Average Error: 0.3 → 0.1
Time: 46.2s
Precision: 64
\[\left(re \cdot re\right) - \left(im \cdot im\right)\]
↓
\[\left(re - im\right) \cdot \left(im + re\right)\]
\left(re \cdot re\right) - \left(im \cdot im\right)
↓
\left(re - im\right) \cdot \left(im + re\right)
double f(double re, double im) {
        double r310918 = re;
        double r310919 = r310918 * r310918;
        double r310920 = im;
        double r310921 = r310920 * r310920;
        double r310922 = r310919 - r310921;
        return r310922;
}
↓
double f(double re, double im) {
        double r310923 = re;
        double r310924 = im;
        double r310925 = r310923 - r310924;
        double r310926 = r310924 + r310923;
        double r310927 = r310925 * r310926;
        return r310927;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.3

    \[\left(re \cdot re\right) - \left(im \cdot im\right)\]

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))