Average Error: 0.1 → 0.1
Time: 7.1s
Precision: 64
Internal Precision: 128
\[-\log \left(\frac{1}{x} - 1\right)\]
↓
\[-\left(\log \left(\sqrt{\frac{1}{x}} - 1\right) + \log_* (1 + \sqrt{\frac{1}{x}})\right)\]

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

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

  4. Applied difference-of-sqr-10.1

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

  5. Applied log-prod0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x)
  :name "neg log"
  (- (log (- (/ 1 x) 1))))