Average Error: 0.4 → 0.4
Time: 9.5s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[x.re \cdot \left(\left(x.im + x.re\right) \cdot x.re + \left(x.im + x.re\right) \cdot \left(-x.im\right)\right) - x.im \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)\]
\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.im + x.re\right) \cdot x.re + \left(x.im + x.re\right) \cdot \left(-x.im\right)\right) - x.im \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)
double f(double x_re, double x_im) {
        double r275547 = x_re;
        double r275548 = r275547 * r275547;
        double r275549 = x_im;
        double r275550 = r275549 * r275549;
        double r275551 = r275548 - r275550;
        double r275552 = r275551 * r275547;
        double r275553 = r275547 * r275549;
        double r275554 = r275549 * r275547;
        double r275555 = r275553 + r275554;
        double r275556 = r275555 * r275549;
        double r275557 = r275552 - r275556;
        return r275557;
}
↓
double f(double x_re, double x_im) {
        double r275558 = x_re;
        double r275559 = x_im;
        double r275560 = r275559 + r275558;
        double r275561 = r275560 * r275558;
        double r275562 = -r275559;
        double r275563 = r275560 * r275562;
        double r275564 = r275561 + r275563;
        double r275565 = r275558 * r275564;
        double r275566 = r275559 + r275559;
        double r275567 = r275566 * r275558;
        double r275568 = r275559 * r275567;
        double r275569 = r275565 - r275568;
        return r275569;
}

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

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

  4. Applied sub-neg0.4

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

  5. Applied distribute-lft-in0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019120 
(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)))