Average Error: 1.0 → 0.0
Time: 5.0s
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 r260447 = 4.0;
        double r260448 = 3.0;
        double r260449 = atan2(1.0, 0.0);
        double r260450 = r260448 * r260449;
        double r260451 = 1.0;
        double r260452 = v;
        double r260453 = r260452 * r260452;
        double r260454 = r260451 - r260453;
        double r260455 = r260450 * r260454;
        double r260456 = 2.0;
        double r260457 = 6.0;
        double r260458 = r260457 * r260453;
        double r260459 = r260456 - r260458;
        double r260460 = sqrt(r260459);
        double r260461 = r260455 * r260460;
        double r260462 = r260447 / r260461;
        return r260462;
}
↓
double f(double v) {
        double r260463 = 4.0;
        double r260464 = 3.0;
        double r260465 = atan2(1.0, 0.0);
        double r260466 = r260464 * r260465;
        double r260467 = 1.0;
        double r260468 = v;
        double r260469 = r260468 * r260468;
        double r260470 = r260467 - r260469;
        double r260471 = r260466 * r260470;
        double r260472 = 2.0;
        double r260473 = 6.0;
        double r260474 = r260473 * r260469;
        double r260475 = r260472 - r260474;
        double r260476 = sqrt(r260475);
        double r260477 = r260471 * r260476;
        double r260478 = r260463 / r260477;
        double r260479 = 3.0;
        double r260480 = pow(r260478, r260479);
        double r260481 = cbrt(r260480);
        return r260481;
}

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 2020039 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))