Average Error: 7.0 → 0.6
Time: 25.0s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\sqrt[3]{x.re \cdot x.im + x.re \cdot x.im} \cdot \sqrt[3]{x.re \cdot x.im + x.re \cdot x.im}\right) \cdot \left(x.re \cdot \left(\sqrt[3]{x.re \cdot x.im} \cdot \sqrt[3]{2}\right)\right) + \left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right)\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re
↓
\left(\sqrt[3]{x.re \cdot x.im + x.re \cdot x.im} \cdot \sqrt[3]{x.re \cdot x.im + x.re \cdot x.im}\right) \cdot \left(x.re \cdot \left(\sqrt[3]{x.re \cdot x.im} \cdot \sqrt[3]{2}\right)\right) + \left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right)
double f(double x_re, double x_im) {
        double r5959348 = x_re;
        double r5959349 = r5959348 * r5959348;
        double r5959350 = x_im;
        double r5959351 = r5959350 * r5959350;
        double r5959352 = r5959349 - r5959351;
        double r5959353 = r5959352 * r5959350;
        double r5959354 = r5959348 * r5959350;
        double r5959355 = r5959350 * r5959348;
        double r5959356 = r5959354 + r5959355;
        double r5959357 = r5959356 * r5959348;
        double r5959358 = r5959353 + r5959357;
        return r5959358;
}
↓
double f(double x_re, double x_im) {
        double r5959359 = x_re;
        double r5959360 = x_im;
        double r5959361 = r5959359 * r5959360;
        double r5959362 = r5959361 + r5959361;
        double r5959363 = cbrt(r5959362);
        double r5959364 = r5959363 * r5959363;
        double r5959365 = cbrt(r5959361);
        double r5959366 = 2.0;
        double r5959367 = cbrt(r5959366);
        double r5959368 = r5959365 * r5959367;
        double r5959369 = r5959359 * r5959368;
        double r5959370 = r5959364 * r5959369;
        double r5959371 = r5959359 - r5959360;
        double r5959372 = r5959371 * r5959360;
        double r5959373 = r5959360 + r5959359;
        double r5959374 = r5959372 * r5959373;
        double r5959375 = r5959370 + r5959374;
        return r5959375;
}

Error

Bits error versus x.re

Bits error versus x.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.0
Target0.2
Herbie0.6
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 7.0

    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
  2. Using strategy rm

  3. Applied difference-of-squares7.0

    \[\leadsto \color{blue}{\left(\left(x.re + x.im\right) \cdot \left(x.re - x.im\right)\right)} \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

  4. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)} + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.6

    \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{\left(\left(\sqrt[3]{x.re \cdot x.im + x.im \cdot x.re} \cdot \sqrt[3]{x.re \cdot x.im + x.im \cdot x.re}\right) \cdot \sqrt[3]{x.re \cdot x.im + x.im \cdot x.re}\right)} \cdot x.re\]

  7. Applied associate-*l*0.6

    \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \color{blue}{\left(\sqrt[3]{x.re \cdot x.im + x.im \cdot x.re} \cdot \sqrt[3]{x.re \cdot x.im + x.im \cdot x.re}\right) \cdot \left(\sqrt[3]{x.re \cdot x.im + x.im \cdot x.re} \cdot x.re\right)}\]

  8. Taylor expanded around 0 47.3

    \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(\sqrt[3]{x.re \cdot x.im + x.im \cdot x.re} \cdot \sqrt[3]{x.re \cdot x.im + x.im \cdot x.re}\right) \cdot \left(\color{blue}{\left(e^{\frac{1}{3} \cdot \left(\log x.im + \log x.re\right)} \cdot \sqrt[3]{2}\right)} \cdot x.re\right)\]

  9. Simplified0.6

    \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(\sqrt[3]{x.re \cdot x.im + x.im \cdot x.re} \cdot \sqrt[3]{x.re \cdot x.im + x.im \cdot x.re}\right) \cdot \left(\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{x.re \cdot x.im}\right)} \cdot x.re\right)\]

  10. Final simplification0.6

    \[\leadsto \left(\sqrt[3]{x.re \cdot x.im + x.re \cdot x.im} \cdot \sqrt[3]{x.re \cdot x.im + x.re \cdot x.im}\right) \cdot \left(x.re \cdot \left(\sqrt[3]{x.re \cdot x.im} \cdot \sqrt[3]{2}\right)\right) + \left(\left(x.re - x.im\right) \cdot x.im\right) \cdot \left(x.im + x.re\right)\]

Reproduce

herbie shell --seed 2019142 
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"

  :herbie-target
  (+ (* (* x.re x.im) (* 2 x.re)) (* (* x.im (- x.re x.im)) (+ x.re x.im)))

  (+ (* (- (* x.re x.re) (* x.im x.im)) x.im) (* (+ (* x.re x.im) (* x.im x.re)) x.re)))