Average Error: 0.0 → 0.0
Time: 19.5s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{\left(\log \left(e \cdot e^{-3 \cdot \left(v \cdot v\right)}\right) \cdot \left(\frac{1}{32} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1 - \left(-3 \cdot \left(v \cdot v\right)\right) \cdot \left(-3 \cdot \left(v \cdot v\right)\right)}{1 - -3 \cdot \left(v \cdot v\right)}}} \cdot \left(1 - v \cdot v\right)\]
\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]{\left(\log \left(e \cdot e^{-3 \cdot \left(v \cdot v\right)}\right) \cdot \left(\frac{1}{32} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1 - \left(-3 \cdot \left(v \cdot v\right)\right) \cdot \left(-3 \cdot \left(v \cdot v\right)\right)}{1 - -3 \cdot \left(v \cdot v\right)}}} \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r6281714 = 2.0;
        double r6281715 = sqrt(r6281714);
        double r6281716 = 4.0;
        double r6281717 = r6281715 / r6281716;
        double r6281718 = 1.0;
        double r6281719 = 3.0;
        double r6281720 = v;
        double r6281721 = r6281720 * r6281720;
        double r6281722 = r6281719 * r6281721;
        double r6281723 = r6281718 - r6281722;
        double r6281724 = sqrt(r6281723);
        double r6281725 = r6281717 * r6281724;
        double r6281726 = r6281718 - r6281721;
        double r6281727 = r6281725 * r6281726;
        return r6281727;
}
↓
double f(double v) {
        double r6281728 = exp(1.0);
        double r6281729 = -3.0;
        double r6281730 = v;
        double r6281731 = r6281730 * r6281730;
        double r6281732 = r6281729 * r6281731;
        double r6281733 = exp(r6281732);
        double r6281734 = r6281728 * r6281733;
        double r6281735 = log(r6281734);
        double r6281736 = 0.03125;
        double r6281737 = 2.0;
        double r6281738 = sqrt(r6281737);
        double r6281739 = r6281736 * r6281738;
        double r6281740 = r6281735 * r6281739;
        double r6281741 = 1.0;
        double r6281742 = r6281732 * r6281732;
        double r6281743 = r6281741 - r6281742;
        double r6281744 = r6281741 - r6281732;
        double r6281745 = r6281743 / r6281744;
        double r6281746 = sqrt(r6281745);
        double r6281747 = r6281740 * r6281746;
        double r6281748 = cbrt(r6281747);
        double r6281749 = r6281741 - r6281731;
        double r6281750 = r6281748 * r6281749;
        return r6281750;
}

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-cbrt-cube0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube1.0

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

  6. Applied cbrt-undiv0.0

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

  7. Applied cbrt-unprod0.0

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

  8. Simplified0.0

    \[\leadsto \sqrt[3]{\color{blue}{\left(\left(\frac{1}{32} \cdot \sqrt{2}\right) \cdot \left(1 + -3 \cdot \left(v \cdot v\right)\right)\right) \cdot \sqrt{1 + -3 \cdot \left(v \cdot v\right)}}} \cdot \left(1 - v \cdot v\right)\]
  9. Using strategy rm

  10. Applied add-log-exp0.0

    \[\leadsto \sqrt[3]{\left(\left(\frac{1}{32} \cdot \sqrt{2}\right) \cdot \left(1 + \color{blue}{\log \left(e^{-3 \cdot \left(v \cdot v\right)}\right)}\right)\right) \cdot \sqrt{1 + -3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)\]

  11. Applied add-log-exp0.0

    \[\leadsto \sqrt[3]{\left(\left(\frac{1}{32} \cdot \sqrt{2}\right) \cdot \left(\color{blue}{\log \left(e^{1}\right)} + \log \left(e^{-3 \cdot \left(v \cdot v\right)}\right)\right)\right) \cdot \sqrt{1 + -3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)\]

  12. Applied sum-log0.0

    \[\leadsto \sqrt[3]{\left(\left(\frac{1}{32} \cdot \sqrt{2}\right) \cdot \color{blue}{\log \left(e^{1} \cdot e^{-3 \cdot \left(v \cdot v\right)}\right)}\right) \cdot \sqrt{1 + -3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)\]
  13. Using strategy rm

  14. Applied flip-+0.0

    \[\leadsto \sqrt[3]{\left(\left(\frac{1}{32} \cdot \sqrt{2}\right) \cdot \log \left(e^{1} \cdot e^{-3 \cdot \left(v \cdot v\right)}\right)\right) \cdot \sqrt{\color{blue}{\frac{1 \cdot 1 - \left(-3 \cdot \left(v \cdot v\right)\right) \cdot \left(-3 \cdot \left(v \cdot v\right)\right)}{1 - -3 \cdot \left(v \cdot v\right)}}}} \cdot \left(1 - v \cdot v\right)\]

  15. Final simplification0.0

    \[\leadsto \sqrt[3]{\left(\log \left(e \cdot e^{-3 \cdot \left(v \cdot v\right)}\right) \cdot \left(\frac{1}{32} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1 - \left(-3 \cdot \left(v \cdot v\right)\right) \cdot \left(-3 \cdot \left(v \cdot v\right)\right)}{1 - -3 \cdot \left(v \cdot v\right)}}} \cdot \left(1 - v \cdot v\right)\]

Reproduce

herbie shell --seed 2019163 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))