Average Error: 0.5 → 0.5
Time: 16.2s
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 r142223 = 1.0;
        double r142224 = 5.0;
        double r142225 = v;
        double r142226 = r142225 * r142225;
        double r142227 = r142224 * r142226;
        double r142228 = r142223 - r142227;
        double r142229 = r142226 - r142223;
        double r142230 = r142228 / r142229;
        double r142231 = acos(r142230);
        return r142231;
}
↓
double f(double v) {
        double r142232 = 1.0;
        double r142233 = v;
        double r142234 = r142233 * r142233;
        double r142235 = 5.0;
        double r142236 = r142234 * r142235;
        double r142237 = r142232 - r142236;
        double r142238 = r142234 - r142232;
        double r142239 = r142237 / r142238;
        double r142240 = acos(r142239);
        double r142241 = log(r142240);
        double r142242 = sqrt(r142241);
        double r142243 = exp(r142242);
        double r142244 = 2.0;
        double r142245 = r142235 * r142233;
        double r142246 = exp(r142245);
        double r142247 = cbrt(r142246);
        double r142248 = log(r142247);
        double r142249 = r142244 * r142248;
        double r142250 = r142249 * r142233;
        double r142251 = pow(r142247, r142233);
        double r142252 = log(r142251);
        double r142253 = r142250 + r142252;
        double r142254 = r142232 - r142253;
        double r142255 = pow(r142233, r142244);
        double r142256 = r142255 - r142232;
        double r142257 = r142254 / r142256;
        double r142258 = acos(r142257);
        double r142259 = log(r142258);
        double r142260 = sqrt(r142259);
        double r142261 = pow(r142243, r142260);
        return r142261;
}

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))))