Average Error: 0.4 → 0.3
Time: 25.4s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.im \cdot \left(x.re + x.re\right)\right), x.re\right)\right)\]
\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}
↓
\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.im \cdot \left(x.re + x.re\right)\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r1947890 = x_re;
        double r1947891 = r1947890 * r1947890;
        double r1947892 = x_im;
        double r1947893 = r1947892 * r1947892;
        double r1947894 = r1947891 - r1947893;
        double r1947895 = r1947894 * r1947892;
        double r1947896 = r1947890 * r1947892;
        double r1947897 = r1947892 * r1947890;
        double r1947898 = r1947896 + r1947897;
        double r1947899 = r1947898 * r1947890;
        double r1947900 = r1947895 + r1947899;
        return r1947900;
}
↓
double f(double x_re, double x_im) {
        double r1947901 = x_im;
        double r1947902 = x_re;
        double r1947903 = r1947902 - r1947901;
        double r1947904 = r1947901 * r1947903;
        double r1947905 = r1947901 + r1947902;
        double r1947906 = r1947904 * r1947905;
        double r1947907 = /*Error: no posit support in C */;
        double r1947908 = r1947902 + r1947902;
        double r1947909 = r1947901 * r1947908;
        double r1947910 = /*Error: no posit support in C */;
        double r1947911 = /*Error: no posit support in C */;
        return r1947911;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

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

  3. Applied introduce-quire0.4

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

  4. Applied insert-quire-fdp-add0.3

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

  5. Simplified0.3

    \[\leadsto \color{blue}{\left(\mathsf{qma}\left(\left(\left(x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(\frac{x.im}{x.re}\right)\right)\right)\right), \left(x.im \cdot \left(\frac{x.re}{x.re}\right)\right), x.re\right)\right)}\]
  6. Using strategy rm

  7. Applied associate-*r*0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x.re x.im)
  :name "math.cube on complex, imaginary part"
  (+.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.im) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.re)))