math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 8.2s

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 r317585 = re;
        double r317586 = r317585 * r317585;
        double r317587 = im;
        double r317588 = r317587 * r317587;
        double r317589 = r317586 - r317588;
        return r317589;
}

↓

double f(double re, double im) {
        double r317590 = re;
        double r317591 = im;
        double r317592 = r317590 - r317591;
        double r317593 = r317591 + r317590;
        double r317594 = r317592 * r317593;
        return r317594;
}

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 2019107 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))