Average Error: 0.4 → 0.6
Time: 30.7s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1}{t}}{\sqrt{2} \cdot \pi} + \left(\frac{\frac{27}{2}}{t} \cdot \frac{{v}^{4}}{{\left(\sqrt{2}\right)}^{5} \cdot \pi} + \left(\frac{\frac{v \cdot v}{t} \cdot 3}{\left(2 \cdot \sqrt{2}\right) \cdot \pi} - \mathsf{fma}\left(4, \frac{{v}^{4}}{\sqrt{2} \cdot \left(\pi \cdot t\right)}, \mathsf{fma}\left(12, \frac{\frac{{v}^{4}}{\left(2 \cdot \sqrt{2}\right) \cdot \pi}}{t}, \frac{4}{\frac{\sqrt{2} \cdot \left(\pi \cdot t\right)}{v \cdot v}}\right)\right)\right)\right)\]
\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 \pi} + \left(\frac{\frac{27}{2}}{t} \cdot \frac{{v}^{4}}{{\left(\sqrt{2}\right)}^{5} \cdot \pi} + \left(\frac{\frac{v \cdot v}{t} \cdot 3}{\left(2 \cdot \sqrt{2}\right) \cdot \pi} - \mathsf{fma}\left(4, \frac{{v}^{4}}{\sqrt{2} \cdot \left(\pi \cdot t\right)}, \mathsf{fma}\left(12, \frac{\frac{{v}^{4}}{\left(2 \cdot \sqrt{2}\right) \cdot \pi}}{t}, \frac{4}{\frac{\sqrt{2} \cdot \left(\pi \cdot t\right)}{v \cdot v}}\right)\right)\right)\right)
double f(double v, double t) {
        double r4697863 = 1.0;
        double r4697864 = 5.0;
        double r4697865 = v;
        double r4697866 = r4697865 * r4697865;
        double r4697867 = r4697864 * r4697866;
        double r4697868 = r4697863 - r4697867;
        double r4697869 = atan2(1.0, 0.0);
        double r4697870 = t;
        double r4697871 = r4697869 * r4697870;
        double r4697872 = 2.0;
        double r4697873 = 3.0;
        double r4697874 = r4697873 * r4697866;
        double r4697875 = r4697863 - r4697874;
        double r4697876 = r4697872 * r4697875;
        double r4697877 = sqrt(r4697876);
        double r4697878 = r4697871 * r4697877;
        double r4697879 = r4697863 - r4697866;
        double r4697880 = r4697878 * r4697879;
        double r4697881 = r4697868 / r4697880;
        return r4697881;
}
↓
double f(double v, double t) {
        double r4697882 = 1.0;
        double r4697883 = t;
        double r4697884 = r4697882 / r4697883;
        double r4697885 = 2.0;
        double r4697886 = sqrt(r4697885);
        double r4697887 = atan2(1.0, 0.0);
        double r4697888 = r4697886 * r4697887;
        double r4697889 = r4697884 / r4697888;
        double r4697890 = 13.5;
        double r4697891 = r4697890 / r4697883;
        double r4697892 = v;
        double r4697893 = 4.0;
        double r4697894 = pow(r4697892, r4697893);
        double r4697895 = 5.0;
        double r4697896 = pow(r4697886, r4697895);
        double r4697897 = r4697896 * r4697887;
        double r4697898 = r4697894 / r4697897;
        double r4697899 = r4697891 * r4697898;
        double r4697900 = r4697892 * r4697892;
        double r4697901 = r4697900 / r4697883;
        double r4697902 = 3.0;
        double r4697903 = r4697901 * r4697902;
        double r4697904 = r4697885 * r4697886;
        double r4697905 = r4697904 * r4697887;
        double r4697906 = r4697903 / r4697905;
        double r4697907 = r4697887 * r4697883;
        double r4697908 = r4697886 * r4697907;
        double r4697909 = r4697894 / r4697908;
        double r4697910 = 12.0;
        double r4697911 = r4697894 / r4697905;
        double r4697912 = r4697911 / r4697883;
        double r4697913 = r4697908 / r4697900;
        double r4697914 = r4697893 / r4697913;
        double r4697915 = fma(r4697910, r4697912, r4697914);
        double r4697916 = fma(r4697893, r4697909, r4697915);
        double r4697917 = r4697906 - r4697916;
        double r4697918 = r4697899 + r4697917;
        double r4697919 = r4697889 + r4697918;
        return r4697919;
}

Error

Bits error versus v

Bits error versus t

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. Simplified0.4

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

  3. Taylor expanded around 0 0.5

    \[\leadsto \color{blue}{\left(\frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \left(3 \cdot \frac{{v}^{2}}{t \cdot \left({\left(\sqrt{2}\right)}^{3} \cdot \pi\right)} + \frac{27}{2} \cdot \frac{{v}^{4}}{t \cdot \left({\left(\sqrt{2}\right)}^{5} \cdot \pi\right)}\right)\right) - \left(4 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \left(4 \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + 12 \cdot \frac{{v}^{4}}{t \cdot \left({\left(\sqrt{2}\right)}^{3} \cdot \pi\right)}\right)\right)}\]

  4. Simplified0.6

    \[\leadsto \color{blue}{\frac{\frac{1}{t}}{\sqrt{2} \cdot \pi} + \left(\frac{\frac{27}{2}}{t} \cdot \frac{{v}^{4}}{\pi \cdot {\left(\sqrt{2}\right)}^{5}} + \left(\frac{\frac{v \cdot v}{t} \cdot 3}{\pi \cdot \left(2 \cdot \sqrt{2}\right)} - \mathsf{fma}\left(4, \frac{{v}^{4}}{\sqrt{2} \cdot \left(t \cdot \pi\right)}, \mathsf{fma}\left(12, \frac{\frac{{v}^{4}}{\pi \cdot \left(2 \cdot \sqrt{2}\right)}}{t}, \frac{4}{\frac{\sqrt{2} \cdot \left(t \cdot \pi\right)}{v \cdot v}}\right)\right)\right)\right)}\]

  5. Final simplification0.6

    \[\leadsto \frac{\frac{1}{t}}{\sqrt{2} \cdot \pi} + \left(\frac{\frac{27}{2}}{t} \cdot \frac{{v}^{4}}{{\left(\sqrt{2}\right)}^{5} \cdot \pi} + \left(\frac{\frac{v \cdot v}{t} \cdot 3}{\left(2 \cdot \sqrt{2}\right) \cdot \pi} - \mathsf{fma}\left(4, \frac{{v}^{4}}{\sqrt{2} \cdot \left(\pi \cdot t\right)}, \mathsf{fma}\left(12, \frac{\frac{{v}^{4}}{\left(2 \cdot \sqrt{2}\right) \cdot \pi}}{t}, \frac{4}{\frac{\sqrt{2} \cdot \left(\pi \cdot t\right)}{v \cdot v}}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019134 +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)))))