Average Error: 1.0 → 0.0
Time: 7.3s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\sqrt[3]{\left(\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\right) \cdot \frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}}\]
\frac{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)}} \cdot \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) \cdot \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)}}}
double f(double v) {
        double r250535 = 4.0;
        double r250536 = 3.0;
        double r250537 = atan2(1.0, 0.0);
        double r250538 = r250536 * r250537;
        double r250539 = 1.0;
        double r250540 = v;
        double r250541 = r250540 * r250540;
        double r250542 = r250539 - r250541;
        double r250543 = r250538 * r250542;
        double r250544 = 2.0;
        double r250545 = 6.0;
        double r250546 = r250545 * r250541;
        double r250547 = r250544 - r250546;
        double r250548 = sqrt(r250547);
        double r250549 = r250543 * r250548;
        double r250550 = r250535 / r250549;
        return r250550;
}
↓
double f(double v) {
        double r250551 = 4.0;
        double r250552 = 3.0;
        double r250553 = atan2(1.0, 0.0);
        double r250554 = r250552 * r250553;
        double r250555 = 1.0;
        double r250556 = v;
        double r250557 = r250556 * r250556;
        double r250558 = r250555 - r250557;
        double r250559 = r250554 * r250558;
        double r250560 = 2.0;
        double r250561 = 6.0;
        double r250562 = r250561 * r250557;
        double r250563 = r250560 - r250562;
        double r250564 = sqrt(r250563);
        double r250565 = r250559 * r250564;
        double r250566 = r250551 / r250565;
        double r250567 = r250566 * r250566;
        double r250568 = r250567 * r250566;
        double r250569 = cbrt(r250568);
        return r250569;
}

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 add-cbrt-cube0.0

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

  15. Applied rem-cube-cbrt0.0

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

  16. Final simplification0.0

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))