Average Error: 0.4 → 0.4
Time: 15.6s
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 r1953683 = x_re;
        double r1953684 = r1953683 * r1953683;
        double r1953685 = x_im;
        double r1953686 = r1953685 * r1953685;
        double r1953687 = r1953684 - r1953686;
        double r1953688 = r1953687 * r1953685;
        double r1953689 = r1953683 * r1953685;
        double r1953690 = r1953685 * r1953683;
        double r1953691 = r1953689 + r1953690;
        double r1953692 = r1953691 * r1953683;
        double r1953693 = r1953688 + r1953692;
        return r1953693;
}
↓
double f(double x_re, double x_im) {
        double r1953694 = x_im;
        double r1953695 = x_re;
        double r1953696 = r1953695 - r1953694;
        double r1953697 = r1953694 + r1953695;
        double r1953698 = r1953696 * r1953697;
        double r1953699 = r1953695 + r1953695;
        double r1953700 = r1953699 * r1953695;
        double r1953701 = r1953698 + r1953700;
        double r1953702 = r1953694 * r1953701;
        return r1953702;
}

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 
(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)))