Average Error: 1.0 → 0.0
Time: 21.9s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\left(\sqrt[3]{\frac{4}{3}} \cdot \sqrt[3]{\frac{4}{3}}\right) \cdot \frac{\frac{\sqrt[3]{\frac{4}{3}}}{\pi}}{\sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} - \left(v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)}}\]
\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
↓
\left(\sqrt[3]{\frac{4}{3}} \cdot \sqrt[3]{\frac{4}{3}}\right) \cdot \frac{\frac{\sqrt[3]{\frac{4}{3}}}{\pi}}{\sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} - \left(v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)}}
double f(double v) {
        double r4304860 = 4.0;
        double r4304861 = 3.0;
        double r4304862 = atan2(1.0, 0.0);
        double r4304863 = r4304861 * r4304862;
        double r4304864 = 1.0;
        double r4304865 = v;
        double r4304866 = r4304865 * r4304865;
        double r4304867 = r4304864 - r4304866;
        double r4304868 = r4304863 * r4304867;
        double r4304869 = 2.0;
        double r4304870 = 6.0;
        double r4304871 = r4304870 * r4304866;
        double r4304872 = r4304869 - r4304871;
        double r4304873 = sqrt(r4304872);
        double r4304874 = r4304868 * r4304873;
        double r4304875 = r4304860 / r4304874;
        return r4304875;
}
↓
double f(double v) {
        double r4304876 = 1.3333333333333333;
        double r4304877 = cbrt(r4304876);
        double r4304878 = r4304877 * r4304877;
        double r4304879 = atan2(1.0, 0.0);
        double r4304880 = r4304877 / r4304879;
        double r4304881 = -6.0;
        double r4304882 = v;
        double r4304883 = r4304882 * r4304882;
        double r4304884 = 2.0;
        double r4304885 = fma(r4304881, r4304883, r4304884);
        double r4304886 = sqrt(r4304885);
        double r4304887 = r4304883 * r4304886;
        double r4304888 = r4304886 - r4304887;
        double r4304889 = r4304880 / r4304888;
        double r4304890 = r4304878 * r4304889;
        return r4304890;
}

Error

Bits error versus v

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\pi}}{\sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} - \sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} \cdot \left(v \cdot v\right)}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.0

    \[\leadsto \frac{\frac{\frac{4}{3}}{\pi}}{\color{blue}{1 \cdot \left(\sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} - \sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} \cdot \left(v \cdot v\right)\right)}}\]

  5. Applied *-un-lft-identity0.0

    \[\leadsto \frac{\frac{\frac{4}{3}}{\color{blue}{1 \cdot \pi}}}{1 \cdot \left(\sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} - \sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} \cdot \left(v \cdot v\right)\right)}\]

  6. Applied add-cube-cbrt0.0

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{4}{3}} \cdot \sqrt[3]{\frac{4}{3}}\right) \cdot \sqrt[3]{\frac{4}{3}}}}{1 \cdot \pi}}{1 \cdot \left(\sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} - \sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} \cdot \left(v \cdot v\right)\right)}\]

  7. Applied times-frac0.0

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{4}{3}} \cdot \sqrt[3]{\frac{4}{3}}}{1} \cdot \frac{\sqrt[3]{\frac{4}{3}}}{\pi}}}{1 \cdot \left(\sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} - \sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} \cdot \left(v \cdot v\right)\right)}\]

  8. Applied times-frac0.0

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{4}{3}} \cdot \sqrt[3]{\frac{4}{3}}}{1}}{1} \cdot \frac{\frac{\sqrt[3]{\frac{4}{3}}}{\pi}}{\sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} - \sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} \cdot \left(v \cdot v\right)}}\]

  9. Simplified0.0

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{4}{3}} \cdot \sqrt[3]{\frac{4}{3}}\right)} \cdot \frac{\frac{\sqrt[3]{\frac{4}{3}}}{\pi}}{\sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} - \sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} \cdot \left(v \cdot v\right)}\]

  10. Final simplification0.0

    \[\leadsto \left(\sqrt[3]{\frac{4}{3}} \cdot \sqrt[3]{\frac{4}{3}}\right) \cdot \frac{\frac{\sqrt[3]{\frac{4}{3}}}{\pi}}{\sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)} - \left(v \cdot v\right) \cdot \sqrt{\mathsf{fma}\left(-6, \left(v \cdot v\right), 2\right)}}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))