Average Error: 0.0 → 0.0
Time: 17.5s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{\frac{\left(\left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \sqrt{2}\right)\right) \cdot \left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \sqrt{2}\right)\right)}{\left(\left(4 \cdot \left(1 + v \cdot v\right)\right) \cdot \left(4 \cdot \left(1 + v \cdot v\right)\right)\right) \cdot \left(4 \cdot \left(1 + v \cdot v\right)\right)}}\]
\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{\left(\left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \sqrt{2}\right)\right) \cdot \left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \sqrt{2}\right)\right)\right) \cdot \left(\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\sqrt{1 - \left(v \cdot v\right) \cdot 3} \cdot \sqrt{2}\right)\right)}{\left(\left(4 \cdot \left(1 + v \cdot v\right)\right) \cdot \left(4 \cdot \left(1 + v \cdot v\right)\right)\right) \cdot \left(4 \cdot \left(1 + v \cdot v\right)\right)}}
double f(double v) {
        double r16173354 = 2.0;
        double r16173355 = sqrt(r16173354);
        double r16173356 = 4.0;
        double r16173357 = r16173355 / r16173356;
        double r16173358 = 1.0;
        double r16173359 = 3.0;
        double r16173360 = v;
        double r16173361 = r16173360 * r16173360;
        double r16173362 = r16173359 * r16173361;
        double r16173363 = r16173358 - r16173362;
        double r16173364 = sqrt(r16173363);
        double r16173365 = r16173357 * r16173364;
        double r16173366 = r16173358 - r16173361;
        double r16173367 = r16173365 * r16173366;
        return r16173367;
}
↓
double f(double v) {
        double r16173368 = 1.0;
        double r16173369 = v;
        double r16173370 = r16173369 * r16173369;
        double r16173371 = r16173370 * r16173370;
        double r16173372 = r16173368 - r16173371;
        double r16173373 = 3.0;
        double r16173374 = r16173370 * r16173373;
        double r16173375 = r16173368 - r16173374;
        double r16173376 = sqrt(r16173375);
        double r16173377 = 2.0;
        double r16173378 = sqrt(r16173377);
        double r16173379 = r16173376 * r16173378;
        double r16173380 = r16173372 * r16173379;
        double r16173381 = r16173380 * r16173380;
        double r16173382 = r16173381 * r16173380;
        double r16173383 = 4.0;
        double r16173384 = r16173368 + r16173370;
        double r16173385 = r16173383 * r16173384;
        double r16173386 = r16173385 * r16173385;
        double r16173387 = r16173386 * r16173385;
        double r16173388 = r16173382 / r16173387;
        double r16173389 = cbrt(r16173388);
        return r16173389;
}

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 flip--0.0

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

  4. Applied associate-*l/0.0

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

  5. Applied frac-times0.0

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

  6. Simplified0.0

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

  8. Applied add-cbrt-cube0.0

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

  9. Applied add-cbrt-cube1.0

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

  10. Applied cbrt-undiv0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019162 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))