Average Error: 7.7 → 0.2
Time: 25.6s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(x.im, -3 \cdot \left(x.im \cdot x.re\right), {x.re}^{3}\right)\]
\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
↓
\mathsf{fma}\left(x.im, -3 \cdot \left(x.im \cdot x.re\right), {x.re}^{3}\right)
double f(double x_re, double x_im) {
        double r175927 = x_re;
        double r175928 = r175927 * r175927;
        double r175929 = x_im;
        double r175930 = r175929 * r175929;
        double r175931 = r175928 - r175930;
        double r175932 = r175931 * r175927;
        double r175933 = r175927 * r175929;
        double r175934 = r175929 * r175927;
        double r175935 = r175933 + r175934;
        double r175936 = r175935 * r175929;
        double r175937 = r175932 - r175936;
        return r175937;
}
↓
double f(double x_re, double x_im) {
        double r175938 = x_im;
        double r175939 = -3.0;
        double r175940 = x_re;
        double r175941 = r175938 * r175940;
        double r175942 = r175939 * r175941;
        double r175943 = 3.0;
        double r175944 = pow(r175940, r175943);
        double r175945 = fma(r175938, r175942, r175944);
        return r175945;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.7
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 7.7

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

  2. Simplified0.2

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

  3. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(x.im, -3 \cdot \left(x.im \cdot x.re\right), {x.re}^{3}\right)\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  :precision binary64

  :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)))