Average Error: 1.0 → 0.0
Time: 17.8s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{3 \cdot \pi}}{\left({1}^{3} - {v}^{6}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \left(1 \cdot 1 + \left(v \cdot v\right) \cdot \left(1 + v \cdot v\right)\right)\]
\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{\frac{4}{3 \cdot \pi}}{\left({1}^{3} - {v}^{6}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \cdot \left(1 \cdot 1 + \left(v \cdot v\right) \cdot \left(1 + v \cdot v\right)\right)
double f(double v) {
        double r135357 = 4.0;
        double r135358 = 3.0;
        double r135359 = atan2(1.0, 0.0);
        double r135360 = r135358 * r135359;
        double r135361 = 1.0;
        double r135362 = v;
        double r135363 = r135362 * r135362;
        double r135364 = r135361 - r135363;
        double r135365 = r135360 * r135364;
        double r135366 = 2.0;
        double r135367 = 6.0;
        double r135368 = r135367 * r135363;
        double r135369 = r135366 - r135368;
        double r135370 = sqrt(r135369);
        double r135371 = r135365 * r135370;
        double r135372 = r135357 / r135371;
        return r135372;
}
↓
double f(double v) {
        double r135373 = 4.0;
        double r135374 = 3.0;
        double r135375 = atan2(1.0, 0.0);
        double r135376 = r135374 * r135375;
        double r135377 = r135373 / r135376;
        double r135378 = 1.0;
        double r135379 = 3.0;
        double r135380 = pow(r135378, r135379);
        double r135381 = v;
        double r135382 = 6.0;
        double r135383 = pow(r135381, r135382);
        double r135384 = r135380 - r135383;
        double r135385 = 2.0;
        double r135386 = 6.0;
        double r135387 = r135381 * r135381;
        double r135388 = r135386 * r135387;
        double r135389 = r135385 - r135388;
        double r135390 = sqrt(r135389);
        double r135391 = r135384 * r135390;
        double r135392 = r135377 / r135391;
        double r135393 = r135378 * r135378;
        double r135394 = r135378 + r135387;
        double r135395 = r135387 * r135394;
        double r135396 = r135393 + r135395;
        double r135397 = r135392 * r135396;
        return r135397;
}

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 flip3--0.0

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

  15. Applied associate-*r/0.0

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

  16. Applied associate-*l/0.0

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

  17. Applied associate-/r/0.0

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

  18. Applied unpow-prod-down0.0

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

  19. Applied cbrt-prod0.0

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

  20. Simplified0.0

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

  21. Simplified0.0

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

  22. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))