Average Error: 7.0 → 0.6
Time: 18.6s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \sqrt[3]{\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re} \cdot \left(\sqrt[3]{\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re} \cdot \sqrt[3]{\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re}\right)\]
\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(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right) + \sqrt[3]{\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re} \cdot \left(\sqrt[3]{\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re} \cdot \sqrt[3]{\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re}\right)
double f(double x_re, double x_im) {
        double r5719363 = x_re;
        double r5719364 = r5719363 * r5719363;
        double r5719365 = x_im;
        double r5719366 = r5719365 * r5719365;
        double r5719367 = r5719364 - r5719366;
        double r5719368 = r5719367 * r5719365;
        double r5719369 = r5719363 * r5719365;
        double r5719370 = r5719365 * r5719363;
        double r5719371 = r5719369 + r5719370;
        double r5719372 = r5719371 * r5719363;
        double r5719373 = r5719368 + r5719372;
        return r5719373;
}
↓
double f(double x_re, double x_im) {
        double r5719374 = x_im;
        double r5719375 = x_re;
        double r5719376 = r5719375 + r5719374;
        double r5719377 = r5719374 * r5719376;
        double r5719378 = r5719375 - r5719374;
        double r5719379 = r5719377 * r5719378;
        double r5719380 = r5719374 * r5719375;
        double r5719381 = r5719380 + r5719380;
        double r5719382 = r5719381 * r5719375;
        double r5719383 = cbrt(r5719382);
        double r5719384 = r5719383 * r5719383;
        double r5719385 = r5719383 * r5719384;
        double r5719386 = r5719379 + r5719385;
        return r5719386;
}

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. Taylor expanded around inf 7.0

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

  3. Simplified0.2

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

  5. Applied add-cube-cbrt0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019132 
(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)))