Average Error: 0.4 → 0.4
Time: 14.7s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right) + \left(x.re + x.re\right) \cdot x.re\right)\]
\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)}
↓
x.im \cdot \left(\left(x.re - x.im\right) \cdot \left(x.im + x.re\right) + \left(x.re + x.re\right) \cdot x.re\right)
double f(double x_re, double x_im) {
        double r1770122 = x_re;
        double r1770123 = r1770122 * r1770122;
        double r1770124 = x_im;
        double r1770125 = r1770124 * r1770124;
        double r1770126 = r1770123 - r1770125;
        double r1770127 = r1770126 * r1770124;
        double r1770128 = r1770122 * r1770124;
        double r1770129 = r1770124 * r1770122;
        double r1770130 = r1770128 + r1770129;
        double r1770131 = r1770130 * r1770122;
        double r1770132 = r1770127 + r1770131;
        return r1770132;
}
↓
double f(double x_re, double x_im) {
        double r1770133 = x_im;
        double r1770134 = x_re;
        double r1770135 = r1770134 - r1770133;
        double r1770136 = r1770133 + r1770134;
        double r1770137 = r1770135 * r1770136;
        double r1770138 = r1770134 + r1770134;
        double r1770139 = r1770138 * r1770134;
        double r1770140 = r1770137 + r1770139;
        double r1770141 = r1770133 * r1770140;
        return r1770141;
}

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. Simplified0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019128 +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)))