Average Error: 0.0 → 0.0
Time: 22.5s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{\frac{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}{\left(4 \cdot 4\right) \cdot 4} \cdot \left(\left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt[3]{\frac{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}{\left(4 \cdot 4\right) \cdot 4} \cdot \left(\left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)}
double f(double v) {
        double r281738 = 2.0;
        double r281739 = sqrt(r281738);
        double r281740 = 4.0;
        double r281741 = r281739 / r281740;
        double r281742 = 1.0;
        double r281743 = 3.0;
        double r281744 = v;
        double r281745 = r281744 * r281744;
        double r281746 = r281743 * r281745;
        double r281747 = r281742 - r281746;
        double r281748 = sqrt(r281747);
        double r281749 = r281741 * r281748;
        double r281750 = r281742 - r281745;
        double r281751 = r281749 * r281750;
        return r281751;
}
↓
double f(double v) {
        double r281752 = 2.0;
        double r281753 = sqrt(r281752);
        double r281754 = r281753 * r281753;
        double r281755 = r281754 * r281753;
        double r281756 = 4.0;
        double r281757 = r281756 * r281756;
        double r281758 = r281757 * r281756;
        double r281759 = r281755 / r281758;
        double r281760 = 1.0;
        double r281761 = v;
        double r281762 = r281761 * r281761;
        double r281763 = r281760 - r281762;
        double r281764 = r281763 * r281763;
        double r281765 = r281764 * r281763;
        double r281766 = 3.0;
        double r281767 = r281766 * r281762;
        double r281768 = r281760 - r281767;
        double r281769 = sqrt(r281768);
        double r281770 = r281769 * r281769;
        double r281771 = r281770 * r281769;
        double r281772 = r281765 * r281771;
        double r281773 = r281759 * r281772;
        double r281774 = cbrt(r281773);
        return r281774;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
  2. Using strategy rm

  3. Applied add-log-exp0.0

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \color{blue}{\log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}}\right) \cdot \left(1 - v \cdot v\right)\]
  4. Using strategy rm

  5. Applied add-cbrt-cube0.0

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}\right) \cdot \color{blue}{\sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)}}\]

  6. Applied add-cbrt-cube0.0

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)} \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}\right) \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}}}\right) \cdot \sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)}\]

  7. Applied add-cbrt-cube0.0

    \[\leadsto \left(\frac{\sqrt{2}}{\color{blue}{\sqrt[3]{\left(4 \cdot 4\right) \cdot 4}}} \cdot \sqrt[3]{\left(\sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)} \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}\right) \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}}\right) \cdot \sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)}\]

  8. Applied add-cbrt-cube1.0

    \[\leadsto \left(\frac{\color{blue}{\sqrt[3]{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}}}{\sqrt[3]{\left(4 \cdot 4\right) \cdot 4}} \cdot \sqrt[3]{\left(\sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)} \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}\right) \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}}\right) \cdot \sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)}\]

  9. Applied cbrt-undiv0.0

    \[\leadsto \left(\color{blue}{\sqrt[3]{\frac{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}{\left(4 \cdot 4\right) \cdot 4}}} \cdot \sqrt[3]{\left(\sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)} \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}\right) \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}}\right) \cdot \sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)}\]

  10. Applied cbrt-unprod0.0

    \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}{\left(4 \cdot 4\right) \cdot 4} \cdot \left(\left(\sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)} \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}\right) \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}\right)}} \cdot \sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)}\]

  11. Applied cbrt-unprod0.0

    \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}{\left(4 \cdot 4\right) \cdot 4} \cdot \left(\left(\sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)} \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}\right) \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}\right)\right) \cdot \left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right)}}\]

  12. Simplified0.0

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\sqrt{2}}{4} \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)}^{3}}}\]
  13. Using strategy rm

  14. Applied add-cbrt-cube0.0

    \[\leadsto \sqrt[3]{{\left(\frac{\sqrt{2}}{4} \cdot \left(\left(1 - v \cdot v\right) \cdot \color{blue}{\sqrt[3]{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}\right)\right)}^{3}}\]

  15. Applied add-cbrt-cube0.0

    \[\leadsto \sqrt[3]{{\left(\frac{\sqrt{2}}{4} \cdot \left(\color{blue}{\sqrt[3]{\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)}} \cdot \sqrt[3]{\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right)\right)}^{3}}\]

  16. Applied cbrt-unprod0.0

    \[\leadsto \sqrt[3]{{\left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\sqrt[3]{\left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}}\right)}^{3}}\]

  17. Applied add-cbrt-cube0.0

    \[\leadsto \sqrt[3]{{\left(\frac{\sqrt{2}}{\color{blue}{\sqrt[3]{\left(4 \cdot 4\right) \cdot 4}}} \cdot \sqrt[3]{\left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}\right)}^{3}}\]

  18. Applied add-cbrt-cube1.0

    \[\leadsto \sqrt[3]{{\left(\frac{\color{blue}{\sqrt[3]{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}}}{\sqrt[3]{\left(4 \cdot 4\right) \cdot 4}} \cdot \sqrt[3]{\left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}\right)}^{3}}\]

  19. Applied cbrt-undiv0.0

    \[\leadsto \sqrt[3]{{\left(\color{blue}{\sqrt[3]{\frac{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}{\left(4 \cdot 4\right) \cdot 4}}} \cdot \sqrt[3]{\left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}\right)}^{3}}\]

  20. Applied cbrt-unprod0.0

    \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt[3]{\frac{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}{\left(4 \cdot 4\right) \cdot 4} \cdot \left(\left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)}\right)}}^{3}}\]

  21. Applied rem-cube-cbrt0.0

    \[\leadsto \sqrt[3]{\color{blue}{\frac{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}{\left(4 \cdot 4\right) \cdot 4} \cdot \left(\left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)}}\]

  22. Final simplification0.0

    \[\leadsto \sqrt[3]{\frac{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}}{\left(4 \cdot 4\right) \cdot 4} \cdot \left(\left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \left(\left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\right)}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))