Average Error: 0.4 → 0.3
Time: 22.0s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\mathsf{qma}\left(\left(\left(x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\right)\right)\right), \left(x.re \cdot \left(x.im + x.im\right)\right), x.re\right)\right)\]
\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(x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right)\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 r1787962 = x_re;
        double r1787963 = r1787962 * r1787962;
        double r1787964 = x_im;
        double r1787965 = r1787964 * r1787964;
        double r1787966 = r1787963 - r1787965;
        double r1787967 = r1787966 * r1787964;
        double r1787968 = r1787962 * r1787964;
        double r1787969 = r1787964 * r1787962;
        double r1787970 = r1787968 + r1787969;
        double r1787971 = r1787970 * r1787962;
        double r1787972 = r1787967 + r1787971;
        return r1787972;
}
↓
double f(double x_re, double x_im) {
        double r1787973 = x_im;
        double r1787974 = x_re;
        double r1787975 = r1787974 - r1787973;
        double r1787976 = r1787973 + r1787974;
        double r1787977 = r1787975 * r1787976;
        double r1787978 = r1787973 * r1787977;
        double r1787979 = /*Error: no posit support in C */;
        double r1787980 = r1787973 + r1787973;
        double r1787981 = r1787974 * r1787980;
        double r1787982 = /*Error: no posit support in C */;
        double r1787983 = /*Error: no posit support in C */;
        return r1787983;
}

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

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019165 
(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)))