Average Error: 7.0 → 0.7
Time: 20.8s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(x.im \cdot x.re + x.im \cdot x.re\right) \cdot x.re + \left(\sqrt[3]{\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)} \cdot \sqrt[3]{\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) \cdot \sqrt[3]{\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\]
\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 x.re + x.im \cdot x.re\right) \cdot x.re + \left(\sqrt[3]{\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)} \cdot \sqrt[3]{\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}\right) \cdot \sqrt[3]{\left(x.im \cdot \left(x.re + x.im\right)\right) \cdot \left(x.re - x.im\right)}
double f(double x_re, double x_im) {
        double r8315720 = x_re;
        double r8315721 = r8315720 * r8315720;
        double r8315722 = x_im;
        double r8315723 = r8315722 * r8315722;
        double r8315724 = r8315721 - r8315723;
        double r8315725 = r8315724 * r8315722;
        double r8315726 = r8315720 * r8315722;
        double r8315727 = r8315722 * r8315720;
        double r8315728 = r8315726 + r8315727;
        double r8315729 = r8315728 * r8315720;
        double r8315730 = r8315725 + r8315729;
        return r8315730;
}
↓
double f(double x_re, double x_im) {
        double r8315731 = x_im;
        double r8315732 = x_re;
        double r8315733 = r8315731 * r8315732;
        double r8315734 = r8315733 + r8315733;
        double r8315735 = r8315734 * r8315732;
        double r8315736 = r8315732 + r8315731;
        double r8315737 = r8315731 * r8315736;
        double r8315738 = r8315732 - r8315731;
        double r8315739 = r8315737 * r8315738;
        double r8315740 = cbrt(r8315739);
        double r8315741 = r8315740 * r8315740;
        double r8315742 = r8315741 * r8315740;
        double r8315743 = r8315735 + r8315742;
        return r8315743;
}

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
Herbie0.7
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

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 6.9

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

  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-cbrt0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019135 
(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)))