Average Error: 6.6 → 0.6
Time: 54.9s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(x.im + x.re\right) \cdot x.im\right) \cdot \left(x.re - x.im\right) + \sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \left(\sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)} \cdot \sqrt[3]{x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)}\right)\]
double f(double x_re, double x_im) {
        double r27010971 = x_re;
        double r27010972 = r27010971 * r27010971;
        double r27010973 = x_im;
        double r27010974 = r27010973 * r27010973;
        double r27010975 = r27010972 - r27010974;
        double r27010976 = r27010975 * r27010973;
        double r27010977 = r27010971 * r27010973;
        double r27010978 = r27010973 * r27010971;
        double r27010979 = r27010977 + r27010978;
        double r27010980 = r27010979 * r27010971;
        double r27010981 = r27010976 + r27010980;
        return r27010981;
}
↓
double f(double x_re, double x_im) {
        double r27010982 = x_im;
        double r27010983 = x_re;
        double r27010984 = r27010982 + r27010983;
        double r27010985 = r27010984 * r27010982;
        double r27010986 = r27010983 - r27010982;
        double r27010987 = r27010985 * r27010986;
        double r27010988 = r27010983 * r27010982;
        double r27010989 = r27010988 + r27010988;
        double r27010990 = r27010983 * r27010989;
        double r27010991 = cbrt(r27010990);
        double r27010992 = r27010991 * r27010991;
        double r27010993 = r27010991 * r27010992;
        double r27010994 = r27010987 + r27010993;
        return r27010994;
}


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

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.6
Target0.2
Herbie0.6
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

Derivation

  1. Initial program 6.6

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

  2. Taylor expanded around -inf 6.5

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

  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. Final simplification0.6

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

Reproduce

herbie shell --seed 2019102 
(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)))