math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 7.7s

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 r303706 = re;
        double r303707 = r303706 * r303706;
        double r303708 = im;
        double r303709 = r303708 * r303708;
        double r303710 = r303707 - r303709;
        return r303710;
}

↓

double f(double re, double im) {
        double r303711 = re;
        double r303712 = im;
        double r303713 = r303711 - r303712;
        double r303714 = r303712 + r303711;
        double r303715 = r303713 * r303714;
        return r303715;
}

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 2019125 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))