Average Error: 6.7 → 0.2
Time: 52.5s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\left(\left(x.im \cdot \left(-x.re\right)\right) \cdot \left(x.re + x.im\right) - x.im \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right) + \left(x.re + x.im\right) \cdot \left(x.re \cdot x.re\right)\]
double f(double x_re, double x_im) {
        double r43759976 = x_re;
        double r43759977 = r43759976 * r43759976;
        double r43759978 = x_im;
        double r43759979 = r43759978 * r43759978;
        double r43759980 = r43759977 - r43759979;
        double r43759981 = r43759980 * r43759976;
        double r43759982 = r43759976 * r43759978;
        double r43759983 = r43759978 * r43759976;
        double r43759984 = r43759982 + r43759983;
        double r43759985 = r43759984 * r43759978;
        double r43759986 = r43759981 - r43759985;
        return r43759986;
}
↓
double f(double x_re, double x_im) {
        double r43759987 = x_im;
        double r43759988 = x_re;
        double r43759989 = -r43759988;
        double r43759990 = r43759987 * r43759989;
        double r43759991 = r43759988 + r43759987;
        double r43759992 = r43759990 * r43759991;
        double r43759993 = r43759987 * r43759988;
        double r43759994 = r43759993 + r43759993;
        double r43759995 = r43759987 * r43759994;
        double r43759996 = r43759992 - r43759995;
        double r43759997 = r43759988 * r43759988;
        double r43759998 = r43759991 * r43759997;
        double r43759999 = r43759996 + r43759998;
        return r43759999;
}


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

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.7
Target0.3
Herbie0.2
\[\left(x.re \cdot x.re\right) \cdot \left(x.re - x.im\right) + \left(x.re \cdot x.im\right) \cdot \left(x.re - 3 \cdot x.im\right)\]

Derivation

  1. Initial program 6.7

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

  2. Taylor expanded around 0 6.7

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

  3. Simplified0.2

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

  5. Applied sub-neg0.2

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

  6. Applied distribute-rgt-in0.2

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

  7. Applied distribute-lft-in0.2

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

  8. Applied associate--l+0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019102 
(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 x.im))))

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