Average Error: 6.8 → 0.2
Time: 49.4s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\left(\left(x.im + x.re\right) \cdot x.im\right) \cdot \left(x.re - x.im\right) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)\]
double f(double x_re, double x_im) {
        double r43655978 = x_re;
        double r43655979 = r43655978 * r43655978;
        double r43655980 = x_im;
        double r43655981 = r43655980 * r43655980;
        double r43655982 = r43655979 - r43655981;
        double r43655983 = r43655982 * r43655980;
        double r43655984 = r43655978 * r43655980;
        double r43655985 = r43655980 * r43655978;
        double r43655986 = r43655984 + r43655985;
        double r43655987 = r43655986 * r43655978;
        double r43655988 = r43655983 + r43655987;
        return r43655988;
}
↓
double f(double x_re, double x_im) {
        double r43655989 = x_im;
        double r43655990 = x_re;
        double r43655991 = r43655989 + r43655990;
        double r43655992 = r43655991 * r43655989;
        double r43655993 = r43655990 - r43655989;
        double r43655994 = r43655992 * r43655993;
        double r43655995 = r43655990 * r43655989;
        double r43655996 = r43655995 + r43655995;
        double r43655997 = r43655990 * r43655996;
        double r43655998 = r43655994 + r43655997;
        return r43655998;
}


\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) + x.re \cdot \left(x.re \cdot x.im + x.re \cdot x.im\right)

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.8
Target0.2
Herbie0.2
\[\left(x.re \cdot x.im\right) \cdot \left(2 \cdot x.re\right) + \left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.re + x.im\right)\]

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. Taylor expanded around inf 6.7

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

  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019101 
(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)))