Stable curve (x³y + xy³ = k)

For assets expected to trade near 1:1 (for example USDC/USDT), constant product wastes liquidity on prices that never happen. The stable invariant is much flatter around the peg:

Amounts are first normalized to 18 decimals, so a 6-decimal USDC and an 18-decimal DAI compare fairly.

Solving for the output

The implementation solves the integer invariant with bounded bisection. The derivative explains the curve, but is not the executable solver:

Bisection selects the smallest retained output reserve whose integer invariant meets the target. Stable normalized reserves are bounded at 10^28; this costs more iterations than Newton but avoids zero derivatives and overshoot. See NeuraMath and compatibility.

Volatile vs stable, same liquidity

 price   │ volatile  ╲   │            ╲__   │ stable  ──────────────╮   │                        ╲___   └──────────────────────────────── amount swapped

Near the peg the stable curve gives far less slippage; far from it, slippage rises sharply so the pool cannot be drained at a bad price.

Safety rule

The first deposit into a stable pool must be balanced in normalized units, and $k$ must exceed a minimum. Initialization alone is insufficient: stable LP locking uses a decimal-aware normalized floor, and post-update invariant must remain above 1,000, including after burns. These are tested boundaries, not a guarantee against all attacks.