Average Error: 7.0 → 0.2
Time: 11.7s
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\mathsf{fma}\left(x.im + x.re, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)\]
\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
↓
\mathsf{fma}\left(x.im + x.re, \left(x.re - x.im\right) \cdot x.im, \left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)
double f(double x_re, double x_im) {
        double r3071358 = x_re;
        double r3071359 = r3071358 * r3071358;
        double r3071360 = x_im;
        double r3071361 = r3071360 * r3071360;
        double r3071362 = r3071359 - r3071361;
        double r3071363 = r3071362 * r3071360;
        double r3071364 = r3071358 * r3071360;
        double r3071365 = r3071360 * r3071358;
        double r3071366 = r3071364 + r3071365;
        double r3071367 = r3071366 * r3071358;
        double r3071368 = r3071363 + r3071367;
        return r3071368;
}
↓
double f(double x_re, double x_im) {
        double r3071369 = x_im;
        double r3071370 = x_re;
        double r3071371 = r3071369 + r3071370;
        double r3071372 = r3071370 - r3071369;
        double r3071373 = r3071372 * r3071369;
        double r3071374 = r3071370 * r3071369;
        double r3071375 = r3071374 + r3071374;
        double r3071376 = r3071375 * r3071370;
        double r3071377 = fma(r3071371, r3071373, r3071376);
        return r3071377;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original7.0
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 7.0

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

  3. Applied difference-of-squares7.0

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

  4. Applied associate-*l*0.2

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

  6. Applied fma-def0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(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)))