math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 47.2s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r310858 = re;
        double r310859 = r310858 * r310858;
        double r310860 = im;
        double r310861 = r310860 * r310860;
        double r310862 = r310859 - r310861;
        return r310862;
}

↓

double f(double re, double im) {
        double r310863 = re;
        double r310864 = im;
        double r310865 = r310863 - r310864;
        double r310866 = r310864 + r310863;
        double r310867 = r310865 * r310866;
        return r310867;
}

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. Simplified 0.1

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

3. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))