Math

Fixed-point arithmetic libraries operating at 1e18 scale — the arithmetic backbone of the Fira protocol.

Source: src/fira_bonding/libraries/math/PMath.sol, src/fira_bonding/libraries/math/LogExpMath.sol

PMath

Core fixed-point math used throughout the protocol. Constants: ONE = 1e18, IONE = int256(1e18).

Key operations

Function	Description	Rounding
mulDown(a, b)	(a × b) / 1e18	Toward zero (conservative for user)
divDown(a, b)	(a × 1e18) / b	Toward zero
rawDivUp(a, b)	(a + b - 1) / b	Up (conservative for protocol)
abs(a)	Absolute value of int256	—
neg(a)	Negation (BT/FW flows are signed)	—
subNoNeg(a, b)	a - b, reverts if negative	—
sqrt(x)	Babylonian method integer square root	—
Int() / Uint()	Safe int256 ↔ uint256 casting	—
Int128() / Uint96()	Narrowing casts for storage packing	—

LogExpMath

Ported from Balancer's math library. Implements ln(x) and exp(x) for signed 256-bit integers at 1e18 scale.

Usage in the protocol

• exp(x) — Converts between implied rates and exchange rates: exchangeRate = exp(lnImpliedRate × timeToExpiry / YEAR)

• ln(x) — After computing a post-trade exchange rate, converts back to log-space implied rate for storage and TWAP tracking

Both functions use lookup tables and polynomial approximations. They revert on out-of-bounds inputs (e.g., ln(0) or overflow in exp). Precision is critical — rounding errors in exp compound over many trades and could allow arbitrage.