Average Error: 7.1 → 0.7
Time: 22.7s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(x.im \cdot \sqrt[3]{x.re \cdot x.im + x.re \cdot x.im}\right) \cdot \left(\sqrt[3]{x.re \cdot x.im + x.re \cdot x.im} \cdot \left(\sqrt[3]{\sqrt[3]{x.re \cdot x.im + x.re \cdot x.im} \cdot \sqrt[3]{x.re \cdot x.im + x.re \cdot x.im}} \cdot \sqrt[3]{\sqrt[3]{x.re \cdot x.im + x.re \cdot x.im}}\right)\right)\]
\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
↓
\left(\left(x.re - x.im\right) \cdot x.re\right) \cdot \left(x.im + x.re\right) - \left(x.im \cdot \sqrt[3]{x.re \cdot x.im + x.re \cdot x.im}\right) \cdot \left(\sqrt[3]{x.re \cdot x.im + x.re \cdot x.im} \cdot \left(\sqrt[3]{\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}} \cdot \sqrt[3]{\sqrt[3]{x.re \cdot x.im + x.re \cdot x.im}}\right)\right)
double f(double x_re, double x_im) {
        double r7905861 = x_re;
        double r7905862 = r7905861 * r7905861;
        double r7905863 = x_im;
        double r7905864 = r7905863 * r7905863;
        double r7905865 = r7905862 - r7905864;
        double r7905866 = r7905865 * r7905861;
        double r7905867 = r7905861 * r7905863;
        double r7905868 = r7905863 * r7905861;
        double r7905869 = r7905867 + r7905868;
        double r7905870 = r7905869 * r7905863;
        double r7905871 = r7905866 - r7905870;
        return r7905871;
}
↓
double f(double x_re, double x_im) {
        double r7905872 = x_re;
        double r7905873 = x_im;
        double r7905874 = r7905872 - r7905873;
        double r7905875 = r7905874 * r7905872;
        double r7905876 = r7905873 + r7905872;
        double r7905877 = r7905875 * r7905876;
        double r7905878 = r7905872 * r7905873;
        double r7905879 = r7905878 + r7905878;
        double r7905880 = cbrt(r7905879);
        double r7905881 = r7905873 * r7905880;
        double r7905882 = r7905880 * r7905880;
        double r7905883 = cbrt(r7905882);
        double r7905884 = cbrt(r7905880);
        double r7905885 = r7905883 * r7905884;
        double r7905886 = r7905880 * r7905885;
        double r7905887 = r7905881 * r7905886;
        double r7905888 = r7905877 - r7905887;
        return r7905888;
}

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

Original7.1
Target0.3
Herbie0.7
\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

  1. Initial program 7.1

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

  2. Taylor expanded around 0 7.0

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

  3. Simplified0.3

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

  5. Applied add-cube-cbrt0.6

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

  6. Applied associate-*l*0.6

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

  8. Applied add-cube-cbrt0.6

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

  9. Applied cbrt-prod0.7

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

  10. Final simplification0.7

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"

  :herbie-target
  (+ (* (* x.re x.re) (- x.re x.im)) (* (* x.re x.im) (- x.re (* 3.0 x.im))))

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