Average Error: 0.0 → 0.2
Time: 18.6s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\frac{{v}^{4} \cdot \left(0.375 \cdot \sqrt{2}\right)}{\sqrt{1}} - \left(\frac{0.375 \cdot \sqrt{2}}{\frac{\sqrt{1}}{v \cdot v}} + \left(0.25 \cdot \left(\sqrt{1} \cdot \left(\sqrt{2} \cdot \left(v \cdot v\right)\right)\right) + \frac{\sqrt{2}}{\frac{\sqrt{1}}{{v}^{4}}} \cdot \frac{0.28125}{1}\right)\right)\right) + \left(0.25 \cdot \sqrt{2}\right) \cdot \sqrt{1}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(\frac{{v}^{4} \cdot \left(0.375 \cdot \sqrt{2}\right)}{\sqrt{1}} - \left(\frac{0.375 \cdot \sqrt{2}}{\frac{\sqrt{1}}{v \cdot v}} + \left(0.25 \cdot \left(\sqrt{1} \cdot \left(\sqrt{2} \cdot \left(v \cdot v\right)\right)\right) + \frac{\sqrt{2}}{\frac{\sqrt{1}}{{v}^{4}}} \cdot \frac{0.28125}{1}\right)\right)\right) + \left(0.25 \cdot \sqrt{2}\right) \cdot \sqrt{1}
double f(double v) {
        double r155732 = 2.0;
        double r155733 = sqrt(r155732);
        double r155734 = 4.0;
        double r155735 = r155733 / r155734;
        double r155736 = 1.0;
        double r155737 = 3.0;
        double r155738 = v;
        double r155739 = r155738 * r155738;
        double r155740 = r155737 * r155739;
        double r155741 = r155736 - r155740;
        double r155742 = sqrt(r155741);
        double r155743 = r155735 * r155742;
        double r155744 = r155736 - r155739;
        double r155745 = r155743 * r155744;
        return r155745;
}
↓
double f(double v) {
        double r155746 = v;
        double r155747 = 4.0;
        double r155748 = pow(r155746, r155747);
        double r155749 = 0.375;
        double r155750 = 2.0;
        double r155751 = sqrt(r155750);
        double r155752 = r155749 * r155751;
        double r155753 = r155748 * r155752;
        double r155754 = 1.0;
        double r155755 = sqrt(r155754);
        double r155756 = r155753 / r155755;
        double r155757 = r155746 * r155746;
        double r155758 = r155755 / r155757;
        double r155759 = r155752 / r155758;
        double r155760 = 0.25;
        double r155761 = r155751 * r155757;
        double r155762 = r155755 * r155761;
        double r155763 = r155760 * r155762;
        double r155764 = r155755 / r155748;
        double r155765 = r155751 / r155764;
        double r155766 = 0.28125;
        double r155767 = r155766 / r155754;
        double r155768 = r155765 * r155767;
        double r155769 = r155763 + r155768;
        double r155770 = r155759 + r155769;
        double r155771 = r155756 - r155770;
        double r155772 = r155760 * r155751;
        double r155773 = r155772 * r155755;
        double r155774 = r155771 + r155773;
        return r155774;
}

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. Simplified0.0

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

  3. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{\left(0.375 \cdot \frac{\sqrt{2} \cdot {v}^{4}}{\sqrt{1}} + 0.25 \cdot \left(\sqrt{2} \cdot \sqrt{1}\right)\right) - \left(0.28125 \cdot \frac{\sqrt{2} \cdot {v}^{4}}{{\left(\sqrt{1}\right)}^{3}} + \left(0.25 \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot {v}^{2}\right)\right) + 0.375 \cdot \frac{\sqrt{2} \cdot {v}^{2}}{\sqrt{1}}\right)\right)}\]

  4. Simplified0.2

    \[\leadsto \color{blue}{\left(0.25 \cdot \sqrt{2}\right) \cdot \sqrt{1} + \left(\frac{\left(0.375 \cdot \sqrt{2}\right) \cdot {v}^{4}}{\sqrt{1}} - \left(\frac{0.375 \cdot \sqrt{2}}{\frac{\sqrt{1}}{v \cdot v}} + \left(0.25 \cdot \left(\sqrt{1} \cdot \left(\left(v \cdot v\right) \cdot \sqrt{2}\right)\right) + \frac{0.28125}{1} \cdot \frac{\sqrt{2}}{\frac{\sqrt{1}}{{v}^{4}}}\right)\right)\right)}\]

  5. Final simplification0.2

    \[\leadsto \left(\frac{{v}^{4} \cdot \left(0.375 \cdot \sqrt{2}\right)}{\sqrt{1}} - \left(\frac{0.375 \cdot \sqrt{2}}{\frac{\sqrt{1}}{v \cdot v}} + \left(0.25 \cdot \left(\sqrt{1} \cdot \left(\sqrt{2} \cdot \left(v \cdot v\right)\right)\right) + \frac{\sqrt{2}}{\frac{\sqrt{1}}{{v}^{4}}} \cdot \frac{0.28125}{1}\right)\right)\right) + \left(0.25 \cdot \sqrt{2}\right) \cdot \sqrt{1}\]

Reproduce

herbie shell --seed 2019196 
(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))))