Average Error: 0.4 → 0.4
Time: 9.5s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[x.re \cdot \left(\left(x.im + x.re\right) \cdot \left(x.re - x.im\right)\right) - x.im \cdot \left(\left(x.im + x.im\right) \cdot x.re\right)\]
\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 \left(x.re - 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 r375404 = x_re;
        double r375405 = r375404 * r375404;
        double r375406 = x_im;
        double r375407 = r375406 * r375406;
        double r375408 = r375405 - r375407;
        double r375409 = r375408 * r375404;
        double r375410 = r375404 * r375406;
        double r375411 = r375406 * r375404;
        double r375412 = r375410 + r375411;
        double r375413 = r375412 * r375406;
        double r375414 = r375409 - r375413;
        return r375414;
}
↓
double f(double x_re, double x_im) {
        double r375415 = x_re;
        double r375416 = x_im;
        double r375417 = r375416 + r375415;
        double r375418 = r375415 - r375416;
        double r375419 = r375417 * r375418;
        double r375420 = r375415 * r375419;
        double r375421 = r375416 + r375416;
        double r375422 = r375421 * r375415;
        double r375423 = r375416 * r375422;
        double r375424 = r375420 - r375423;
        return r375424;
}

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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019121 +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)))