Average Error: 0.4 → 0.4
Time: 12.7s
Precision: 64
\[\frac{\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.im\right)}{\left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.re\right)}\]
↓
\[\left(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - x.im \cdot x.im\right) \cdot x.im\]
\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(\left(\left(x.re + x.re\right) \cdot x.re + x.re \cdot x.re\right) - x.im \cdot x.im\right) \cdot x.im
double f(double x_re, double x_im) {
        double r1599863 = x_re;
        double r1599864 = r1599863 * r1599863;
        double r1599865 = x_im;
        double r1599866 = r1599865 * r1599865;
        double r1599867 = r1599864 - r1599866;
        double r1599868 = r1599867 * r1599865;
        double r1599869 = r1599863 * r1599865;
        double r1599870 = r1599865 * r1599863;
        double r1599871 = r1599869 + r1599870;
        double r1599872 = r1599871 * r1599863;
        double r1599873 = r1599868 + r1599872;
        return r1599873;
}
↓
double f(double x_re, double x_im) {
        double r1599874 = x_re;
        double r1599875 = r1599874 + r1599874;
        double r1599876 = r1599875 * r1599874;
        double r1599877 = r1599874 * r1599874;
        double r1599878 = r1599876 + r1599877;
        double r1599879 = x_im;
        double r1599880 = r1599879 * r1599879;
        double r1599881 = r1599878 - r1599880;
        double r1599882 = r1599881 * r1599879;
        return r1599882;
}

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

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

  3. Simplified0.4

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

  5. Applied distribute-rgt-in0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019119 +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)))