math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 33.5s

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 r444867 = re;
        double r444868 = r444867 * r444867;
        double r444869 = im;
        double r444870 = r444869 * r444869;
        double r444871 = r444868 - r444870;
        return r444871;
}

↓

double f(double re, double im) {
        double r444872 = re;
        double r444873 = im;
        double r444874 = r444872 - r444873;
        double r444875 = r444873 + r444872;
        double r444876 = r444874 * r444875;
        return r444876;
}

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 2019149 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))