Average Error: 0.4 → 0.6
Time: 21.6s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1}{t}}{\sqrt{2}} \cdot \frac{1}{\pi} - \left(\frac{\left(v \cdot v\right) \cdot \frac{5}{2}}{\left(\sqrt{2} \cdot \pi\right) \cdot t} + \frac{\frac{53}{8}}{t} \cdot \frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{\sqrt{2} \cdot \pi}\right)\]
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
↓
\frac{\frac{1}{t}}{\sqrt{2}} \cdot \frac{1}{\pi} - \left(\frac{\left(v \cdot v\right) \cdot \frac{5}{2}}{\left(\sqrt{2} \cdot \pi\right) \cdot t} + \frac{\frac{53}{8}}{t} \cdot \frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{\sqrt{2} \cdot \pi}\right)
double f(double v, double t) {
        double r4962771 = 1.0;
        double r4962772 = 5.0;
        double r4962773 = v;
        double r4962774 = r4962773 * r4962773;
        double r4962775 = r4962772 * r4962774;
        double r4962776 = r4962771 - r4962775;
        double r4962777 = atan2(1.0, 0.0);
        double r4962778 = t;
        double r4962779 = r4962777 * r4962778;
        double r4962780 = 2.0;
        double r4962781 = 3.0;
        double r4962782 = r4962781 * r4962774;
        double r4962783 = r4962771 - r4962782;
        double r4962784 = r4962780 * r4962783;
        double r4962785 = sqrt(r4962784);
        double r4962786 = r4962779 * r4962785;
        double r4962787 = r4962771 - r4962774;
        double r4962788 = r4962786 * r4962787;
        double r4962789 = r4962776 / r4962788;
        return r4962789;
}
↓
double f(double v, double t) {
        double r4962790 = 1.0;
        double r4962791 = t;
        double r4962792 = r4962790 / r4962791;
        double r4962793 = 2.0;
        double r4962794 = sqrt(r4962793);
        double r4962795 = r4962792 / r4962794;
        double r4962796 = atan2(1.0, 0.0);
        double r4962797 = r4962790 / r4962796;
        double r4962798 = r4962795 * r4962797;
        double r4962799 = v;
        double r4962800 = r4962799 * r4962799;
        double r4962801 = 2.5;
        double r4962802 = r4962800 * r4962801;
        double r4962803 = r4962794 * r4962796;
        double r4962804 = r4962803 * r4962791;
        double r4962805 = r4962802 / r4962804;
        double r4962806 = 6.625;
        double r4962807 = r4962806 / r4962791;
        double r4962808 = r4962800 * r4962800;
        double r4962809 = r4962808 / r4962803;
        double r4962810 = r4962807 * r4962809;
        double r4962811 = r4962805 + r4962810;
        double r4962812 = r4962798 - r4962811;
        return r4962812;
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\frac{53}{8} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)}\]

  3. Simplified0.6

    \[\leadsto \color{blue}{\frac{\frac{1}{t}}{\pi \cdot \sqrt{2}} - \left(\frac{\frac{53}{8}}{t} \cdot \frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{\pi \cdot \sqrt{2}} + \frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{\left(\pi \cdot \sqrt{2}\right) \cdot t}\right)}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity0.6

    \[\leadsto \frac{\frac{1}{\color{blue}{1 \cdot t}}}{\pi \cdot \sqrt{2}} - \left(\frac{\frac{53}{8}}{t} \cdot \frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{\pi \cdot \sqrt{2}} + \frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{\left(\pi \cdot \sqrt{2}\right) \cdot t}\right)\]

  6. Applied add-cube-cbrt0.6

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot t}}{\pi \cdot \sqrt{2}} - \left(\frac{\frac{53}{8}}{t} \cdot \frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{\pi \cdot \sqrt{2}} + \frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{\left(\pi \cdot \sqrt{2}\right) \cdot t}\right)\]

  7. Applied times-frac0.6

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{t}}}{\pi \cdot \sqrt{2}} - \left(\frac{\frac{53}{8}}{t} \cdot \frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{\pi \cdot \sqrt{2}} + \frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{\left(\pi \cdot \sqrt{2}\right) \cdot t}\right)\]

  8. Applied times-frac0.6

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{\pi} \cdot \frac{\frac{\sqrt[3]{1}}{t}}{\sqrt{2}}} - \left(\frac{\frac{53}{8}}{t} \cdot \frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{\pi \cdot \sqrt{2}} + \frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{\left(\pi \cdot \sqrt{2}\right) \cdot t}\right)\]

  9. Simplified0.6

    \[\leadsto \color{blue}{\frac{1}{\pi}} \cdot \frac{\frac{\sqrt[3]{1}}{t}}{\sqrt{2}} - \left(\frac{\frac{53}{8}}{t} \cdot \frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{\pi \cdot \sqrt{2}} + \frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{\left(\pi \cdot \sqrt{2}\right) \cdot t}\right)\]

  10. Simplified0.6

    \[\leadsto \frac{1}{\pi} \cdot \color{blue}{\frac{\frac{1}{t}}{\sqrt{2}}} - \left(\frac{\frac{53}{8}}{t} \cdot \frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{\pi \cdot \sqrt{2}} + \frac{\frac{5}{2} \cdot \left(v \cdot v\right)}{\left(\pi \cdot \sqrt{2}\right) \cdot t}\right)\]

  11. Final simplification0.6

    \[\leadsto \frac{\frac{1}{t}}{\sqrt{2}} \cdot \frac{1}{\pi} - \left(\frac{\left(v \cdot v\right) \cdot \frac{5}{2}}{\left(\sqrt{2} \cdot \pi\right) \cdot t} + \frac{\frac{53}{8}}{t} \cdot \frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right)}{\sqrt{2} \cdot \pi}\right)\]

Reproduce

herbie shell --seed 2019155 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))