math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 33.7s

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 r444876 = re;
        double r444877 = r444876 * r444876;
        double r444878 = im;
        double r444879 = r444878 * r444878;
        double r444880 = r444877 - r444879;
        return r444880;
}

↓

double f(double re, double im) {
        double r444881 = re;
        double r444882 = im;
        double r444883 = r444881 - r444882;
        double r444884 = r444882 + r444881;
        double r444885 = r444883 * r444884;
        return r444885;
}

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 2019138 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))