math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 7.0s

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 r23540 = re;
        double r23541 = r23540 * r23540;
        double r23542 = im;
        double r23543 = r23542 * r23542;
        double r23544 = r23541 - r23543;
        return r23544;
}

↓

double f(double re, double im) {
        double r23545 = re;
        double r23546 = im;
        double r23547 = r23545 - r23546;
        double r23548 = r23546 + r23545;
        double r23549 = r23547 * r23548;
        return r23549;
}

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