Error

Derivation

1. Initial program 0.0

   \[-\log \left(\frac{1}{x} - 1\right)\]
2. Using strategy rm

3. Applied add-sqr-sqrt0.0

   \[\leadsto -\log \color{blue}{\left(\sqrt{\frac{1}{x} - 1} \cdot \sqrt{\frac{1}{x} - 1}\right)}\]

4. Applied log-prod0.0

   \[\leadsto -\color{blue}{\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\sqrt{\frac{1}{x} - 1}\right)\right)}\]
5. Using strategy rm

6. Applied *-un-lft-identity0.0

   \[\leadsto -\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\sqrt{\frac{1}{x} - \color{blue}{1 \cdot 1}}\right)\right)\]

7. Applied add-sqr-sqrt0.0

   \[\leadsto -\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\sqrt{\frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} - 1 \cdot 1}\right)\right)\]

8. Applied *-un-lft-identity0.0

   \[\leadsto -\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\sqrt{\frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x}} - 1 \cdot 1}\right)\right)\]

9. Applied times-frac0.0

   \[\leadsto -\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\sqrt{\color{blue}{\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x}}} - 1 \cdot 1}\right)\right)\]

10. Applied difference-of-squares0.0

    \[\leadsto -\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\sqrt{\color{blue}{\left(\frac{1}{\sqrt{x}} + 1\right) \cdot \left(\frac{1}{\sqrt{x}} - 1\right)}}\right)\right)\]

11. Applied sqrt-prod0.0

    \[\leadsto -\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \color{blue}{\left(\sqrt{\frac{1}{\sqrt{x}} + 1} \cdot \sqrt{\frac{1}{\sqrt{x}} - 1}\right)}\right)\]

12. Final simplification0.0

    \[\leadsto -\left(\log \left(\sqrt{\frac{1}{\sqrt{x}} - 1} \cdot \sqrt{1 + \frac{1}{\sqrt{x}}}\right) + \log \left(\sqrt{\frac{1}{x} - 1}\right)\right)\]

Reproduce

herbie shell --seed 2019088 
(FPCore (x)
  :name "neg log"
  (- (log (- (/ 1 x) 1))))