Can someone explain Uniswap's v2 getAmountIn?

Question

Code can be seen here: https://github.com/Uniswap/v2-periphery/blob/master/contracts/libraries/UniswapV2Library.sol#L53

What I don't understand is that if you use the constant product formula, you arrive at a different formula:

• invariant formula: resIn * resOut = k
• therefore, (resIn+amountIn)*(resOut-amountOut)=k
• solving for amountIn, we get amountIn = (resIn*resOut) / (resOut-amountOut) - resIn
• But the contract has a version that's not equivalent, even after accounting for fees.

Note: the above doesn’t include fees

    function getAmountIn(uint amountOut, uint reserveIn, uint reserveOut) internal pure returns (uint amountIn) {
        require(amountOut > 0, 'UniswapV2Library: INSUFFICIENT_OUTPUT_AMOUNT');
        require(reserveIn > 0 && reserveOut > 0, 'UniswapV2Library: INSUFFICIENT_LIQUIDITY');
        uint numerator = reserveIn.mul(amountOut).mul(1000);
        uint denominator = reserveOut.sub(amountOut).mul(997);
        amountIn = (numerator / denominator).add(1);
    }

Answer 1

You can take a look at this calculation for the version of the contract:

what is math for uniswap calculates the amountout and amountin why 997 and 1000

In addition, the contract's version also added +1 .add(1) to round up as Uniswap pool strictly enforces k and make you pay one more wei to the LPs:

https://www.reddit.com/r/UniSwap/comments/rjzkcj/why_does_it_have_add1_uniswapv2library_getamountin/