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 r653907 = re;
        double r653908 = r653907 * r653907;
        double r653909 = im;
        double r653910 = r653909 * r653909;
        double r653911 = r653908 - r653910;
        return r653911;
}

↓

double f(double re, double im) {
        double r653912 = re;
        double r653913 = im;
        double r653914 = r653912 - r653913;
        double r653915 = r653913 + r653912;
        double r653916 = r653914 * r653915;
        return r653916;
}

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 2019163 
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))