Average Error: 0.4 → 0.3
Time: 23.4s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
\[\left(\mathsf{qma}\left(\left(\left(\left(x.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.im\right)\right)\right), \left(\left(x.im + x.im\right) \cdot x.re\right), x.re\right)\right)\]
\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.re - x.im\right) \cdot \left(\left(x.im + x.re\right) \cdot x.im\right)\right)\right), \left(\left(x.im + x.im\right) \cdot x.re\right), x.re\right)\right)
double f(double x_re, double x_im) {
        double r754143 = x_re;
        double r754144 = r754143 * r754143;
        double r754145 = x_im;
        double r754146 = r754145 * r754145;
        double r754147 = r754144 - r754146;
        double r754148 = r754147 * r754145;
        double r754149 = r754143 * r754145;
        double r754150 = r754145 * r754143;
        double r754151 = r754149 + r754150;
        double r754152 = r754151 * r754143;
        double r754153 = r754148 + r754152;
        return r754153;
}

double f(double x_re, double x_im) {
        double r754154 = x_re;
        double r754155 = x_im;
        double r754156 = r754154 - r754155;
        double r754157 = r754155 + r754154;
        double r754158 = r754157 * r754155;
        double r754159 = r754156 * r754158;
        double r754160 = /*Error: no posit support in C */;
        double r754161 = r754155 + r754155;
        double r754162 = r754161 * r754154;
        double r754163 = /*Error: no posit support in C */;
        double r754164 = /*Error: no posit support in C */;
        return r754164;
}

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

    \[\leadsto \color{blue}{\left(\mathsf{qma}\left(\left(\left(\left(x.re - x.im\right) \cdot \left(\left(\frac{x.im}{x.re}\right) \cdot x.im\right)\right)\right), \left(\left(\frac{x.im}{x.im}\right) \cdot x.re\right), x.re\right)\right)}\]
  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019151 
(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)))