Average Error: 0.4 → 0.4
Time: 25.2s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right) - \left(x.im + x.im\right) \cdot x.im\right)\]
\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)
↓
x.re \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right) - \left(x.im + x.im\right) \cdot x.im\right)
double f(double x_re, double x_im) {
        double r2902135 = x_re;
        double r2902136 = r2902135 * r2902135;
        double r2902137 = x_im;
        double r2902138 = r2902137 * r2902137;
        double r2902139 = r2902136 - r2902138;
        double r2902140 = r2902139 * r2902135;
        double r2902141 = r2902135 * r2902137;
        double r2902142 = r2902137 * r2902135;
        double r2902143 = r2902141 + r2902142;
        double r2902144 = r2902143 * r2902137;
        double r2902145 = r2902140 - r2902144;
        return r2902145;
}
↓
double f(double x_re, double x_im) {
        double r2902146 = x_re;
        double r2902147 = x_im;
        double r2902148 = r2902146 - r2902147;
        double r2902149 = r2902147 + r2902146;
        double r2902150 = r2902148 * r2902149;
        double r2902151 = r2902147 + r2902147;
        double r2902152 = r2902151 * r2902147;
        double r2902153 = r2902150 - r2902152;
        double r2902154 = r2902146 * r2902153;
        return r2902154;
}

Error

Bits error versus x.re

Bits error versus x.im

Derivation

  1. Initial program 0.4

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

  2. Simplified0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019134 
(FPCore (x.re x.im)
  :name "math.cube on complex, real part"
  (-.p16 (*.p16 (-.p16 (*.p16 x.re x.re) (*.p16 x.im x.im)) x.re) (*.p16 (+.p16 (*.p16 x.re x.im) (*.p16 x.im x.re)) x.im)))