math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 8.8s

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 r358841 = re;
        double r358842 = r358841 * r358841;
        double r358843 = im;
        double r358844 = r358843 * r358843;
        double r358845 = r358842 - r358844;
        return r358845;
}

↓

double f(double re, double im) {
        double r358846 = re;
        double r358847 = im;
        double r358848 = r358846 - r358847;
        double r358849 = r358847 + r358846;
        double r358850 = r358848 * r358849;
        return r358850;
}

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 2019128 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))