doc for withdraw imbalance formula (#30)

* docs for withdraw imbalance * update docs * update docs --------- Co-authored-by: h2physics <[email protected]>
minswap · Feb 15, 2024 · d51628e · d51628e
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diff --git a/amm-v2-docs/formula.md b/amm-v2-docs/formula.md
@@ -70,5 +70,40 @@ $$\Delta y = \frac{\Delta L}{L} * y_{0}$$
 In case users want to zap out to Asset B, $\Delta x$ will be swapped, then total amount users receive is
 $$Out = \Delta y + \frac{(f_{d} - f_{n}) * \Delta x * y_{0}}{x_{0} * f_{d} + (f_{d} - f_{n}) * \Delta x}$$
 
-### 5. Withdraw Imbalance
-TODO
+### 6. Withdraw Imbalance
+
+Users want to withdraw with a ratio $A/B$. 
+
+We have the basic withdrawal formulas:
+
+$$\Delta x = \frac{\Delta L}{L} * x_{0}$$
+$$\Delta y = \frac{\Delta L}{L} * y_{0}$$
+
+Suppose we need to swap some in $\Delta x$ to get $\frac{\Delta x'}{\Delta y'} = \frac{A}{B}$.
+
+So we have the formula:
+$$\frac{\Delta x - swap_{x}}{\Delta y + receive_{y}} =\frac{A}{B} (1)$$
+
+
+$$ receive_{y} = \frac{(f_{d} - f_{n}) * swap_{x} * y_{0}}{x_{0} * f_{d} + (f_{d} - f_{n}) * swap_{x}}(2)$$
+
+(With $x_{0}$ and $y_{0}$ being $x$ and $y$ after withdrawal, $swap_{x}$ is the amount need to be swapped to adapt the ratio $A/B$ that users expect).
+
+Combination of formula (1) and (2), we have:
+
+$$a * swap_{x} ^ 2 + b * swap_{x} + c = 0$$
+where 
+$$a = (f_{d} - f_{n}) * B$$
+$$b = A*(f_{d} - f_{n})*(y_{0}+\Delta y) + B *(f_{d} * x_{0} - (f_{d} - f_{n})*\Delta x)$$
+$$ c =f_{d} * x_{0} *(A * \Delta y - B * \Delta x) $$
+
+### 7. Partial Swap
+Allow users swap only if price is exactly matched.
+
+In case users want to swap with price $A/B$
+We have 2 formulas: 
+$$\frac{\Delta x}{\Delta y} = \frac{A}{B}  (1)$$
+$$ \Delta y = \frac{(f_{d} - f_{n}) * \Delta x * y_{0}}{x_{0} * f_{d} + (f_{d} - f_{n}) * \Delta x} (2)$$
+We can calculate $\Delta x$: 
+$$\Delta x = \frac{A * (f_{d} - f_{n}) * y_{0} - B * f_{d} * x_{0}}{(f_{d} - f_{n}) * B}$$
+where $\Delta x$ is the maximum amount can be swapped to adapt $A/B$ ratio
diff --git a/lib/amm_dex_v2/math.ak b/lib/amm_dex_v2/math.ak
@@ -358,8 +358,8 @@ pub fn calculate_zap_out(
 // a * swap_amount_in ^ 2 + b * swap_amount_in + c = 0
 // Where:
 // - a = fee_diff * expect_io_ratio_denominator
-// - b = expect_io_ratio_numerator * fee_diff ( reserve_out + amount_out ) * expect_io_ratio_denominator * (fee_denominator * reserve_in - fee_diff * amount_in)
-// - c = fee_denominator * (expect_io_ratio_numerator * amount_out - expect_io_ratio_denominator * amount_in) 
+// - b = expect_io_ratio_numerator * fee_diff ( reserve_out + amount_out ) + expect_io_ratio_denominator * (fee_denominator * reserve_in - fee_diff * amount_in)
+// - c = fee_denominator * reserve_in* (expect_io_ratio_numerator * amount_out - expect_io_ratio_denominator * amount_in) 
 fn calculate_withdraw_swap_amount(
   amount_in: Int,
   amount_out: Int,