Average Error: 1.0 → 0.0
Time: 3.9s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt{4}}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)} \cdot \frac{\sqrt{4}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{\sqrt{4}}{\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)} \cdot \frac{\sqrt{4}}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r218883 = 4.0;
        double r218884 = 3.0;
        double r218885 = atan2(1.0, 0.0);
        double r218886 = r218884 * r218885;
        double r218887 = 1.0;
        double r218888 = v;
        double r218889 = r218888 * r218888;
        double r218890 = r218887 - r218889;
        double r218891 = r218886 * r218890;
        double r218892 = 2.0;
        double r218893 = 6.0;
        double r218894 = r218893 * r218889;
        double r218895 = r218892 - r218894;
        double r218896 = sqrt(r218895);
        double r218897 = r218891 * r218896;
        double r218898 = r218883 / r218897;
        return r218898;
}
↓
double f(double v) {
        double r218899 = 4.0;
        double r218900 = sqrt(r218899);
        double r218901 = 3.0;
        double r218902 = atan2(1.0, 0.0);
        double r218903 = r218901 * r218902;
        double r218904 = 1.0;
        double r218905 = v;
        double r218906 = r218905 * r218905;
        double r218907 = r218904 - r218906;
        double r218908 = r218903 * r218907;
        double r218909 = r218900 / r218908;
        double r218910 = 2.0;
        double r218911 = 6.0;
        double r218912 = r218911 * r218906;
        double r218913 = r218910 - r218912;
        double r218914 = sqrt(r218913);
        double r218915 = r218900 / r218914;
        double r218916 = r218909 * r218915;
        return r218916;
}

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 add-sqr-sqrt1.0

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

  4. Applied times-frac0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))