Average Error: 0.4 → 0.4
Time: 21.0s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[x.re \cdot \frac{x.re + x.im}{\frac{1.0}{x.re - x.im}} - x.im \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)\]
\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 \frac{x.re + x.im}{\frac{1.0}{x.re - x.im}} - x.im \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)
double f(double x_re, double x_im) {
        double r1655578 = x_re;
        double r1655579 = r1655578 * r1655578;
        double r1655580 = x_im;
        double r1655581 = r1655580 * r1655580;
        double r1655582 = r1655579 - r1655581;
        double r1655583 = r1655582 * r1655578;
        double r1655584 = r1655578 * r1655580;
        double r1655585 = r1655580 * r1655578;
        double r1655586 = r1655584 + r1655585;
        double r1655587 = r1655586 * r1655580;
        double r1655588 = r1655583 - r1655587;
        return r1655588;
}
↓
double f(double x_re, double x_im) {
        double r1655589 = x_re;
        double r1655590 = x_im;
        double r1655591 = r1655589 + r1655590;
        double r1655592 = 1.0;
        double r1655593 = r1655589 - r1655590;
        double r1655594 = r1655592 / r1655593;
        double r1655595 = r1655591 / r1655594;
        double r1655596 = r1655589 * r1655595;
        double r1655597 = r1655590 + r1655590;
        double r1655598 = r1655597 * r1655589;
        double r1655599 = r1655590 * r1655598;
        double r1655600 = r1655596 - r1655599;
        return r1655600;
}

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 p16-flip--0.4

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

  5. Applied associate-*r/0.5

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

  6. Simplified0.5

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

  8. Applied associate-/l*0.4

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

  9. Simplified0.4

    \[\leadsto \left(x.re \cdot \left(\frac{\left(\frac{x.re}{x.im}\right)}{\color{blue}{\left(\frac{\left(1.0\right)}{\left(x.re - x.im\right)}\right)}}\right)\right) - \left(x.im \cdot \left(\left(\frac{x.im}{x.im}\right) \cdot x.re\right)\right)\]

  10. Final simplification0.4

    \[\leadsto x.re \cdot \frac{x.re + x.im}{\frac{1.0}{x.re - x.im}} - x.im \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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)))