Average Error: 0.4 → 0.3
Time: 46.8s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(\left(x.im \cdot \left(x.re - x.im\right)\right) \cdot \left(x.im + x.re\right)\right)\right), \left(x.re \cdot \left(x.im + x.im\right)\right), x.re\right)\right)\]
\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.re \cdot \left(x.im + x.im\right)\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r4209933 = x_re;
        double r4209934 = r4209933 * r4209933;
        double r4209935 = x_im;
        double r4209936 = r4209935 * r4209935;
        double r4209937 = r4209934 - r4209936;
        double r4209938 = r4209937 * r4209935;
        double r4209939 = r4209933 * r4209935;
        double r4209940 = r4209935 * r4209933;
        double r4209941 = r4209939 + r4209940;
        double r4209942 = r4209941 * r4209933;
        double r4209943 = r4209938 + r4209942;
        return r4209943;
}
↓
double f(double x_re, double x_im) {
        double r4209944 = x_im;
        double r4209945 = x_re;
        double r4209946 = r4209945 - r4209944;
        double r4209947 = r4209944 * r4209946;
        double r4209948 = r4209944 + r4209945;
        double r4209949 = r4209947 * r4209948;
        double r4209950 = /*Error: no posit support in C */;
        double r4209951 = r4209944 + r4209944;
        double r4209952 = r4209945 * r4209951;
        double r4209953 = /*Error: no posit support in C */;
        double r4209954 = /*Error: no posit support in C */;
        return r4209954;
}

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

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019142 
(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)))