Average Error: 1.0 → 0.0
Time: 7.4s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{1}{\frac{\left(3 \cdot \pi\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}{1 + v \cdot v}} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{1}{\frac{\left(3 \cdot \pi\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}{1 + v \cdot v}} \cdot \frac{4}{\sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r315905 = 4.0;
        double r315906 = 3.0;
        double r315907 = atan2(1.0, 0.0);
        double r315908 = r315906 * r315907;
        double r315909 = 1.0;
        double r315910 = v;
        double r315911 = r315910 * r315910;
        double r315912 = r315909 - r315911;
        double r315913 = r315908 * r315912;
        double r315914 = 2.0;
        double r315915 = 6.0;
        double r315916 = r315915 * r315911;
        double r315917 = r315914 - r315916;
        double r315918 = sqrt(r315917);
        double r315919 = r315913 * r315918;
        double r315920 = r315905 / r315919;
        return r315920;
}
↓
double f(double v) {
        double r315921 = 1.0;
        double r315922 = 3.0;
        double r315923 = atan2(1.0, 0.0);
        double r315924 = r315922 * r315923;
        double r315925 = 1.0;
        double r315926 = r315925 * r315925;
        double r315927 = v;
        double r315928 = r315927 * r315927;
        double r315929 = r315928 * r315928;
        double r315930 = r315926 - r315929;
        double r315931 = r315924 * r315930;
        double r315932 = r315925 + r315928;
        double r315933 = r315931 / r315932;
        double r315934 = r315921 / r315933;
        double r315935 = 4.0;
        double r315936 = 2.0;
        double r315937 = 6.0;
        double r315938 = r315937 * r315928;
        double r315939 = r315936 - r315938;
        double r315940 = sqrt(r315939);
        double r315941 = r315935 / r315940;
        double r315942 = r315934 * r315941;
        return r315942;
}

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 *-un-lft-identity1.0

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

  4. Applied times-frac0.0

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

  6. Applied flip--0.0

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

  7. Applied associate-*r/0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019356 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))