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 r282633 = re;
        double r282634 = r282633 * r282633;
        double r282635 = im;
        double r282636 = r282635 * r282635;
        double r282637 = r282634 - r282636;
        return r282637;
}

↓

double f(double re, double im) {
        double r282638 = re;
        double r282639 = im;
        double r282640 = r282638 - r282639;
        double r282641 = r282639 + r282638;
        double r282642 = r282640 * r282641;
        return r282642;
}

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 2019121 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))