Average Error: 1.0 → 0.0
Time: 7.7s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}\]
\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)}}
↓
\frac{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}
double f(double v) {
        double r3487914 = 4.0;
        double r3487915 = 3.0;
        double r3487916 = atan2(1.0, 0.0);
        double r3487917 = r3487915 * r3487916;
        double r3487918 = 1.0;
        double r3487919 = v;
        double r3487920 = r3487919 * r3487919;
        double r3487921 = r3487918 - r3487920;
        double r3487922 = r3487917 * r3487921;
        double r3487923 = 2.0;
        double r3487924 = 6.0;
        double r3487925 = r3487924 * r3487920;
        double r3487926 = r3487923 - r3487925;
        double r3487927 = sqrt(r3487926);
        double r3487928 = r3487922 * r3487927;
        double r3487929 = r3487914 / r3487928;
        return r3487929;
}
↓
double f(double v) {
        double r3487930 = 1.3333333333333333;
        double r3487931 = atan2(1.0, 0.0);
        double r3487932 = v;
        double r3487933 = r3487932 * r3487932;
        double r3487934 = r3487933 * r3487931;
        double r3487935 = r3487931 - r3487934;
        double r3487936 = r3487930 / r3487935;
        double r3487937 = -6.0;
        double r3487938 = 2.0;
        double r3487939 = fma(r3487937, r3487933, r3487938);
        double r3487940 = sqrt(r3487939);
        double r3487941 = r3487936 / r3487940;
        return r3487941;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))