Average Error: 1.0 → 0.0
Time: 16.6s
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 r6573865 = 4.0;
        double r6573866 = 3.0;
        double r6573867 = atan2(1.0, 0.0);
        double r6573868 = r6573866 * r6573867;
        double r6573869 = 1.0;
        double r6573870 = v;
        double r6573871 = r6573870 * r6573870;
        double r6573872 = r6573869 - r6573871;
        double r6573873 = r6573868 * r6573872;
        double r6573874 = 2.0;
        double r6573875 = 6.0;
        double r6573876 = r6573875 * r6573871;
        double r6573877 = r6573874 - r6573876;
        double r6573878 = sqrt(r6573877);
        double r6573879 = r6573873 * r6573878;
        double r6573880 = r6573865 / r6573879;
        return r6573880;
}
↓
double f(double v) {
        double r6573881 = 1.3333333333333333;
        double r6573882 = atan2(1.0, 0.0);
        double r6573883 = v;
        double r6573884 = r6573883 * r6573883;
        double r6573885 = r6573884 * r6573882;
        double r6573886 = r6573882 - r6573885;
        double r6573887 = r6573881 / r6573886;
        double r6573888 = -6.0;
        double r6573889 = 2.0;
        double r6573890 = fma(r6573888, r6573884, r6573889);
        double r6573891 = sqrt(r6573890);
        double r6573892 = r6573887 / r6573891;
        return r6573892;
}

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 2019151 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))