to find the best path to trade a particular option/pair, but routing can be handled with help from the UI of DAPP.

The marginal price offered by SFSwap (with zero trades) at time can be computed by dividing the reserves of asset $a$ by the reserves of asset $b$.

$$p_t = \frac{r^a_t}{r^b_t}$$

But SFSwap improves this oracle functionality by measuring and recording the price before the first trade of each block (or equivalently, after the last trade of the previous block). This price is more difficult to manipulate than prices during a block. If the attacker submits a transaction that attempts to manipulate the price at the end of a block, some other arbitrageur may be able to submit another transaction to trade back immediately afterward in the same block. A miner (or an attacker who uses enough gas to fill an entire block) could manipulate the price at the end of a block, but unless they mine the next block as well, they may not have a particular advantage in arbitraging the trade back.

Specifically, SFSwap accumulates this price, by keeping track of the cumulative sum of prices at the beginning of each block in which someone interacts with the contract. Each price is weighted by the amount of time that has passed since the last block in which it was updated, according to the block timestamp. This means that the accumulator value at any given time (after being updated) should be the sum of the spot price at each second in the history of the contract.

$$a_t = \sum_{i=1}^t{p_i}$$

To estimate the time-weighted average price from time t1 to t2, an external caller can checkpoint the accumulator’s value at t1 and then again at t2, subtract the first value from the second, and divide by the number of seconds elapsed. (Note that the contract itself does not store historical values for this accumulator—the caller has to call the contract at the beginning of the period to read and store this value.)

$$p_{t_1,t_2} = \frac{\sum_{i=t_1}^{t_2} p_i}{t_2 - t_1} = \frac{\sum_{i=1}^{t_2} p_i - \sum_{i=1}^{t_1} p_i}{t_2 - t_1} = \frac{a_{t_2} - a_{t_1}}{t_2 - t_1}$$

Users of the oracle can choose when to start and end this period. Choosing a longer period makes it more expensive for an attacker to manipulate the TWAP, although it results in a less up-to-date price.

One complication: should we measure the price of asset A in terms of asset B, or the price of asset B in terms of asset A? While the spot price of A in terms of B is always the reciprocal of the spot price of B in terms of A, the mean price of asset A in terms of asset B over a particular period of time is not equal to the reciprocal of the mean price of asset B in terms of asset A.3 For example, if the USD/ETH price is 100 in block 1 and 300 in block 2, the average USD/ETH price will be 200 USD/ETH, but the average ETH/USD price will be 1/150 ETH/USD. Since the contract cannot know which of the two assets users would want to use as the unit of account, Uniswap v2 tracks both prices.

Another complication is that it is possible for someone to send assets to the pair contract—and thus change its balances and marginal price—without interacting with it, and thus without triggering an oracle update. If the contract simply checked its own balances and updated the oracle based on the current price, an attacker could manipulate the oracle by sending an asset to the contract immediately before calling it for the first time in a block. If the last trade was in a block whose timestamp was X seconds ago, the contract would incorrectly multiply the new price by X before accumulating it, even though nobody has had an opportunity to trade at that price. To prevent this, the core contract caches its reserves after each interaction, and updates the oracle using the price derived from the cached reserves rather than the current reserves. In addition to protecting the oracle from manipulation, this change enables the contract re-architecture described below in section 3.2.

Because Solidity does not have first-class support for non-integer numeric data types, the Uniswap v2 uses a simple binary fixed point format to encode and manipulate prices. Specifically, prices at a given moment are stored as UQ112.112 numbers, meaning that 112 fractional bits of precision are specified on either side of the decimal point, with no sign.

These numbers have a range of $[0, 2^{112} - 1]$ and a precision of $\frac {1} {2^{112}}$.

The UQ112.112 format was chosen for a pragmatic reason — because these numbers can be stored in a uint224, this leaves 32 bits of a 256 bit storage slot free. It also happens that the reserves, each stored in a uint112, also leave 32 bits free in a (packed) 256 bit storage slot. These free spaces are used for the accumulation process described above. Specifically, the reserves are stored alongside the timestamp of the most recent block with at least one trade, modded with $2^{32}$ so that it fits into 32 bits. Additionally, although the price at any given moment (stored as a UQ112.112 number) is guaranteed to fit in 224 bits, the accumulation of this price over an interval is not. The extra 32 bits on the end of the storage slots for the accumulated price of A/B and B/A are used to store overflow bits resulting from repeated summations of prices. This design means that the price oracle only adds an additional three SSTORE operations (a current cost of about 15,000 gas) to the first trade in each block.

The primary downside is that 32 bits isn’t quite enough to store timestamp values that will reasonably never overflow. In fact, the date when the Unix timestamp overflows a uint32 is 02/07/2106. To ensure that this system continues to function properly after this date, and every multiple of 232 − 1 seconds thereafter, oracles are simply required to check prices at least once per interval (approximately 136 years). This is because the core method of accumulation (and modding of timestamp), is actually overflow-safe, meaning that trades across overflow intervals can be appropriately accounted for given that oracles are using the proper (simple) overflow arithmetic to compute deltas.