Formula for optimal in incorrect, doesn't pass simple test.

[tomekozlowski]

I've made a simplest python test based on the formulas here:

import math as m

r01_0 = 10*10**9
r01_1 = 10*10**9
r10_0 =   5*10**9
r10_1 =  15*10**9


a1 = r01_0
b1 = r01_1

a2 = r10_0
b2 = r10_1

a = a1 * b1 - a2 * b2
b = 2 * b1 * b2 * (a1 + a2)
c = b1 * b2 * (a1 * b2 - a2 * b1)

m_ = b**2 - 4 * a * c

sqrtM = m.sqrt(m_);

x1 = (-b + sqrtM) / (2 * a);
x2 = (-b - sqrtM) / (2 * a);


if x1 > 0 and x1 < b1 and x1 < b2:
    amount = x1
else:
    amount = x2

print( f'opt_in: {amount:.2f}')


def getAmountOut(
    amountIn,
    reserveIn,
    reserveOut
):
    amountInWithFee = amountIn * 997
    numerator = amountInWithFee * reserveOut
    denominator = (reserveIn * 1000) + amountInWithFee
    amountOut = numerator / denominator
    return amountOut

amountz1 = getAmountOut(amount, r01_0, r01_1)
amountz2 = getAmountOut(amountz1, r10_0, r10_1)


print("amnz out: ", amountz1)
print("amnz out: ", amountz2)

pr = amountz2 - amount
print("profitz: ", pr)


print("[!!] verifying +")
print("if amounts were higher by 1000, the profits should be less!")
print("profits are...")

amountz1 = getAmountOut(amount+1000, r01_0, r01_1)
amountz2 = getAmountOut(amountz1, r10_0, r10_1)
pr = amountz2 - (amount+1000)

print(f"...{pr}")

print("[!!] verifying -")
print("if amounts were lower by 1000, the profits should be less!")
print("profits are...")

amountz1 = getAmountOut(amount-1000, r01_0, r01_1)
amountz2 = getAmountOut(amountz1, r10_0, r10_1)
pr = amountz2 - (amount-1000)

print(f"...{pr}")

Output of this:

opt_in: -176602540378.44
amnz out:  10602145808.88565
amnz out:  10183149306.464203
profitz:  186785689684.90808
[!!] verifying +
if amounts were higher by 1000, the profits should be less!
profits are...
...186785688686.02304
[!!] verifying -
if amounts were lower by 1000, the profits should be less!
profits are...
...186785690683.79312