neg log

Average Error: 0.0 → 0.0

Time: 2.7s

Precision: 64

Math

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

↓

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

Error

Bits error versus x

Derivation

1. Initial program 0.0

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

3. Applied add-sqr-sqrt 0.0

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

4. Applied add-sqr-sqrt 0.0

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

5. Applied add-sqr-sqrt 0.0

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

6. Applied times-frac 0.0

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

7. Applied difference-of-squares 0.0

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

8. Applied log-prod 0.0

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

9. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020124 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1.0 x) 1.0))))