I am maintaining an old piece of financial software. In the source code I have found an implementation of the following formula:
$$p2 \over (p1 + 1) - (p1 * p2)$$
The formula is used as part of some sort of financial tax calculation. If it is of any help, both p1 and p2 are percentages expressed as factors. For instance, if p1 represents 5% then the actual value of p1 is 0.05. In addition, I can say that p1 represents the tax percentage, whereas the meaning of p2 is not clear to me - the only thing I know is that it is another percentage.
In order to understand the meaning of p2 and what the software does (it's undocumented, of course), it might help if I understood what the formula expresses. So my question to this community: Does anyone here recognize the purpose of the formula?
The formula rewritten as a transformation of $p_2$ looks like
$$p_2'={p_2\over 1+p_1-p_1\cdot p_2}$$
and is not immediately recognizable as a "typical" result in any particular field that I am aware of. The effect is, as you probably know, that $p_2$ is modified according to the current value of $p_1$. Moreover, this effect is that $p_2'$ is smaller than $p_2$ when $p_1\gt 0, p_2\in[0,1]$, which would be the typical case for positive tax percentages.
The formula creates the "simple" ratio and difference relationships
$${p_2\over p_2'}=1+p_1-p_1\cdot p_2\\ p_2-p_2'={p_1\cdot p_2-p_1\cdot p_2^2\over 1+p_1-p_1\cdot p_2}$$
If it were possible that the formula were "wrong" and needed to be corrected, I would guess that a term $-p_2$ was missing from the denominator, like
$$p_2''={p_2\over 1+p_1-p_2-p_1\cdot p_2}={p_2\over (1+p_1)(1-p_2)}$$
which at the very least makes for a more symmetric formula.
Otherwise, "nope, never seen that before..."