Average Error: 0.0 → 0.0
Time: 24.3s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{\frac{\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \sqrt{2}}{4} \cdot \frac{\frac{\left(\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot 2\right) \cdot \left(\left(1 - \left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)\right) \cdot \left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right)\right)}{4}}{4 \cdot \left(\left(\left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) + 1\right) \cdot \left(1 + v \cdot v\right)\right)}}\]
\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]{\frac{\sqrt{1 + \left(v \cdot v\right) \cdot -3} \cdot \sqrt{2}}{4} \cdot \frac{\frac{\left(\left(1 + \left(v \cdot v\right) \cdot -3\right) \cdot 2\right) \cdot \left(\left(1 - \left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)\right) \cdot \left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(1 - v \cdot v\right)\right)\right)}{4}}{4 \cdot \left(\left(\left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) + 1\right) \cdot \left(1 + v \cdot v\right)\right)}}
double f(double v) {
        double r6441291 = 2.0;
        double r6441292 = sqrt(r6441291);
        double r6441293 = 4.0;
        double r6441294 = r6441292 / r6441293;
        double r6441295 = 1.0;
        double r6441296 = 3.0;
        double r6441297 = v;
        double r6441298 = r6441297 * r6441297;
        double r6441299 = r6441296 * r6441298;
        double r6441300 = r6441295 - r6441299;
        double r6441301 = sqrt(r6441300);
        double r6441302 = r6441294 * r6441301;
        double r6441303 = r6441295 - r6441298;
        double r6441304 = r6441302 * r6441303;
        return r6441304;
}
↓
double f(double v) {
        double r6441305 = 1.0;
        double r6441306 = v;
        double r6441307 = r6441306 * r6441306;
        double r6441308 = -3.0;
        double r6441309 = r6441307 * r6441308;
        double r6441310 = r6441305 + r6441309;
        double r6441311 = sqrt(r6441310);
        double r6441312 = 2.0;
        double r6441313 = sqrt(r6441312);
        double r6441314 = r6441311 * r6441313;
        double r6441315 = 4.0;
        double r6441316 = r6441314 / r6441315;
        double r6441317 = r6441310 * r6441312;
        double r6441318 = r6441306 * r6441307;
        double r6441319 = r6441318 * r6441318;
        double r6441320 = r6441305 - r6441319;
        double r6441321 = r6441307 * r6441307;
        double r6441322 = r6441305 - r6441321;
        double r6441323 = r6441305 - r6441307;
        double r6441324 = r6441322 * r6441323;
        double r6441325 = r6441320 * r6441324;
        double r6441326 = r6441317 * r6441325;
        double r6441327 = r6441326 / r6441315;
        double r6441328 = r6441307 + r6441321;
        double r6441329 = r6441328 + r6441305;
        double r6441330 = r6441305 + r6441307;
        double r6441331 = r6441329 * r6441330;
        double r6441332 = r6441315 * r6441331;
        double r6441333 = r6441327 / r6441332;
        double r6441334 = r6441316 * r6441333;
        double r6441335 = cbrt(r6441334);
        return r6441335;
}

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. Using strategy rm

  3. Applied add-cbrt-cube0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied cbrt-unprod0.0

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

  7. Applied cbrt-unprod0.0

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

  8. Simplified0.0

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

  10. Applied flip3--0.0

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

  11. Applied associate-*l/0.0

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

  12. Applied flip--0.0

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

  13. Applied frac-times0.0

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

  14. Applied frac-times0.0

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

  15. Applied associate-*r/0.0

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

  16. Simplified0.0

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

  17. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))