Math and ChatGPT

Performing symbolic math steps is often related to pattern recognition. In theory, ChatGPT could be doing a good job here. I wanted to find the inverse of $$f(x) = {\mathrm{sign}}(x) \log(1+|x|)$$ This function is a form of a signed logarithmic scaling. So, let's see what ChatGPT is telling us:

The idea to split this into two branches is excellent. Strictly speaking, we should handle $x=0$ also in one of the branches, but let's ignore that for now. More worisome, somehow, it looses the ${\mathrm{sign}}(x)$ function in step 2 (needed for the $x\lt 0$ branch). So, let's inform ChatGPT about this:

It says, "Let's include the ${\mathrm{sign}}(x)$ factor," but then it does not actually do that.

Doing the inverse by hand yields: $$f^{-1}(x) = {\mathrm{sign}}(x)(\exp(|x|)-1)$$

Hmm. ChatGPT does not really understand much of my feedback. It is rather polite, but it is just ignoring what I say.

Update 1: Wolfram Alpha

Wolfram Alpha [1] is based on the same symbolic math foundations as Mathematica. I tried the command:

inverse of sign(x)*log(1+|x|)

The response is:

This does not look too good either:

• The inverse function is incorrect 
• and the blue line is not the plot of $e^x-1$. 

It is noted that ChatGPT can use Wolfram Alpha [2].

Update 2: Sage

Sage is a well-known open-source symbolic math package. An online version is available [3]. This does not seem to work either:

Sage cannot solve this. I tried doing it for the branches $x\ge 0$ and $g \lt 0$ separately. That worked fine. The complete function, however, seems out of reach.

Update 3: Google Gemini

Not much better:

Also wrong!! The correct answer is: $$f^{-1}(y) = \begin{cases}e^y-1 & \text{if $y>0$} \\ 1-e^{-y} & \text{if $y<0$} \\  0 & \text{if $y=0$} \end{cases}$$

Conclusion

A pen and a piece of paper is sometimes the most reliable tool to do math.

References

1. https://www.wolframalpha.com/
2. ChatGPT Gets Its “Wolfram Superpowers”!, https://writings.stephenwolfram.com/2023/03/chatgpt-gets-its-wolfram-superpowers/
3. https://sagecell.sagemath.org/