The constant-product formula x·y=k is just one point on a spectrum of AMM invariant curves. Different invariants trade off price impact, capital efficiency, and slippage in distinct ways.

Constant Sum (x+y=k): zero slippage but drains completely if prices diverge — used rarely, mainly in stable-asset pools with tight bands.
StableSwap (Curve): a hybrid that behaves like constant-sum near the peg and constant-product far from it, using an amplification coefficient A.
Concentrated Liquidity (Uniswap v3): LPs choose price ranges, multiplying capital efficiency by 10–4000x versus v2.
Weighted Pools (Balancer): generalize to x^w1 · y^w2 = k, enabling pools like 80/20 that reduce impermanent loss.

Invariant	Slippage	Capital Efficiency
x·y=k	Moderate	Baseline (1x)
StableSwap	Very low near peg	10–100x for stables
Concentrated	Range-dependent	Up to 4000x