Average Error: 6.5 → 0.5
Time: 58.6s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\sqrt[3]{x.re} \cdot \left(\sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \sqrt[3]{x.re \cdot x.im + x.re \cdot x.im}\right)\right) \cdot \sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} + \left(\left(x.im + x.re\right) \cdot x.im\right) \cdot \left(x.re - x.im\right)\]
\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(\sqrt[3]{x.re} \cdot \left(\sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \sqrt[3]{x.re \cdot x.im + x.re \cdot x.im}\right)\right) \cdot \sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} + \left(\left(x.im + x.re\right) \cdot x.im\right) \cdot \left(x.re - x.im\right)
double f(double x_re, double x_im) {
        double r30151808 = x_re;
        double r30151809 = r30151808 * r30151808;
        double r30151810 = x_im;
        double r30151811 = r30151810 * r30151810;
        double r30151812 = r30151809 - r30151811;
        double r30151813 = r30151812 * r30151810;
        double r30151814 = r30151808 * r30151810;
        double r30151815 = r30151810 * r30151808;
        double r30151816 = r30151814 + r30151815;
        double r30151817 = r30151816 * r30151808;
        double r30151818 = r30151813 + r30151817;
        return r30151818;
}
↓
double f(double x_re, double x_im) {
        double r30151819 = x_re;
        double r30151820 = cbrt(r30151819);
        double r30151821 = x_im;
        double r30151822 = r30151819 * r30151821;
        double r30151823 = r30151822 + r30151822;
        double r30151824 = r30151819 * r30151823;
        double r30151825 = cbrt(r30151824);
        double r30151826 = cbrt(r30151823);
        double r30151827 = r30151825 * r30151826;
        double r30151828 = r30151820 * r30151827;
        double r30151829 = r30151828 * r30151825;
        double r30151830 = r30151821 + r30151819;
        double r30151831 = r30151830 * r30151821;
        double r30151832 = r30151819 - r30151821;
        double r30151833 = r30151831 * r30151832;
        double r30151834 = r30151829 + r30151833;
        return r30151834;
}

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

Original6.5
Target0.2
Herbie0.5
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 6.5

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

  2. Taylor expanded around 0 6.4

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

  3. Simplified0.2

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

  5. Applied add-cube-cbrt0.6

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

  7. Applied cbrt-prod0.5

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

  8. Applied associate-*r*0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019124 
(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)))