Average Error: 1.0 → 0.0
Time: 4.6s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\sqrt[3]{{\left(\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\right)}^{3}}\]
\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)}}
↓
\sqrt[3]{{\left(\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)}}\right)}^{3}}
double f(double v) {
        double r293929 = 4.0;
        double r293930 = 3.0;
        double r293931 = atan2(1.0, 0.0);
        double r293932 = r293930 * r293931;
        double r293933 = 1.0;
        double r293934 = v;
        double r293935 = r293934 * r293934;
        double r293936 = r293933 - r293935;
        double r293937 = r293932 * r293936;
        double r293938 = 2.0;
        double r293939 = 6.0;
        double r293940 = r293939 * r293935;
        double r293941 = r293938 - r293940;
        double r293942 = sqrt(r293941);
        double r293943 = r293937 * r293942;
        double r293944 = r293929 / r293943;
        return r293944;
}
↓
double f(double v) {
        double r293945 = 4.0;
        double r293946 = 3.0;
        double r293947 = atan2(1.0, 0.0);
        double r293948 = r293946 * r293947;
        double r293949 = 1.0;
        double r293950 = v;
        double r293951 = r293950 * r293950;
        double r293952 = r293949 - r293951;
        double r293953 = r293948 * r293952;
        double r293954 = 2.0;
        double r293955 = 6.0;
        double r293956 = r293955 * r293951;
        double r293957 = r293954 - r293956;
        double r293958 = sqrt(r293957);
        double r293959 = r293953 * r293958;
        double r293960 = r293945 / r293959;
        double r293961 = 3.0;
        double r293962 = pow(r293960, r293961);
        double r293963 = cbrt(r293962);
        return r293963;
}

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-cbrt-cube1.0

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

  4. Applied add-cbrt-cube1.0

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

  5. Applied add-cbrt-cube1.6

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

  6. Applied add-cbrt-cube1.6

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

  7. Applied cbrt-unprod1.0

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

  8. Applied cbrt-unprod1.0

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

  9. Applied cbrt-unprod1.0

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

  10. Applied add-cbrt-cube1.0

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

  11. Applied cbrt-undiv0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2020059 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))