Average Error: 0.0 → 0.0
Time: 23.9s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{{\left(\left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)}^{3}}\]
\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(\left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}}\right) \cdot \left(1 - v \cdot v\right)\right)}^{3}}
double f(double v) {
        double r200668 = 2.0;
        double r200669 = sqrt(r200668);
        double r200670 = 4.0;
        double r200671 = r200669 / r200670;
        double r200672 = 1.0;
        double r200673 = 3.0;
        double r200674 = v;
        double r200675 = r200674 * r200674;
        double r200676 = r200673 * r200675;
        double r200677 = r200672 - r200676;
        double r200678 = sqrt(r200677);
        double r200679 = r200671 * r200678;
        double r200680 = r200672 - r200675;
        double r200681 = r200679 * r200680;
        return r200681;
}
↓
double f(double v) {
        double r200682 = 2.0;
        double r200683 = sqrt(r200682);
        double r200684 = 4.0;
        double r200685 = r200683 / r200684;
        double r200686 = 1.0;
        double r200687 = r200686 * r200686;
        double r200688 = 3.0;
        double r200689 = r200688 * r200688;
        double r200690 = v;
        double r200691 = 4.0;
        double r200692 = pow(r200690, r200691);
        double r200693 = r200689 * r200692;
        double r200694 = r200687 - r200693;
        double r200695 = sqrt(r200694);
        double r200696 = r200690 * r200690;
        double r200697 = r200688 * r200696;
        double r200698 = r200686 + r200697;
        double r200699 = sqrt(r200698);
        double r200700 = r200695 / r200699;
        double r200701 = r200685 * r200700;
        double r200702 = r200686 - r200696;
        double r200703 = r200701 * r200702;
        double r200704 = 3.0;
        double r200705 = pow(r200703, r200704);
        double r200706 = cbrt(r200705);
        return r200706;
}

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

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

  4. Simplified0.0

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

  6. Applied flip--0.0

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

  7. Applied sqrt-div0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019199 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))