Average Error: 1.0 → 0.0
Time: 51.2s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3}}{\pi - v \cdot \left(v \cdot \pi\right)}}{\sqrt{(\left(-6 \cdot v\right) \cdot v + 2)_*}}\]
double f(double v) {
        double r11582892 = 4.0;
        double r11582893 = 3.0;
        double r11582894 = atan2(1.0, 0.0);
        double r11582895 = r11582893 * r11582894;
        double r11582896 = 1.0;
        double r11582897 = v;
        double r11582898 = r11582897 * r11582897;
        double r11582899 = r11582896 - r11582898;
        double r11582900 = r11582895 * r11582899;
        double r11582901 = 2.0;
        double r11582902 = 6.0;
        double r11582903 = r11582902 * r11582898;
        double r11582904 = r11582901 - r11582903;
        double r11582905 = sqrt(r11582904);
        double r11582906 = r11582900 * r11582905;
        double r11582907 = r11582892 / r11582906;
        return r11582907;
}
↓
double f(double v) {
        double r11582908 = 1.3333333333333333;
        double r11582909 = atan2(1.0, 0.0);
        double r11582910 = v;
        double r11582911 = r11582910 * r11582909;
        double r11582912 = r11582910 * r11582911;
        double r11582913 = r11582909 - r11582912;
        double r11582914 = r11582908 / r11582913;
        double r11582915 = -6.0;
        double r11582916 = r11582915 * r11582910;
        double r11582917 = 2.0;
        double r11582918 = fma(r11582916, r11582910, r11582917);
        double r11582919 = sqrt(r11582918);
        double r11582920 = r11582914 / r11582919;
        return r11582920;
}


\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)}}
↓
\frac{\frac{\frac{4}{3}}{\pi - v \cdot \left(v \cdot \pi\right)}}{\sqrt{(\left(-6 \cdot v\right) \cdot v + 2)_*}}

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

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\pi - v \cdot \left(\pi \cdot v\right)}}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}}\]

  3. Final simplification0.0

    \[\leadsto \frac{\frac{\frac{4}{3}}{\pi - v \cdot \left(v \cdot \pi\right)}}{\sqrt{(\left(-6 \cdot v\right) \cdot v + 2)_*}}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))