Average Error: 1.0 → 0.0
Time: 9.4s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\sqrt[3]{{\left(\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)\right) \cdot \sqrt{2 \cdot 2 - \left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right)}}\right)}^{3} \cdot {\left(\sqrt{2 + 6 \cdot \left(v \cdot v\right)} \cdot \left({v}^{2} \cdot \left({v}^{2} + 1\right) + 1 \cdot 1\right)\right)}^{3}}\]
\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}^{3} - {\left(v \cdot v\right)}^{3}\right)\right) \cdot \sqrt{2 \cdot 2 - \left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right)}}\right)}^{3} \cdot {\left(\sqrt{2 + 6 \cdot \left(v \cdot v\right)} \cdot \left({v}^{2} \cdot \left({v}^{2} + 1\right) + 1 \cdot 1\right)\right)}^{3}}
double f(double v) {
        double r230046 = 4.0;
        double r230047 = 3.0;
        double r230048 = atan2(1.0, 0.0);
        double r230049 = r230047 * r230048;
        double r230050 = 1.0;
        double r230051 = v;
        double r230052 = r230051 * r230051;
        double r230053 = r230050 - r230052;
        double r230054 = r230049 * r230053;
        double r230055 = 2.0;
        double r230056 = 6.0;
        double r230057 = r230056 * r230052;
        double r230058 = r230055 - r230057;
        double r230059 = sqrt(r230058);
        double r230060 = r230054 * r230059;
        double r230061 = r230046 / r230060;
        return r230061;
}
↓
double f(double v) {
        double r230062 = 4.0;
        double r230063 = 3.0;
        double r230064 = atan2(1.0, 0.0);
        double r230065 = r230063 * r230064;
        double r230066 = 1.0;
        double r230067 = 3.0;
        double r230068 = pow(r230066, r230067);
        double r230069 = v;
        double r230070 = r230069 * r230069;
        double r230071 = pow(r230070, r230067);
        double r230072 = r230068 - r230071;
        double r230073 = r230065 * r230072;
        double r230074 = 2.0;
        double r230075 = r230074 * r230074;
        double r230076 = 6.0;
        double r230077 = r230076 * r230070;
        double r230078 = r230077 * r230077;
        double r230079 = r230075 - r230078;
        double r230080 = sqrt(r230079);
        double r230081 = r230073 * r230080;
        double r230082 = r230062 / r230081;
        double r230083 = pow(r230082, r230067);
        double r230084 = r230074 + r230077;
        double r230085 = sqrt(r230084);
        double r230086 = 2.0;
        double r230087 = pow(r230069, r230086);
        double r230088 = r230087 + r230066;
        double r230089 = r230087 * r230088;
        double r230090 = r230066 * r230066;
        double r230091 = r230089 + r230090;
        double r230092 = r230085 * r230091;
        double r230093 = pow(r230092, r230067);
        double r230094 = r230083 * r230093;
        double r230095 = cbrt(r230094);
        return r230095;
}

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. Using strategy rm

  14. Applied flip--0.0

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

  15. Applied sqrt-div0.0

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

  16. Applied flip3--0.0

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

  17. Applied associate-*r/0.0

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

  18. Applied frac-times0.0

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

  19. Applied associate-/r/0.0

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

  20. Applied unpow-prod-down0.0

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

  21. Simplified0.0

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

  22. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))