Average Error: 0.3 → 0.1
Time: 9.0s
Precision: 64
\[re \cdot re - im \cdot im\]
↓
\[\left(re - im\right) \cdot \left(im + re\right)\]
double f(double re, double im) {
        double r322161 = re;
        double r322162 = r322161 * r322161;
        double r322163 = im;
        double r322164 = r322163 * r322163;
        double r322165 = r322162 - r322164;
        return r322165;
}
↓
double f(double re, double im) {
        double r322166 = re;
        double r322167 = im;
        double r322168 = r322166 - r322167;
        double r322169 = r322167 + r322166;
        double r322170 = r322168 * r322169;
        return r322170;
}


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 2019101 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))