Average Error: 0.0 → 0.0
Time: 1.0m
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{\frac{\sqrt{2}}{4} \cdot \left(\left(\left(\left(\left(1 - \left(v \cdot 3\right) \cdot v\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot 3\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \frac{\sqrt{2}}{4}\right) \cdot \frac{\sqrt{2}}{4}\right)}\]
\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{2}}{4} \cdot \left(\left(\left(\left(\left(1 - \left(v \cdot 3\right) \cdot v\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot 3\right) \cdot v} \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \frac{\sqrt{2}}{4}\right) \cdot \frac{\sqrt{2}}{4}\right)}
double f(double v) {
        double r7212452 = 2.0;
        double r7212453 = sqrt(r7212452);
        double r7212454 = 4.0;
        double r7212455 = r7212453 / r7212454;
        double r7212456 = 1.0;
        double r7212457 = 3.0;
        double r7212458 = v;
        double r7212459 = r7212458 * r7212458;
        double r7212460 = r7212457 * r7212459;
        double r7212461 = r7212456 - r7212460;
        double r7212462 = sqrt(r7212461);
        double r7212463 = r7212455 * r7212462;
        double r7212464 = r7212456 - r7212459;
        double r7212465 = r7212463 * r7212464;
        return r7212465;
}
↓
double f(double v) {
        double r7212466 = 2.0;
        double r7212467 = sqrt(r7212466);
        double r7212468 = 4.0;
        double r7212469 = r7212467 / r7212468;
        double r7212470 = 1.0;
        double r7212471 = v;
        double r7212472 = 3.0;
        double r7212473 = r7212471 * r7212472;
        double r7212474 = r7212473 * r7212471;
        double r7212475 = r7212470 - r7212474;
        double r7212476 = r7212471 * r7212471;
        double r7212477 = r7212470 - r7212476;
        double r7212478 = r7212477 * r7212477;
        double r7212479 = r7212475 * r7212478;
        double r7212480 = sqrt(r7212475);
        double r7212481 = r7212480 * r7212477;
        double r7212482 = r7212479 * r7212481;
        double r7212483 = r7212482 * r7212469;
        double r7212484 = r7212483 * r7212469;
        double r7212485 = r7212469 * r7212484;
        double r7212486 = cbrt(r7212485);
        return r7212486;
}

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 associate-*l*0.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied add-cbrt-cube0.0

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

  7. Applied cbrt-unprod0.0

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

  8. Applied add-cbrt-cube0.0

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

  9. Applied add-cbrt-cube1.0

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

  10. Applied cbrt-undiv0.0

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

  11. Applied cbrt-unprod0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(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))))