Average Error: 1.0 → 0.0
Time: 18.3s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3 \cdot \pi}}{1 - v \cdot v}}{\sqrt{\left(2 \cdot 2\right) \cdot 2 - \left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(v \cdot v\right) \cdot 6\right)\right)}} \cdot \sqrt{2 \cdot 2 + \left(\left(\left(v \cdot v\right) \cdot 6\right) \cdot 2 + \left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(v \cdot v\right) \cdot 6\right)\right)}\]
\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 \cdot \pi}}{1 - v \cdot v}}{\sqrt{\left(2 \cdot 2\right) \cdot 2 - \left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(v \cdot v\right) \cdot 6\right)\right)}} \cdot \sqrt{2 \cdot 2 + \left(\left(\left(v \cdot v\right) \cdot 6\right) \cdot 2 + \left(\left(v \cdot v\right) \cdot 6\right) \cdot \left(\left(v \cdot v\right) \cdot 6\right)\right)}
double f(double v) {
        double r6269890 = 4.0;
        double r6269891 = 3.0;
        double r6269892 = atan2(1.0, 0.0);
        double r6269893 = r6269891 * r6269892;
        double r6269894 = 1.0;
        double r6269895 = v;
        double r6269896 = r6269895 * r6269895;
        double r6269897 = r6269894 - r6269896;
        double r6269898 = r6269893 * r6269897;
        double r6269899 = 2.0;
        double r6269900 = 6.0;
        double r6269901 = r6269900 * r6269896;
        double r6269902 = r6269899 - r6269901;
        double r6269903 = sqrt(r6269902);
        double r6269904 = r6269898 * r6269903;
        double r6269905 = r6269890 / r6269904;
        return r6269905;
}
↓
double f(double v) {
        double r6269906 = 4.0;
        double r6269907 = 3.0;
        double r6269908 = atan2(1.0, 0.0);
        double r6269909 = r6269907 * r6269908;
        double r6269910 = r6269906 / r6269909;
        double r6269911 = 1.0;
        double r6269912 = v;
        double r6269913 = r6269912 * r6269912;
        double r6269914 = r6269911 - r6269913;
        double r6269915 = r6269910 / r6269914;
        double r6269916 = 2.0;
        double r6269917 = r6269916 * r6269916;
        double r6269918 = r6269917 * r6269916;
        double r6269919 = 6.0;
        double r6269920 = r6269913 * r6269919;
        double r6269921 = r6269920 * r6269920;
        double r6269922 = r6269920 * r6269921;
        double r6269923 = r6269918 - r6269922;
        double r6269924 = sqrt(r6269923);
        double r6269925 = r6269915 / r6269924;
        double r6269926 = r6269920 * r6269916;
        double r6269927 = r6269926 + r6269921;
        double r6269928 = r6269917 + r6269927;
        double r6269929 = sqrt(r6269928);
        double r6269930 = r6269925 * r6269929;
        return r6269930;
}

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 flip3--1.0

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

  4. Applied sqrt-div1.0

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

  5. Applied associate-*r/1.0

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

  6. Applied associate-/r/1.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))