Average Error: 1.0 → 0.0
Time: 8.8s
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 r2792967 = 4.0;
        double r2792968 = 3.0;
        double r2792969 = atan2(1.0, 0.0);
        double r2792970 = r2792968 * r2792969;
        double r2792971 = 1.0;
        double r2792972 = v;
        double r2792973 = r2792972 * r2792972;
        double r2792974 = r2792971 - r2792973;
        double r2792975 = r2792970 * r2792974;
        double r2792976 = 2.0;
        double r2792977 = 6.0;
        double r2792978 = r2792977 * r2792973;
        double r2792979 = r2792976 - r2792978;
        double r2792980 = sqrt(r2792979);
        double r2792981 = r2792975 * r2792980;
        double r2792982 = r2792967 / r2792981;
        return r2792982;
}
↓
double f(double v) {
        double r2792983 = 1.3333333333333333;
        double r2792984 = atan2(1.0, 0.0);
        double r2792985 = v;
        double r2792986 = r2792985 * r2792985;
        double r2792987 = r2792986 * r2792984;
        double r2792988 = r2792984 - r2792987;
        double r2792989 = r2792983 / r2792988;
        double r2792990 = -6.0;
        double r2792991 = 2.0;
        double r2792992 = fma(r2792990, r2792986, r2792991);
        double r2792993 = sqrt(r2792992);
        double r2792994 = r2792989 / r2792993;
        return r2792994;
}

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 2019142 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))