Average Error: 0.3 → 0.1
Time: 7.5s
Precision: 64
\[re \cdot re - im \cdot im\]
↓
\[\left(re - im\right) \cdot \left(im + re\right)\]
double f(double re, double im) {
        double r282676 = re;
        double r282677 = r282676 * r282676;
        double r282678 = im;
        double r282679 = r282678 * r282678;
        double r282680 = r282677 - r282679;
        return r282680;
}
↓
double f(double re, double im) {
        double r282681 = re;
        double r282682 = im;
        double r282683 = r282681 - r282682;
        double r282684 = r282682 + r282681;
        double r282685 = r282683 * r282684;
        return r282685;
}


re \cdot re - im \cdot im
↓
\left(re - im\right) \cdot \left(im + re\right)

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 2019102 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))