy)
const newSpotPrice = divDown(numerator, denominator);
return newSpotPrice;
}
// /////////
// /// Derivatives of spotPriceAfterSwap
// /////////
// SwapType = 'swapExactIn'
export function _derivativeSpotPriceAfterSwapExactTokenInForTokenOut(
balances: BigNumber[],
outAmount: BigNumber,
virtualParamOut: BigNumber
): BigNumber {
/**********************************************************************************************
// dy = incrY = amountOut < 0 //
//
// y = balanceOut y' = y + virtualParamY = virtOut //
// //
// / 1 \ //
// (p_y)' = 2 | -------------------------- | //
// \ y' + dy / //
// //
// Note that dy < 0 //
**********************************************************************************************/
const TWO = BigNumber.from(2).mul(ONE);
const virtOut = balances[1].add(virtualParamOut); // y' = y + virtualParamY
const denominator = virtOut.sub(outAmount); // y' + dy
const derivative = divDown(TWO, denominator);
return derivative;
}
// SwapType = 'swapExactOut'
export function _derivativeSpotPriceAfterSwapTokenInForExactTokenOut(
balances: BigNumber[],
inAmount: BigNumber,
outAmount: BigNumber,
virtualParamIn: BigNumber,
virtualParamOut: BigNumber,
swapFee: BigNumber
): BigNumber {
/**********************************************************************************************
// dX = incrX = amountIn > 0 //
// dY = incrY = amountOut < 0 //
// x = balanceIn x' = x + virtualParamX //
// y = balanceOut y' = y + virtualParamY //
// s = swapFee //
// L = inv.Liq 1 / x' + (1 - s) * dx \ //
// p_y = --- (2) | -------------------------- | //
// x' = virtIn 1-s \ (y' + dy)^2 / //
// y' = virtOut //
// Note that dy < 0 < dx. //
**********************************************************************************************/
const TWO = BigNumber.from(2).mul(ONE);
const afterFeeMultiplier = ONE.sub(swapFee); // 1 - s
const virtIn = balances[0].add(virtualParamIn); // x + virtualParamX = x'
const numerator = virtIn.add(mulDown(afterFeeMultiplier, inAmount)); // x' + (1 - s) * dx
const virtOut = balances[1].add(virtualParamOut); // y + virtualParamY = y'
const denominator = mulDown(virtOut.sub(outAmount), virtOut.sub(outAmount)); // (y' + dy)^2
const factor = divDown(TWO, afterFeeMultiplier); // 2 / (1 - s)
const derivative = mulDown(factor, divDown(numerator, denominator));
return derivative;
}
// /////////
// /// Normalized Liquidity measured with respect to the in-asset.
// /////////
export function _getNormalizedLiquidity(
balances: BigNumber[],
virtualParamOut: BigNumber
): BigNumber {
/**********************************************************************************************
// x = balanceOut x' = x + virtualParamOut //
// s = swapFee //
// //
// normalizedLiquidity = 0.5 * x' //
// //
// x' = virtOut //
// Note that balances = [balanceIn, balanceOut]. //
**********************************************************************************************/
const virtOut = balances[1].add(virtualParamOut);
return virtOut.div(2);
}