Tiny Uniswap amounts cause enormous price impact

Question

When exchanging the smallest possible amount of USDC/WETH - 0.000001 USDC - on Uniswap V3 the price impact soars to 99.95%.

Is this accurate, or is this an error in price impact estimation? Why is this high estimation appearing?

The price impact diminishes as the swap amount increases:

USDC amount	price impact
0.000001	99.95%
0.00001	9.0%
0.0001	0.7%
0.001	0.00000000035%
0.01	0.0000000027%	note the price impact grows again as expected
0.1	0.000000026%

Same happens when doing the inverse for WETH/USDC, or for other pairs such as WBTC/DAI, etc.

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I got these estimations from the Uniswap Interface and where it displayed 0.00% I used the price impact estimation logic from the https://uniswap-python.com/

Answer 1

Rounding errors.

Solidity does not have floating-point math, all fractional calculations are done with fixed-point math, meaning that when the amounts are very small, rounding errors become huge. Specifically in the example, USDC ERC-20 token only has 6 decimals, so the 0.000001 USDC value is the smallest value possible - one micro-USDC to be exact.

In order to compute the result amounts in a swap from y to x, formulas 6.11 and 6.12 from the whitepaper are used:

Gamma is the fee paid by the swappers, for example 0.003. In your example then input token is USDC. In the implementation of the delta y calculation, the fee term is rounded up, meaning it can never be zero.

• When y_in = 100 micro-USDC, the expected fee amount is computed by the frontend as 0.3 micro-USDC, but you actually end up paying 1 micro-USD, leading to 1.0 - 0.3 = 0.7% price impact.
• When y_in = 10 micro-USDC, the 1 micro-USDC fee is 10% of the input value, so it shows as nearly 10% price impact.
• When y_in = 1 micro-USDC, the 1 micro-USDC fee is 100% of the input amount, so it shows as nearly 100% price impact. Moreover, you'll get zero wei in the output because all of the input is "eaten" away by the fee!

On the other hand, when y_in = 1000 the expected fee is exactly 3.0 micro-USDC, so there are no rounding errors.