Average Error: 7.0 → 0.2
Time: 1.6m
Precision: 64
\[\left(x.re \cdot x.re - x.im \cdot x.im\right) \cdot x.im + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
↓
\[\mathsf{fma}\left(\left(x.im \cdot \left(x.re + x.im\right)\right), \left(x.re - x.im\right), \left(x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right)\right)\]
\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 \cdot \left(x.re + x.im\right)\right), \left(x.re - x.im\right), \left(x.re \cdot \left(x.im \cdot x.re + x.im \cdot x.re\right)\right)\right)
double f(double x_re, double x_im) {
        double r55344889 = x_re;
        double r55344890 = r55344889 * r55344889;
        double r55344891 = x_im;
        double r55344892 = r55344891 * r55344891;
        double r55344893 = r55344890 - r55344892;
        double r55344894 = r55344893 * r55344891;
        double r55344895 = r55344889 * r55344891;
        double r55344896 = r55344891 * r55344889;
        double r55344897 = r55344895 + r55344896;
        double r55344898 = r55344897 * r55344889;
        double r55344899 = r55344894 + r55344898;
        return r55344899;
}
↓
double f(double x_re, double x_im) {
        double r55344900 = x_im;
        double r55344901 = x_re;
        double r55344902 = r55344901 + r55344900;
        double r55344903 = r55344900 * r55344902;
        double r55344904 = r55344901 - r55344900;
        double r55344905 = r55344900 * r55344901;
        double r55344906 = r55344905 + r55344905;
        double r55344907 = r55344901 * r55344906;
        double r55344908 = fma(r55344903, r55344904, r55344907);
        return r55344908;
}

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. Taylor expanded around 0 6.9

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

  3. Simplified0.2

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

  5. Applied fma-def0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 +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)))