Average Error: 6.8 → 0.7
Time: 47.3s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re + \left(\sqrt[3]{x.im + x.re} \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)\right) \cdot \left(\sqrt[3]{x.im + x.re} \cdot \sqrt[3]{x.im + x.re}\right)\]
\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.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re + \left(\sqrt[3]{x.im + x.re} \cdot \left(\left(x.re - x.im\right) \cdot x.im\right)\right) \cdot \left(\sqrt[3]{x.im + x.re} \cdot \sqrt[3]{x.im + x.re}\right)
double f(double x_re, double x_im) {
        double r29679876 = x_re;
        double r29679877 = r29679876 * r29679876;
        double r29679878 = x_im;
        double r29679879 = r29679878 * r29679878;
        double r29679880 = r29679877 - r29679879;
        double r29679881 = r29679880 * r29679878;
        double r29679882 = r29679876 * r29679878;
        double r29679883 = r29679878 * r29679876;
        double r29679884 = r29679882 + r29679883;
        double r29679885 = r29679884 * r29679876;
        double r29679886 = r29679881 + r29679885;
        return r29679886;
}
↓
double f(double x_re, double x_im) {
        double r29679887 = x_re;
        double r29679888 = x_im;
        double r29679889 = r29679887 * r29679888;
        double r29679890 = r29679889 + r29679889;
        double r29679891 = r29679890 * r29679887;
        double r29679892 = r29679888 + r29679887;
        double r29679893 = cbrt(r29679892);
        double r29679894 = r29679887 - r29679888;
        double r29679895 = r29679894 * r29679888;
        double r29679896 = r29679893 * r29679895;
        double r29679897 = r29679893 * r29679893;
        double r29679898 = r29679896 * r29679897;
        double r29679899 = r29679891 + r29679898;
        return r29679899;
}

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

Original6.8
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 6.8

    \[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
  2. Using strategy rm

  3. Applied difference-of-squares6.8

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

  4. Applied associate-*l*0.2

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

  6. Applied add-cube-cbrt0.7

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

  7. Applied associate-*l*0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(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)))