math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 13.4s

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 r653812 = re;
        double r653813 = r653812 * r653812;
        double r653814 = im;
        double r653815 = r653814 * r653814;
        double r653816 = r653813 - r653815;
        return r653816;
}

↓

double f(double re, double im) {
        double r653817 = re;
        double r653818 = im;
        double r653819 = r653817 - r653818;
        double r653820 = r653818 + r653817;
        double r653821 = r653819 * r653820;
        return r653821;
}

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 2019158 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))