Average Error: 0.4 → 0.4
Time: 10.4s
Precision: 64
\[\left(\left(\left(x.re \cdot x.re\right) - \left(x.im \cdot x.im\right)\right) \cdot x.re\right) - \left(\left(\frac{\left(x.re \cdot x.im\right)}{\left(x.im \cdot x.re\right)}\right) \cdot x.im\right)\]
↓
\[\left(x.re \cdot \left(x.im + x.re\right)\right) \cdot \left(x.re - x.im\right) - \left(x.im \cdot \left(x.im + x.im\right)\right) \cdot x.re\]
\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) - \left(x.im \cdot \left(x.im + x.im\right)\right) \cdot x.re
double f(double x_re, double x_im) {
        double r440423 = x_re;
        double r440424 = r440423 * r440423;
        double r440425 = x_im;
        double r440426 = r440425 * r440425;
        double r440427 = r440424 - r440426;
        double r440428 = r440427 * r440423;
        double r440429 = r440423 * r440425;
        double r440430 = r440425 * r440423;
        double r440431 = r440429 + r440430;
        double r440432 = r440431 * r440425;
        double r440433 = r440428 - r440432;
        return r440433;
}
↓
double f(double x_re, double x_im) {
        double r440434 = x_re;
        double r440435 = x_im;
        double r440436 = r440435 + r440434;
        double r440437 = r440434 * r440436;
        double r440438 = r440434 - r440435;
        double r440439 = r440437 * r440438;
        double r440440 = r440435 + r440435;
        double r440441 = r440435 * r440440;
        double r440442 = r440441 * r440434;
        double r440443 = r440439 - r440442;
        return r440443;
}

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. Using strategy rm

  6. Applied associate-*r*0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019104 
(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)))