Average Error: 0.4 → 0.4
Time: 15.1s
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 r2318089 = x_re;
        double r2318090 = r2318089 * r2318089;
        double r2318091 = x_im;
        double r2318092 = r2318091 * r2318091;
        double r2318093 = r2318090 - r2318092;
        double r2318094 = r2318093 * r2318091;
        double r2318095 = r2318089 * r2318091;
        double r2318096 = r2318091 * r2318089;
        double r2318097 = r2318095 + r2318096;
        double r2318098 = r2318097 * r2318089;
        double r2318099 = r2318094 + r2318098;
        return r2318099;
}
↓
double f(double x_re, double x_im) {
        double r2318100 = x_re;
        double r2318101 = x_im;
        double r2318102 = r2318100 + r2318101;
        double r2318103 = r2318100 - r2318101;
        double r2318104 = r2318103 * r2318101;
        double r2318105 = r2318102 * r2318104;
        double r2318106 = r2318100 * r2318101;
        double r2318107 = r2318101 * r2318100;
        double r2318108 = r2318106 + r2318107;
        double r2318109 = r2318108 * r2318100;
        double r2318110 = r2318105 + r2318109;
        return r2318110;
}

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 2019130 +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)))