Average Error: 0.5 → 0.5
Time: 16.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \left(\left(2 \cdot \log \left(\sqrt[3]{e^{5 \cdot v}}\right)\right) \cdot v + \log \left({\left(\sqrt[3]{e^{5 \cdot v}}\right)}^{v}\right)\right)}{{v}^{2} - 1}\right)\right)}\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \left(\left(2 \cdot \log \left(\sqrt[3]{e^{5 \cdot v}}\right)\right) \cdot v + \log \left({\left(\sqrt[3]{e^{5 \cdot v}}\right)}^{v}\right)\right)}{{v}^{2} - 1}\right)\right)}\right)}
double f(double v) {
        double r227848 = 1.0;
        double r227849 = 5.0;
        double r227850 = v;
        double r227851 = r227850 * r227850;
        double r227852 = r227849 * r227851;
        double r227853 = r227848 - r227852;
        double r227854 = r227851 - r227848;
        double r227855 = r227853 / r227854;
        double r227856 = acos(r227855);
        return r227856;
}
↓
double f(double v) {
        double r227857 = 1.0;
        double r227858 = v;
        double r227859 = r227858 * r227858;
        double r227860 = 5.0;
        double r227861 = r227859 * r227860;
        double r227862 = r227857 - r227861;
        double r227863 = r227859 - r227857;
        double r227864 = r227862 / r227863;
        double r227865 = acos(r227864);
        double r227866 = log(r227865);
        double r227867 = sqrt(r227866);
        double r227868 = exp(r227867);
        double r227869 = 2.0;
        double r227870 = r227860 * r227858;
        double r227871 = exp(r227870);
        double r227872 = cbrt(r227871);
        double r227873 = log(r227872);
        double r227874 = r227869 * r227873;
        double r227875 = r227874 * r227858;
        double r227876 = pow(r227872, r227858);
        double r227877 = log(r227876);
        double r227878 = r227875 + r227877;
        double r227879 = r227857 - r227878;
        double r227880 = pow(r227858, r227869);
        double r227881 = r227880 - r227857;
        double r227882 = r227879 / r227881;
        double r227883 = acos(r227882);
        double r227884 = log(r227883);
        double r227885 = sqrt(r227884);
        double r227886 = pow(r227868, r227885);
        return r227886;
}

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. Simplified0.5

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

  4. Applied add-exp-log0.5

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

  5. Simplified0.5

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied exp-prod0.5

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

  9. Simplified0.5

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

  11. Applied add-log-exp0.5

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

  12. Simplified0.5

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

  14. Applied add-cube-cbrt0.5

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

  15. Applied unpow-prod-down0.5

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

  16. Applied log-prod0.5

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

  17. Simplified0.5

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

  18. Simplified0.5

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

  19. Final simplification0.5

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

Reproduce

herbie shell --seed 2019194 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))