Average Error: 0.0 → 0.0
Time: 5.6s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{{\left(\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\right)}^{3}}\]
\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(\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\right)}^{3}}
double f(double v) {
        double r177353 = 2.0;
        double r177354 = sqrt(r177353);
        double r177355 = 4.0;
        double r177356 = r177354 / r177355;
        double r177357 = 1.0;
        double r177358 = 3.0;
        double r177359 = v;
        double r177360 = r177359 * r177359;
        double r177361 = r177358 * r177360;
        double r177362 = r177357 - r177361;
        double r177363 = sqrt(r177362);
        double r177364 = r177356 * r177363;
        double r177365 = r177357 - r177360;
        double r177366 = r177364 * r177365;
        return r177366;
}
↓
double f(double v) {
        double r177367 = 2.0;
        double r177368 = sqrt(r177367);
        double r177369 = 4.0;
        double r177370 = r177368 / r177369;
        double r177371 = 1.0;
        double r177372 = 3.0;
        double r177373 = v;
        double r177374 = r177373 * r177373;
        double r177375 = r177372 * r177374;
        double r177376 = r177371 - r177375;
        double r177377 = sqrt(r177376);
        double r177378 = r177370 * r177377;
        double r177379 = r177371 - r177374;
        double r177380 = r177378 * r177379;
        double r177381 = 3.0;
        double r177382 = pow(r177380, r177381);
        double r177383 = cbrt(r177382);
        return r177383;
}

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

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

  6. Applied add-cbrt-cube1.0

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

  7. Applied cbrt-undiv0.0

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

  8. Applied cbrt-unprod0.0

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

  9. Applied cbrt-unprod0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))