update docs (#36)

* update docs * update docs * not equal * format 2
minswap · Feb 21, 2024 · 7437012 · 7437012
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diff --git a/amm-v2-docs/amm-v2-specs.md b/amm-v2-docs/amm-v2-specs.md
@@ -543,7 +543,7 @@ Transaction structure:
            - Step:
              - _a_to_b_direction_
              - _swap_amount_
-             - _minimum_receive_ (TODO: Link formula section here)
+             - _minimum_receive_([Formula](./formula.md#1-swap-exact-in))
              - _killable_
            - Value:
              - _batcher_fee_ + additional ADA to cover output (if need)
@@ -552,7 +552,7 @@ Transaction structure:
            - Step:
              - _a_to_b_direction_
              - _swap_amount_
-             - _stop_loss_receive_ (TODO: Link formula section here)
+             - _stop_loss_receive_([Formula](./formula.md#1-swap-exact-in))
              - _killable_
            - Value:
              - _batcher_fee_ + additional ADA to cover output (if need)
@@ -561,17 +561,17 @@ Transaction structure:
            - Step:
              - _a_to_b_direction_
              - _swap_amount_
-             - _minimum_receive_ (TODO: Link formula section here)
-             - _stop_loss_receive_ (TODO: Link formula section here)
+             - _minimum_receive_([Formula](./formula.md#1-swap-exact-in))
+             - _stop_loss_receive_([Formula](./formula.md#1-swap-exact-in))
              - _killable_
            - Value:
              - _batcher_fee_ + additional ADA to cover output (if need)
              - Token is swapping
          - _SwapExactOut_:
            - Step:
              - _a_to_b_direction_
-             - _maximum_swap_amount_
-             - _expected_receive_ (TODO: Link formula section here)
+             - _maximum_swap_amount_([Formula](./formula.md#2-swap-exact-out))
+             - _expected_receive_
              - _killable_
            - Value:
              - _batcher_fee_ + additional ADA to cover output (if need)
@@ -580,15 +580,15 @@ Transaction structure:
            - Step:
              - _deposit_amount_a_
              - _deposit_amount_b_
-             - _minimum_lp_ (TODO: Link formula section here)
+             - _minimum_lp_([Formula](./formula.md#3-deposit))
            - Value:
              - _batcher_fee_ + additional ADA to cover output (if need)
              - Token A and Token B (amount of token A or B can be zero)
          - _Withdraw_:
            - Step:
              - _withdrawal_lp_amount_
-             - _minimum_asset_a_ (TODO: Link formula section here)
-             - _minimum_asset_b_ (TODO: Link formula section here)
+             - _minimum_asset_a_([Formula](./formula.md#4-withdraw))
+             - _minimum_asset_b_([Formula](./formula.md#4-withdraw))
              - _killable_
            - Value:
              - _batcher_fee_ + additional ADA to cover output (if need)
@@ -597,38 +597,38 @@ Transaction structure:
            - Step:
              - _a_to_b_direction_
              - _withdrawal_lp_amount_
-             - _minimum_receive_ (TODO: Link formula section here)
+             - _minimum_receive_([Formula](./formula.md#5-zap-out))
              - _killable_
            - Value:
              - _batcher_fee_ + additional ADA to cover output (if need)
              - LP Asset
          - _WithdrawImbalance_:
            - Step:
              - _withdrawal_lp_amount_
-             - _ratio_asset_a_ (TODO: Link formula section here)
-             - _ratio_asset_b_ (TODO: Link formula section here)
-             - _minimum_asset_a_ (TODO: Link formula section here)
+             - _ratio_asset_a_
+             - _ratio_asset_b_
+             - _minimum_asset_a_([Formula](./formula.md#6-withdraw-imbalance))
              - _killable_
            - Value:
              - _batcher_fee_ + additional ADA to cover output (if need)
              - LP Asset
          - _PartialSwap_:
            - Step:
-             - _a_to_b_direction_ (TODO: Link formula section here)
+             - _a_to_b_direction_([Formula](./formula.md#7-partial-swap))
              - _total_swap_amount_
-             - _io_ratio_numerator_ (TODO: Link formula section here)
-             - _io_ratio_denominator_ (TODO: Link formula section here)
-             - _hops_ (TODO: Link formula section here)
-             - _minimum_swap_amount_required_ (TODO: Link formula section here)
+             - _io_ratio_numerator_([Formula](./formula.md#7-partial-swap))
+             - _io_ratio_denominator_([Formula](./formula.md#7-partial-swap))
+             - _hops_([Formula](./formula.md#7-partial-swap))
+             - _minimum_swap_amount_required_([Formula](./formula.md#7-partial-swap))
              - _max_batcher_fee_each_time_
            - Value:
              - _batcher_fee_ * _hops_ + additional ADA to cover output (if need)
              - Token is swapping and received Token of previous swap (if you are order's creator, you no need to care about _received Token_)
          - _SwapMultiRouting_:
            - Step:
-             - _routings_ (TODO: Link formula section here)
+             - _routings_([Formula](./formula.md#1-swap-exact-in))
              - _swap_amount_
-             - _minimum_receive_ (TODO: Link formula section here)
+             - _minimum_receive_([Formula](./formula.md#1-swap-exact-in))
            - Value:
              - _batcher_fee_ + additional ADA to cover output (if need)
              - Token is swapping

diff --git a/amm-v2-docs/formula.md b/amm-v2-docs/formula.md
@@ -13,7 +13,9 @@ Before we go to details, here are some definitions:
 ## AMM has 2 main formula:
 ### 1. When the price changes (Swap)
 When a Token is swapped, Swap Fee is deducted from the input amount ($\Delta x$) 
+
 $$ k = x_{0} * y_{0}$$
+
 $$ k = (x_{0} + (1 - f) * \Delta x) * (y_{0} - \Delta y)$$
 
 ### 2. When the price is not changes (Add or Remove Liquidity)
@@ -34,7 +36,7 @@ $$\Delta x = \frac{x_{0} * \Delta y * f_{d}}{(f_{d} - f_{n}) * (y_{0} - \Delta y
 ### 3. Deposit
 On the V2, we allow users to deposit with any amount A & B, so there are two cases can happen:
 - $\frac{\Delta x}{x_{0}} = \frac{\Delta y}{y_{0}} \Rightarrow \Delta L = \frac{\Delta x}{x_{0}} * L$
-- $\frac{\Delta x}{x_{0}} \#  \frac{\Delta y}{y_{0}}$:
+- $\frac{\Delta x}{x_{0}} ≠ \frac{\Delta y}{y_{0}}$:
   - $\frac{\Delta x}{x_{0}} >  \frac{\Delta y}{y_{0}}$: We need to Swap a part of X to Y
   - $\frac{\Delta x}{x_{0}} <  \frac{\Delta y}{y_{0}}$: We need to Swap a part of Y to X
 
@@ -94,8 +96,10 @@ Combination of formula (1) and (2), we have:
 $$a * swap_{x} ^ 2 + b * swap_{x} + c = 0$$
 where 
 $$a = (f_{d} - f_{n}) * B$$
+
 $$b = A*(f_{d} - f_{n})*(y_{0}+\Delta y) + B *(f_{d} * x_{0} - (f_{d} - f_{n})*\Delta x)$$
-$$ c =f_{d} * x_{0} *(A * \Delta y - B * \Delta x) $$
+
+$$c =f_{d} * x_{0} *(A * \Delta y - B * \Delta x)$$
 
 ### 7. Partial Swap
 Allow users swap only if price is exactly matched.
@@ -104,6 +108,8 @@ 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
@@ -353,13 +353,13 @@ pub fn calculate_zap_out(
 
 // Base formula:
 // ( amount_in - swap_amount_in) / (amount_out + receive_amount_out) = expect_io_ratio_numerator / expect_io_ratio_denominator (1)
-// receive_amount_out = fee_diff * swap_amount_in * reserve_out / (fee_denominator) * reserve_in + fee_diff * swap_amount_in (2)
+// receive_amount_out = fee_diff * swap_amount_in * reserve_out / (fee_denominator * reserve_in + fee_diff * swap_amount_in) (2)
 // (1) & (2) =>>>>>
 // 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 * reserve_in* (expect_io_ratio_numerator * amount_out - expect_io_ratio_denominator * 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,