Average Error: 0.4 → 0.3
Time: 23.9s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(v \cdot v + 1\right) \cdot \frac{\sqrt{\left(3 \cdot \left(v \cdot v\right) + \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right) + 1} \cdot \frac{\frac{\frac{1 - \left(5 \cdot v\right) \cdot v}{\pi}}{t}}{\sqrt{\left(1 - \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right) \cdot 2}}}{1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)}\]
\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)}
↓
\left(v \cdot v + 1\right) \cdot \frac{\sqrt{\left(3 \cdot \left(v \cdot v\right) + \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right) + 1} \cdot \frac{\frac{\frac{1 - \left(5 \cdot v\right) \cdot v}{\pi}}{t}}{\sqrt{\left(1 - \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right) \cdot 2}}}{1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)}
double f(double v, double t) {
        double r3357829 = 1.0;
        double r3357830 = 5.0;
        double r3357831 = v;
        double r3357832 = r3357831 * r3357831;
        double r3357833 = r3357830 * r3357832;
        double r3357834 = r3357829 - r3357833;
        double r3357835 = atan2(1.0, 0.0);
        double r3357836 = t;
        double r3357837 = r3357835 * r3357836;
        double r3357838 = 2.0;
        double r3357839 = 3.0;
        double r3357840 = r3357839 * r3357832;
        double r3357841 = r3357829 - r3357840;
        double r3357842 = r3357838 * r3357841;
        double r3357843 = sqrt(r3357842);
        double r3357844 = r3357837 * r3357843;
        double r3357845 = r3357829 - r3357832;
        double r3357846 = r3357844 * r3357845;
        double r3357847 = r3357834 / r3357846;
        return r3357847;
}
↓
double f(double v, double t) {
        double r3357848 = v;
        double r3357849 = r3357848 * r3357848;
        double r3357850 = 1.0;
        double r3357851 = r3357849 + r3357850;
        double r3357852 = 3.0;
        double r3357853 = r3357852 * r3357849;
        double r3357854 = r3357853 * r3357853;
        double r3357855 = r3357853 + r3357854;
        double r3357856 = r3357855 + r3357850;
        double r3357857 = sqrt(r3357856);
        double r3357858 = 5.0;
        double r3357859 = r3357858 * r3357848;
        double r3357860 = r3357859 * r3357848;
        double r3357861 = r3357850 - r3357860;
        double r3357862 = atan2(1.0, 0.0);
        double r3357863 = r3357861 / r3357862;
        double r3357864 = t;
        double r3357865 = r3357863 / r3357864;
        double r3357866 = r3357854 * r3357853;
        double r3357867 = r3357850 - r3357866;
        double r3357868 = 2.0;
        double r3357869 = r3357867 * r3357868;
        double r3357870 = sqrt(r3357869);
        double r3357871 = r3357865 / r3357870;
        double r3357872 = r3357857 * r3357871;
        double r3357873 = r3357849 * r3357849;
        double r3357874 = r3357850 - r3357873;
        double r3357875 = r3357872 / r3357874;
        double r3357876 = r3357851 * r3357875;
        return r3357876;
}

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. Using strategy rm

  3. Applied flip--0.4

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

  4. Applied associate-*r/0.4

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

  5. Applied associate-/r/0.4

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

  6. Simplified0.4

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

  8. Applied flip3--0.4

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

  9. Applied associate-*l/0.4

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

  10. Applied sqrt-div0.4

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

  11. Applied associate-/r/0.4

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(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)))))