Average Error: 0.4 → 0.4
Time: 12.5s
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 r1603694 = x_re;
        double r1603695 = r1603694 * r1603694;
        double r1603696 = x_im;
        double r1603697 = r1603696 * r1603696;
        double r1603698 = r1603695 - r1603697;
        double r1603699 = r1603698 * r1603696;
        double r1603700 = r1603694 * r1603696;
        double r1603701 = r1603696 * r1603694;
        double r1603702 = r1603700 + r1603701;
        double r1603703 = r1603702 * r1603694;
        double r1603704 = r1603699 + r1603703;
        return r1603704;
}
↓
double f(double x_re, double x_im) {
        double r1603705 = x_re;
        double r1603706 = r1603705 + r1603705;
        double r1603707 = r1603706 * r1603705;
        double r1603708 = r1603705 * r1603705;
        double r1603709 = r1603707 + r1603708;
        double r1603710 = x_im;
        double r1603711 = r1603710 * r1603710;
        double r1603712 = r1603709 - r1603711;
        double r1603713 = r1603712 * r1603710;
        return r1603713;
}

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 
(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)))