Average Error: 7.2 → 0.2
Time: 54.3s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.re - \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.im\]
↓
\[\mathsf{fma}\left(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.re\right), \left(\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot \left(-x.im\right)\right)\right)\]
\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(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.re\right), \left(\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot \left(-x.im\right)\right)\right)
double f(double x_re, double x_im) {
        double r23071783 = x_re;
        double r23071784 = r23071783 * r23071783;
        double r23071785 = x_im;
        double r23071786 = r23071785 * r23071785;
        double r23071787 = r23071784 - r23071786;
        double r23071788 = r23071787 * r23071783;
        double r23071789 = r23071783 * r23071785;
        double r23071790 = r23071785 * r23071783;
        double r23071791 = r23071789 + r23071790;
        double r23071792 = r23071791 * r23071785;
        double r23071793 = r23071788 - r23071792;
        return r23071793;
}
↓
double f(double x_re, double x_im) {
        double r23071794 = x_im;
        double r23071795 = x_re;
        double r23071796 = r23071794 + r23071795;
        double r23071797 = r23071795 - r23071794;
        double r23071798 = r23071797 * r23071795;
        double r23071799 = r23071795 * r23071794;
        double r23071800 = r23071799 + r23071799;
        double r23071801 = -r23071794;
        double r23071802 = r23071800 * r23071801;
        double r23071803 = fma(r23071796, r23071798, r23071802);
        return r23071803;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.2
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 7.2

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

  3. Applied difference-of-squares7.2

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

  4. Applied associate-*l*0.2

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

  6. Applied fma-neg0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(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)))