Average Error: 0.0 → 0.0
Time: 17.4s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{{\left(\frac{\sqrt{2} \cdot \frac{\left(1 - v \cdot v\right) \cdot \sqrt{{1}^{3} - {\left(\left(3 \cdot v\right) \cdot v\right)}^{3}}}{\sqrt{\left(1 \cdot \left(\left(3 \cdot v\right) \cdot v\right) + \left(\left(3 \cdot v\right) \cdot v\right) \cdot \left(\left(3 \cdot v\right) \cdot v\right)\right) + 1 \cdot 1}}}{4}\right)}^{3}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt[3]{{\left(\frac{\sqrt{2} \cdot \frac{\left(1 - v \cdot v\right) \cdot \sqrt{{1}^{3} - {\left(\left(3 \cdot v\right) \cdot v\right)}^{3}}}{\sqrt{\left(1 \cdot \left(\left(3 \cdot v\right) \cdot v\right) + \left(\left(3 \cdot v\right) \cdot v\right) \cdot \left(\left(3 \cdot v\right) \cdot v\right)\right) + 1 \cdot 1}}}{4}\right)}^{3}}
double f(double v) {
        double r191505 = 2.0;
        double r191506 = sqrt(r191505);
        double r191507 = 4.0;
        double r191508 = r191506 / r191507;
        double r191509 = 1.0;
        double r191510 = 3.0;
        double r191511 = v;
        double r191512 = r191511 * r191511;
        double r191513 = r191510 * r191512;
        double r191514 = r191509 - r191513;
        double r191515 = sqrt(r191514);
        double r191516 = r191508 * r191515;
        double r191517 = r191509 - r191512;
        double r191518 = r191516 * r191517;
        return r191518;
}
↓
double f(double v) {
        double r191519 = 2.0;
        double r191520 = sqrt(r191519);
        double r191521 = 1.0;
        double r191522 = v;
        double r191523 = r191522 * r191522;
        double r191524 = r191521 - r191523;
        double r191525 = 3.0;
        double r191526 = pow(r191521, r191525);
        double r191527 = 3.0;
        double r191528 = r191527 * r191522;
        double r191529 = r191528 * r191522;
        double r191530 = pow(r191529, r191525);
        double r191531 = r191526 - r191530;
        double r191532 = sqrt(r191531);
        double r191533 = r191524 * r191532;
        double r191534 = r191521 * r191529;
        double r191535 = r191529 * r191529;
        double r191536 = r191534 + r191535;
        double r191537 = r191521 * r191521;
        double r191538 = r191536 + r191537;
        double r191539 = sqrt(r191538);
        double r191540 = r191533 / r191539;
        double r191541 = r191520 * r191540;
        double r191542 = 4.0;
        double r191543 = r191541 / r191542;
        double r191544 = pow(r191543, r191525);
        double r191545 = cbrt(r191544);
        return r191545;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \frac{1 - v \cdot v}{4}\right) \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied add-cbrt-cube0.0

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

  7. Applied cbrt-undiv1.0

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

  8. Applied add-cbrt-cube0.0

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

  9. Applied cbrt-unprod0.0

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

  10. Applied cbrt-unprod0.0

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

  11. Simplified0.0

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\left(\sqrt{1 - v \cdot \left(v \cdot 3\right)} \cdot \left(1 - {v}^{2}\right)\right) \cdot \sqrt{2}}{4}\right)}^{3}}}\]
  12. Using strategy rm

  13. Applied flip3--0.0

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

  14. Applied sqrt-div0.0

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

  15. Applied associate-*l/0.0

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

  16. Simplified0.0

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

  17. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))