Average Error: 0.5 → 0.6
Time: 5.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\sqrt[3]{{\left(\sqrt{1} + \sqrt{5} \cdot \sqrt{{v}^{2}}\right)}^{3} \cdot {\left(\frac{\sqrt{1} - \sqrt{5} \cdot \sqrt{{v}^{2}}}{{v}^{2} - 1}\right)}^{3}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\sqrt[3]{{\left(\sqrt{1} + \sqrt{5} \cdot \sqrt{{v}^{2}}\right)}^{3} \cdot {\left(\frac{\sqrt{1} - \sqrt{5} \cdot \sqrt{{v}^{2}}}{{v}^{2} - 1}\right)}^{3}}\right)
double code(double v) {
	return ((double) acos(((double) (((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (v * v)) - 1.0))))));
}
↓
double code(double v) {
	return ((double) acos(((double) cbrt(((double) (((double) pow(((double) (((double) sqrt(1.0)) + ((double) (((double) sqrt(5.0)) * ((double) sqrt(((double) pow(v, 2.0)))))))), 3.0)) * ((double) pow(((double) (((double) (((double) sqrt(1.0)) - ((double) (((double) sqrt(5.0)) * ((double) sqrt(((double) pow(v, 2.0)))))))) / ((double) (((double) pow(v, 2.0)) - 1.0)))), 3.0))))))));
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied add-cbrt-cube0.6

    \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\sqrt[3]{\left(\left(v \cdot v - 1\right) \cdot \left(v \cdot v - 1\right)\right) \cdot \left(v \cdot v - 1\right)}}}\right)\]

  4. Applied add-cbrt-cube0.6

    \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\sqrt[3]{\left(\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \left(1 - 5 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(1 - 5 \cdot \left(v \cdot v\right)\right)}}}{\sqrt[3]{\left(\left(v \cdot v - 1\right) \cdot \left(v \cdot v - 1\right)\right) \cdot \left(v \cdot v - 1\right)}}\right)\]

  5. Applied cbrt-undiv0.6

    \[\leadsto \cos^{-1} \color{blue}{\left(\sqrt[3]{\frac{\left(\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \left(1 - 5 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(1 - 5 \cdot \left(v \cdot v\right)\right)}{\left(\left(v \cdot v - 1\right) \cdot \left(v \cdot v - 1\right)\right) \cdot \left(v \cdot v - 1\right)}}\right)}\]

  6. Simplified0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{\color{blue}{{\left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}^{3}}}\right)\]
  7. Using strategy rm

  8. Applied *-un-lft-identity0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{{\left(\frac{1 - 5 \cdot {v}^{2}}{\color{blue}{1 \cdot \left({v}^{2} - 1\right)}}\right)}^{3}}\right)\]

  9. Applied add-sqr-sqrt0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{{\left(\frac{1 - 5 \cdot \color{blue}{\left(\sqrt{{v}^{2}} \cdot \sqrt{{v}^{2}}\right)}}{1 \cdot \left({v}^{2} - 1\right)}\right)}^{3}}\right)\]

  10. Applied add-sqr-sqrt0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{{\left(\frac{1 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{5}\right)} \cdot \left(\sqrt{{v}^{2}} \cdot \sqrt{{v}^{2}}\right)}{1 \cdot \left({v}^{2} - 1\right)}\right)}^{3}}\right)\]

  11. Applied unswap-sqr0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{{\left(\frac{1 - \color{blue}{\left(\sqrt{5} \cdot \sqrt{{v}^{2}}\right) \cdot \left(\sqrt{5} \cdot \sqrt{{v}^{2}}\right)}}{1 \cdot \left({v}^{2} - 1\right)}\right)}^{3}}\right)\]

  12. Applied add-sqr-sqrt0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{{\left(\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}} - \left(\sqrt{5} \cdot \sqrt{{v}^{2}}\right) \cdot \left(\sqrt{5} \cdot \sqrt{{v}^{2}}\right)}{1 \cdot \left({v}^{2} - 1\right)}\right)}^{3}}\right)\]

  13. Applied difference-of-squares0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{{\left(\frac{\color{blue}{\left(\sqrt{1} + \sqrt{5} \cdot \sqrt{{v}^{2}}\right) \cdot \left(\sqrt{1} - \sqrt{5} \cdot \sqrt{{v}^{2}}\right)}}{1 \cdot \left({v}^{2} - 1\right)}\right)}^{3}}\right)\]

  14. Applied times-frac0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{{\color{blue}{\left(\frac{\sqrt{1} + \sqrt{5} \cdot \sqrt{{v}^{2}}}{1} \cdot \frac{\sqrt{1} - \sqrt{5} \cdot \sqrt{{v}^{2}}}{{v}^{2} - 1}\right)}}^{3}}\right)\]

  15. Applied unpow-prod-down0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{\color{blue}{{\left(\frac{\sqrt{1} + \sqrt{5} \cdot \sqrt{{v}^{2}}}{1}\right)}^{3} \cdot {\left(\frac{\sqrt{1} - \sqrt{5} \cdot \sqrt{{v}^{2}}}{{v}^{2} - 1}\right)}^{3}}}\right)\]

  16. Simplified0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{\color{blue}{{\left(\sqrt{1} + \sqrt{5} \cdot \sqrt{{v}^{2}}\right)}^{3}} \cdot {\left(\frac{\sqrt{1} - \sqrt{5} \cdot \sqrt{{v}^{2}}}{{v}^{2} - 1}\right)}^{3}}\right)\]

  17. Final simplification0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{{\left(\sqrt{1} + \sqrt{5} \cdot \sqrt{{v}^{2}}\right)}^{3} \cdot {\left(\frac{\sqrt{1} - \sqrt{5} \cdot \sqrt{{v}^{2}}}{{v}^{2} - 1}\right)}^{3}}\right)\]

Reproduce

herbie shell --seed 2020113 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))