Average Error: 0.1 → 0.0
Time: 2.5s
Precision: binary64
\[-\log \left(\frac{1}{x} - 1\right)\]
↓
\[\left(-\log \left(\sqrt{1} + \frac{\sqrt{1}}{\sqrt{x}}\right)\right) - \log \left(\frac{\sqrt{1}}{\sqrt{x}} - \sqrt{1}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 0.1

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

  3. Applied add-sqr-sqrt0.1

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

  4. Applied add-sqr-sqrt0.1

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied times-frac0.1

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

  7. Applied difference-of-squares0.1

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

  8. Applied log-prod0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (neg (log (- (/ 1.0 x) 1.0))))