Error

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Derivation

1. Initial program 0.0

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

2. Initial simplification0.0

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

4. Applied div-inv0.0

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

5. Applied distribute-rgt1-in0.0

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

6. Applied log-prod0.2

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

7. Simplified0.2

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

9. Applied add-sqr-sqrt0.2

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

10. Applied log-prod0.2

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

11. Applied associate-+r+0.2

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

12. Final simplification0.2

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

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))

Details

Time bar (total: 6.3s)Debug log