math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 35.8s

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 r444846 = re;
        double r444847 = r444846 * r444846;
        double r444848 = im;
        double r444849 = r444848 * r444848;
        double r444850 = r444847 - r444849;
        return r444850;
}

↓

double f(double re, double im) {
        double r444851 = re;
        double r444852 = im;
        double r444853 = r444851 - r444852;
        double r444854 = r444852 + r444851;
        double r444855 = r444853 * r444854;
        return r444855;
}

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 2019144 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))