Average Error: 6.9 → 0.2
Time: 1.4m
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\mathsf{fma}\left(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right), \left(\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)\right)\]
\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(\left(x.im + x.re\right), \left(\left(x.re - x.im\right) \cdot x.im\right), \left(\left(x.re \cdot x.im + x.re \cdot x.im\right) \cdot x.re\right)\right)
double f(double x_re, double x_im) {
        double r74555585 = x_re;
        double r74555586 = r74555585 * r74555585;
        double r74555587 = x_im;
        double r74555588 = r74555587 * r74555587;
        double r74555589 = r74555586 - r74555588;
        double r74555590 = r74555589 * r74555587;
        double r74555591 = r74555585 * r74555587;
        double r74555592 = r74555587 * r74555585;
        double r74555593 = r74555591 + r74555592;
        double r74555594 = r74555593 * r74555585;
        double r74555595 = r74555590 + r74555594;
        return r74555595;
}
↓
double f(double x_re, double x_im) {
        double r74555596 = x_im;
        double r74555597 = x_re;
        double r74555598 = r74555596 + r74555597;
        double r74555599 = r74555597 - r74555596;
        double r74555600 = r74555599 * r74555596;
        double r74555601 = r74555597 * r74555596;
        double r74555602 = r74555601 + r74555601;
        double r74555603 = r74555602 * r74555597;
        double r74555604 = fma(r74555598, r74555600, r74555603);
        return r74555604;
}

Error

Bits error versus x.re

Bits error versus x.im

Target

Original6.9
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 6.9

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

  3. Applied difference-of-squares6.9

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

  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

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019125 +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)))