Average Error: 0.4 → 0.4
Time: 22.6s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(x.re \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)\]
\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)
↓
\left(x.re \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right) - x.im \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)
double f(double x_re, double x_im) {
        double r1825651 = x_re;
        double r1825652 = r1825651 * r1825651;
        double r1825653 = x_im;
        double r1825654 = r1825653 * r1825653;
        double r1825655 = r1825652 - r1825654;
        double r1825656 = r1825655 * r1825651;
        double r1825657 = r1825651 * r1825653;
        double r1825658 = r1825653 * r1825651;
        double r1825659 = r1825657 + r1825658;
        double r1825660 = r1825659 * r1825653;
        double r1825661 = r1825656 - r1825660;
        return r1825661;
}
↓
double f(double x_re, double x_im) {
        double r1825662 = x_re;
        double r1825663 = x_im;
        double r1825664 = r1825663 + r1825662;
        double r1825665 = r1825662 * r1825664;
        double r1825666 = r1825662 - r1825663;
        double r1825667 = r1825665 * r1825666;
        double r1825668 = r1825663 + r1825663;
        double r1825669 = r1825668 * r1825662;
        double r1825670 = r1825663 * r1825669;
        double r1825671 = r1825667 - r1825670;
        return r1825671;
}

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 associate-*r*0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019125 
(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)))