Hyperbolic arc-(co)secant

Average Error: 0.0 → 0.3

Time: 2.3s

Precision: binary64

Math

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

↓

\[\left(-\left(\log \left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\sqrt{x}\right)\right)\right) + \log \left(\frac{1}{\sqrt{\sqrt[3]{x}}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt{\sqrt[3]{x}}}\right)\]

Error

Bits error versus x

Derivation

1. Initial program 0.0

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

3. Applied add-sqr-sqrt 0.0

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

4. Applied *-un-lft-identity 0.0

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

5. Applied times-frac 0.0

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

6. Applied add-sqr-sqrt 0.0

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

7. Applied *-un-lft-identity 0.0

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

8. Applied times-frac 0.0

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

9. Applied distribute-lft-out 0.0

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

10. Applied log-prod 0.2

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

12. Applied add-cube-cbrt 0.2

    \[\leadsto \log \left(\frac{1}{\sqrt{x}}\right) + \log \left(\frac{1}{\sqrt{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}\right)\]

13. Applied sqrt-prod 0.2

    \[\leadsto \log \left(\frac{1}{\sqrt{x}}\right) + \log \left(\frac{1}{\sqrt{x}} + \frac{\sqrt{1 - x \cdot x}}{\color{blue}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}}}\right)\]

14. Applied *-un-lft-identity 0.2

    \[\leadsto \log \left(\frac{1}{\sqrt{x}}\right) + \log \left(\frac{1}{\sqrt{x}} + \frac{\color{blue}{1 \cdot \sqrt{1 - x \cdot x}}}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}}\right)\]

15. Applied times-frac 0.2

    \[\leadsto \log \left(\frac{1}{\sqrt{x}}\right) + \log \left(\frac{1}{\sqrt{x}} + \color{blue}{\frac{1}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\sqrt{1 - x \cdot x}}{\sqrt{\sqrt[3]{x}}}}\right)\]

16. Applied add-cube-cbrt 0.2

    \[\leadsto \log \left(\frac{1}{\sqrt{x}}\right) + \log \left(\frac{1}{\sqrt{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}} + \frac{1}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\sqrt{1 - x \cdot x}}{\sqrt{\sqrt[3]{x}}}\right)\]

17. Applied sqrt-prod 0.2

    \[\leadsto \log \left(\frac{1}{\sqrt{x}}\right) + \log \left(\frac{1}{\color{blue}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}}} + \frac{1}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\sqrt{1 - x \cdot x}}{\sqrt{\sqrt[3]{x}}}\right)\]

18. Applied *-un-lft-identity 0.2

    \[\leadsto \log \left(\frac{1}{\sqrt{x}}\right) + \log \left(\frac{\color{blue}{1 \cdot 1}}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}} + \frac{1}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\sqrt{1 - x \cdot x}}{\sqrt{\sqrt[3]{x}}}\right)\]

19. Applied times-frac 0.2

    \[\leadsto \log \left(\frac{1}{\sqrt{x}}\right) + \log \left(\color{blue}{\frac{1}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{1}{\sqrt{\sqrt[3]{x}}}} + \frac{1}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\sqrt{1 - x \cdot x}}{\sqrt{\sqrt[3]{x}}}\right)\]

20. Applied distribute-lft-out 0.2

    \[\leadsto \log \left(\frac{1}{\sqrt{x}}\right) + \log \color{blue}{\left(\frac{1}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \left(\frac{1}{\sqrt{\sqrt[3]{x}}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt{\sqrt[3]{x}}}\right)\right)}\]

21. Applied log-prod 0.2

    \[\leadsto \log \left(\frac{1}{\sqrt{x}}\right) + \color{blue}{\left(\log \left(\frac{1}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}}\right) + \log \left(\frac{1}{\sqrt{\sqrt[3]{x}}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt{\sqrt[3]{x}}}\right)\right)}\]

22. Applied associate-+r+ 0.3

    \[\leadsto \color{blue}{\left(\log \left(\frac{1}{\sqrt{x}}\right) + \log \left(\frac{1}{\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}}\right)\right) + \log \left(\frac{1}{\sqrt{\sqrt[3]{x}}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt{\sqrt[3]{x}}}\right)}\]

23. Simplified 0.3

    \[\leadsto \color{blue}{\left(-\left(\log \left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\sqrt{x}\right)\right)\right)} + \log \left(\frac{1}{\sqrt{\sqrt[3]{x}}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt{\sqrt[3]{x}}}\right)\]

24. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020173 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))