Average Error: 7.6 → 0.3
Time: 24.5s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[{x.re}^{3} - \sqrt{3} \cdot \left(\left(\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{3}}\right) \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\left(x.im \cdot x.re\right) \cdot x.im\right)\right)\right)\]
\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im
↓
{x.re}^{3} - \sqrt{3} \cdot \left(\left(\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{3}}\right) \cdot \left(\sqrt[3]{\sqrt{3}} \cdot \left(\left(x.im \cdot x.re\right) \cdot x.im\right)\right)\right)
double f(double x_re, double x_im) {
        double r128678 = x_re;
        double r128679 = r128678 * r128678;
        double r128680 = x_im;
        double r128681 = r128680 * r128680;
        double r128682 = r128679 - r128681;
        double r128683 = r128682 * r128678;
        double r128684 = r128678 * r128680;
        double r128685 = r128680 * r128678;
        double r128686 = r128684 + r128685;
        double r128687 = r128686 * r128680;
        double r128688 = r128683 - r128687;
        return r128688;
}
↓
double f(double x_re, double x_im) {
        double r128689 = x_re;
        double r128690 = 3.0;
        double r128691 = pow(r128689, r128690);
        double r128692 = sqrt(r128690);
        double r128693 = cbrt(r128692);
        double r128694 = r128693 * r128693;
        double r128695 = x_im;
        double r128696 = r128695 * r128689;
        double r128697 = r128696 * r128695;
        double r128698 = r128693 * r128697;
        double r128699 = r128694 * r128698;
        double r128700 = r128692 * r128699;
        double r128701 = r128691 - r128700;
        return r128701;
}

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

Original7.6
Target0.2
Herbie0.3
\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

  1. Initial program 7.6

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

  2. Simplified0.2

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.3

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

  7. Applied add-cube-cbrt0.2

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

  8. Applied associate-*l*0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019322 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3 x.im))))

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