Average Error: 0.4 → 0.4
Time: 11.0s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]
\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(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re
double f(double x_re, double x_im) {
        double r2453142 = x_re;
        double r2453143 = r2453142 * r2453142;
        double r2453144 = x_im;
        double r2453145 = r2453144 * r2453144;
        double r2453146 = r2453143 - r2453145;
        double r2453147 = r2453146 * r2453144;
        double r2453148 = r2453142 * r2453144;
        double r2453149 = r2453144 * r2453142;
        double r2453150 = r2453148 + r2453149;
        double r2453151 = r2453150 * r2453142;
        double r2453152 = r2453147 + r2453151;
        return r2453152;
}
↓
double f(double x_re, double x_im) {
        double r2453153 = x_re;
        double r2453154 = x_im;
        double r2453155 = r2453153 + r2453154;
        double r2453156 = r2453153 - r2453154;
        double r2453157 = r2453156 * r2453154;
        double r2453158 = r2453155 * r2453157;
        double r2453159 = r2453153 * r2453154;
        double r2453160 = r2453154 * r2453153;
        double r2453161 = r2453159 + r2453160;
        double r2453162 = r2453161 * r2453153;
        double r2453163 = r2453158 + r2453162;
        return r2453163;
}

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 difference-of-squares0.4

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

  4. Applied associate-*l*0.4

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

  5. Final simplification0.4

    \[\leadsto \left(x.re + x.im\right) \cdot \left(\left(x.re - x.im\right) \cdot x.im\right) + \left(x.re \cdot x.im + x.im \cdot x.re\right) \cdot x.re\]

Reproduce

herbie shell --seed 2019125 
(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)))