Constant-product AMM lab

A constant-product automated market maker quotes swaps from two token reserves constrained by x × y = k. Change a scenario to see how input size, fee retention, inventory, and price impact connect.

This educational model uses exact decimal arithmetic. It does not model concentrated liquidity, integer rounding, transfer taxes, hooks, or a named protocol implementation.

Model inputs

Controls

Token X reserves

Unit: X · Must be greater than 0

X

Current amount of Token X held by the pool.

Token Y reserves

Unit: Y · Must be greater than 0

Y

Current amount of Token Y held by the pool.

Trade direction

Choose the asset sent to the pool

X to YSwap Token X for Token YY to XSwap Token Y for Token X

Trade size

Unit: X · Must be greater than or equal to 0

X

Gross amount of the input token sent by the trader.

Fee rate

Unit: % · 0% through 5%

%

Percentage charged on the gross input amount.

External market price

Unit: Y/X · Must be greater than 0

Y/X

Reference price in Token Y per Token X.

Reset allShare current state

Reserve geometry

Constant-product price curve

• Before trade
• After trade

The pool moves from 10,000.000000 X and 20,000.000000 Y to 11,500.000000 X and 17,398.112305 Y along the constant-product reserve curve.

Inspect all curve data points

Constant-product curve samples; each row holds the pre-trade reserve product fixed

Point	X reserve	Y reserve
1	4,500.000000 X	44,444.444444 Y
2	4,833.125000 X	41,381.094013 Y
3	5,166.250000 X	38,712.799419 Y
4	5,499.375000 X	36,367.769065 Y
5	5,832.500000 X	34,290.612945 Y
6	6,165.625000 X	32,437.911809 Y
7	6,498.750000 X	30,775.149067 Y
8	6,831.875000 X	29,274.540298 Y
9	7,165.000000 X	27,913.468248 Y
10	7,498.125000 X	26,673.335000 Y
11	7,831.250000 X	25,538.707103 Y
12	8,164.375000 X	24,496.669984 Y
13	8,497.500000 X	23,536.334216 Y
14	8,830.625000 X	22,648.453535 Y
15	9,163.750000 X	21,825.126177 Y
16	9,496.875000 X	21,059.559065 Y
17	9,830.000000 X	20,345.879959 Y
18	10,163.125000 X	19,678.986532 Y
19	10,496.250000 X	19,054.424199 Y
20	10,829.375000 X	18,468.286489 Y
21	11,162.500000 X	17,917.133259 Y
22	11,495.625000 X	17,397.923123 Y
23	11,828.750000 X	16,907.957307 Y
24	12,161.875000 X	16,444.832725 Y
25	12,495.000000 X	16,006.402561 Y
26	12,828.125000 X	15,590.742996 Y
27	13,161.250000 X	15,196.124988 Y
28	13,494.375000 X	14,820.990227 Y
29	13,827.500000 X	14,463.930573 Y
30	14,160.625000 X	14,123.670389 Y
31	14,493.750000 X	13,799.051315 Y
32	14,826.875000 X	13,489.019095 Y
33	15,160.000000 X	13,192.612137 Y
34	15,493.125000 X	12,908.951551 Y
35	15,826.250000 X	12,637.232446 Y
36	16,159.375000 X	12,376.716302 Y
37	16,492.500000 X	12,126.724269 Y
38	16,825.625000 X	11,886.631254 Y
39	17,158.750000 X	11,655.860712 Y
40	17,491.875000 X	11,433.880016 Y
41	17,825.000000 X	11,220.196353 Y

Scenario explanation

What is happening

You send 1,500.000000 X to the pool and receive 2,601.887695 Y. That adds X and removes Y, moving the pool price from 2.000000 to 1.512879 Y per X. The average execution price is 1.734592 Y per X, with 13.2704% price impact. The 4.500000 X fee remains in pool inventory for LP claim holders.

• Capital flow
• Claim flow
• Return flow
• Risk flow

Inventory movement

Reserve bars

Each outlined bar is the reserve before the trade; each solid bar is the reserve after. Labels and values carry the same meaning without color or motion.

• Token X10,000.000000 → 11,500.000000 X
• Token Y20,000.000000 → 17,398.112305 Y

Deterministic output

Results

Constant-product AMM results. Display values are rounded; calculations retain Decimal precision.

Metric	Value	Unit	Meaning
Spot price before	2.000000	Y per X	Y reserve ÷ X reserve
Quoted output	2,601.887695	Y	Amount sent from the pool
Average execution price	1.734592	Y per X	Gross-input execution price
Price impact	13.2704%	percent	Deterioration from spot
Fee paid	4.500000	X	Retained in the pool
X reserve after	11,500.000000	X	Post-trade pool inventory
Y reserve after	17,398.112305	Y	Post-trade pool inventory
Invariant before	200,000,000.000000	X·Y	x × y before
Invariant after	200,078,291.505372	X·Y	Includes retained fee
Pool price after	1.512879	Y per X	Post-trade Y ÷ X
Arbitrage direction	Sell Y, receive X	trade direction	Moves pool toward market
LP inventory change	+1,500.000000 X; -2,601.887695 Y	token amounts	Capital flow, not LP profit

Balance-sheet view

Pool state transition

Pool assets and LP claims before, during, and after the swap. Fee attribution is included in the gross trader flow, not added twice.

Pool account	Before trade	Trader flow	Fee attribution	After trade
Token X asset	10,000.000000 X	+1,500.000000 X	4.500000 X retained	11,500.000000 X
Token Y asset	20,000.000000 Y	-2,601.887695 Y	—	17,398.112305 Y
LP share claims	Pro-rata claim on pool assets	No shares minted or burned	Fee accrues inside pool assets	Supply unchanged; asset mix changed
Invariant (x × y)	200,000,000.000000 X·Y	Pricing curve applied	Retained input fee can increase k	200,078,291.505372 X·Y
Pool price	2.000000 Y per X	Moves against the trade	Fee included in execution	1.512879 Y per X

Guided comparison

Guided scenarios

Load a scenario, then compare its execution price, reserve movement, and invariant.

1. 01Small trade, low impactSee why a trade that is small relative to pool reserves stays close to the starting spot price.
2. 02Medium trade, visible slippageCompare execution price with spot price as the same pool absorbs a larger order.
3. 03Large trade, high impactObserve the nonlinear reserve and price movement created by a trade equal to half of the starting X reserve.