Average Error: 1.0 → 0.0
Time: 14.6s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\left(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{4}{\left(\pi \cdot 3\right) \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r6991874 = 4.0;
        double r6991875 = 3.0;
        double r6991876 = atan2(1.0, 0.0);
        double r6991877 = r6991875 * r6991876;
        double r6991878 = 1.0;
        double r6991879 = v;
        double r6991880 = r6991879 * r6991879;
        double r6991881 = r6991878 - r6991880;
        double r6991882 = r6991877 * r6991881;
        double r6991883 = 2.0;
        double r6991884 = 6.0;
        double r6991885 = r6991884 * r6991880;
        double r6991886 = r6991883 - r6991885;
        double r6991887 = sqrt(r6991886);
        double r6991888 = r6991882 * r6991887;
        double r6991889 = r6991874 / r6991888;
        return r6991889;
}
↓
double f(double v) {
        double r6991890 = 4.0;
        double r6991891 = atan2(1.0, 0.0);
        double r6991892 = 3.0;
        double r6991893 = r6991891 * r6991892;
        double r6991894 = 1.0;
        double r6991895 = v;
        double r6991896 = r6991895 * r6991895;
        double r6991897 = r6991894 - r6991896;
        double r6991898 = r6991893 * r6991897;
        double r6991899 = r6991890 / r6991898;
        double r6991900 = 2.0;
        double r6991901 = 6.0;
        double r6991902 = r6991901 * r6991896;
        double r6991903 = r6991900 - r6991902;
        double r6991904 = sqrt(r6991903);
        double r6991905 = r6991899 / r6991904;
        return r6991905;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied associate-/r*0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))