arccos

Average Error: 0.0 → 0.0

Time: 15.7s

Precision: 64

Math

\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]

↓

\[2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{\frac{\sqrt{\sqrt{1 - x}}}{\sqrt{\sqrt{1 \cdot 1 - x \cdot x}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}}} \cdot \left(\frac{\frac{\sqrt{\sqrt{1 - x}}}{\sqrt{\sqrt{1 \cdot 1 - x \cdot x}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{\sqrt{1 + x}}}}\right)\right) \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

C

double code(double x) {
	return (2.0 * atan(sqrt(((1.0 - x) / (1.0 + x)))));
}

↓

double code(double x) {
	return (2.0 * atan(sqrt(((((sqrt(sqrt((1.0 - x))) / sqrt(sqrt(((1.0 * 1.0) - (x * x))))) / sqrt((cbrt(sqrt(sqrt((1.0 + x)))) * cbrt(sqrt(sqrt((1.0 + x))))))) * (((sqrt(sqrt((1.0 - x))) / sqrt(sqrt(((1.0 * 1.0) - (x * x))))) / sqrt(cbrt(sqrt(sqrt((1.0 + x)))))) * (sqrt((1.0 - x)) / sqrt(sqrt(sqrt((1.0 + x))))))) * (sqrt((1.0 - x)) / sqrt(sqrt((1.0 + x))))))));
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

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

3. Applied add-sqr-sqrt 0.0

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{\color{blue}{\sqrt{1 + x} \cdot \sqrt{1 + x}}}}\right)\]

4. Applied add-sqr-sqrt 0.0

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\color{blue}{\sqrt{1 - x} \cdot \sqrt{1 - x}}}{\sqrt{1 + x} \cdot \sqrt{1 + x}}}\right)\]

5. Applied times-frac 0.0

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{\sqrt{1 - x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{1 - x}}{\sqrt{1 + x}}}}\right)\]
6. Using strategy rm

7. Applied add-sqr-sqrt 0.0

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt{1 - x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\color{blue}{\sqrt{1 + x} \cdot \sqrt{1 + x}}}}}\right)\]

8. Applied sqrt-prod 0.0

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt{1 - x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{1 - x}}{\color{blue}{\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}}}}\right)\]

9. Applied *-un-lft-identity 0.0

   \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt{1 - x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{\color{blue}{1 \cdot \left(1 - x\right)}}}{\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}}}\right)\]

10. Applied sqrt-prod 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt{1 - x}}{\sqrt{1 + x}} \cdot \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1 - x}}}{\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}}}\right)\]

11. Applied times-frac 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\sqrt{1 - x}}{\sqrt{1 + x}} \cdot \color{blue}{\left(\frac{\sqrt{1}}{\sqrt{\sqrt{1 + x}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}\right)}}\right)\]

12. Applied associate-*r* 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\left(\frac{\sqrt{1 - x}}{\sqrt{1 + x}} \cdot \frac{\sqrt{1}}{\sqrt{\sqrt{1 + x}}}\right) \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}}\right)\]

13. Simplified 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\frac{\frac{\sqrt{1 - x}}{\sqrt{1 + x}}}{\sqrt{\sqrt{1 + x}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]
14. Using strategy rm

15. Applied add-sqr-sqrt 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{\sqrt{1 - x}}{\sqrt{1 + x}}}{\sqrt{\sqrt{\color{blue}{\sqrt{1 + x} \cdot \sqrt{1 + x}}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

16. Applied sqrt-prod 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{\sqrt{1 - x}}{\sqrt{1 + x}}}{\sqrt{\color{blue}{\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

17. Applied sqrt-prod 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{\sqrt{1 - x}}{\sqrt{1 + x}}}{\color{blue}{\sqrt{\sqrt{\sqrt{1 + x}}} \cdot \sqrt{\sqrt{\sqrt{1 + x}}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

18. Applied flip-+ 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{\sqrt{1 - x}}{\sqrt{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}}}}}{\sqrt{\sqrt{\sqrt{1 + x}}} \cdot \sqrt{\sqrt{\sqrt{1 + x}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

19. Applied sqrt-div 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\frac{\sqrt{1 - x}}{\color{blue}{\frac{\sqrt{1 \cdot 1 - x \cdot x}}{\sqrt{1 - x}}}}}{\sqrt{\sqrt{\sqrt{1 + x}}} \cdot \sqrt{\sqrt{\sqrt{1 + x}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

20. Applied associate-/r/ 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\frac{\color{blue}{\frac{\sqrt{1 - x}}{\sqrt{1 \cdot 1 - x \cdot x}} \cdot \sqrt{1 - x}}}{\sqrt{\sqrt{\sqrt{1 + x}}} \cdot \sqrt{\sqrt{\sqrt{1 + x}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

21. Applied times-frac 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\left(\frac{\frac{\sqrt{1 - x}}{\sqrt{1 \cdot 1 - x \cdot x}}}{\sqrt{\sqrt{\sqrt{1 + x}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{\sqrt{1 + x}}}}\right)} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]
22. Using strategy rm

23. Applied add-cube-cbrt 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{\frac{\sqrt{1 - x}}{\sqrt{1 \cdot 1 - x \cdot x}}}{\sqrt{\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{\sqrt{1 + x}}}}\right) \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

24. Applied sqrt-prod 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{\frac{\sqrt{1 - x}}{\sqrt{1 \cdot 1 - x \cdot x}}}{\color{blue}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}} \cdot \sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{\sqrt{1 + x}}}}\right) \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

25. Applied add-sqr-sqrt 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{\frac{\sqrt{1 - x}}{\sqrt{\color{blue}{\sqrt{1 \cdot 1 - x \cdot x} \cdot \sqrt{1 \cdot 1 - x \cdot x}}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}} \cdot \sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{\sqrt{1 + x}}}}\right) \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

26. Applied sqrt-prod 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{\frac{\sqrt{1 - x}}{\color{blue}{\sqrt{\sqrt{1 \cdot 1 - x \cdot x}} \cdot \sqrt{\sqrt{1 \cdot 1 - x \cdot x}}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}} \cdot \sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{\sqrt{1 + x}}}}\right) \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

27. Applied add-sqr-sqrt 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{\frac{\sqrt{\color{blue}{\sqrt{1 - x} \cdot \sqrt{1 - x}}}}{\sqrt{\sqrt{1 \cdot 1 - x \cdot x}} \cdot \sqrt{\sqrt{1 \cdot 1 - x \cdot x}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}} \cdot \sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{\sqrt{1 + x}}}}\right) \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

28. Applied sqrt-prod 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{\frac{\color{blue}{\sqrt{\sqrt{1 - x}} \cdot \sqrt{\sqrt{1 - x}}}}{\sqrt{\sqrt{1 \cdot 1 - x \cdot x}} \cdot \sqrt{\sqrt{1 \cdot 1 - x \cdot x}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}} \cdot \sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{\sqrt{1 + x}}}}\right) \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

29. Applied times-frac 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{\color{blue}{\frac{\sqrt{\sqrt{1 - x}}}{\sqrt{\sqrt{1 \cdot 1 - x \cdot x}}} \cdot \frac{\sqrt{\sqrt{1 - x}}}{\sqrt{\sqrt{1 \cdot 1 - x \cdot x}}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}} \cdot \sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{\sqrt{1 + x}}}}\right) \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

30. Applied times-frac 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\left(\color{blue}{\left(\frac{\frac{\sqrt{\sqrt{1 - x}}}{\sqrt{\sqrt{1 \cdot 1 - x \cdot x}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}}} \cdot \frac{\frac{\sqrt{\sqrt{1 - x}}}{\sqrt{\sqrt{1 \cdot 1 - x \cdot x}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}}}}\right)} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{\sqrt{1 + x}}}}\right) \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

31. Applied associate-*l* 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{\left(\frac{\frac{\sqrt{\sqrt{1 - x}}}{\sqrt{\sqrt{1 \cdot 1 - x \cdot x}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}}} \cdot \left(\frac{\frac{\sqrt{\sqrt{1 - x}}}{\sqrt{\sqrt{1 \cdot 1 - x \cdot x}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{\sqrt{1 + x}}}}\right)\right)} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

32. Final simplification 0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\left(\frac{\frac{\sqrt{\sqrt{1 - x}}}{\sqrt{\sqrt{1 \cdot 1 - x \cdot x}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}}} \cdot \left(\frac{\frac{\sqrt{\sqrt{1 - x}}}{\sqrt{\sqrt{1 \cdot 1 - x \cdot x}}}}{\sqrt{\sqrt[3]{\sqrt{\sqrt{1 + x}}}}} \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{\sqrt{1 + x}}}}\right)\right) \cdot \frac{\sqrt{1 - x}}{\sqrt{\sqrt{1 + x}}}}\right)\]

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(FPCore (x)
  :name "arccos"
  :precision binary64
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))