math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 18.2s

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 r1135674 = re;
        double r1135675 = r1135674 * r1135674;
        double r1135676 = im;
        double r1135677 = r1135676 * r1135676;
        double r1135678 = r1135675 - r1135677;
        return r1135678;
}

↓

double f(double re, double im) {
        double r1135679 = re;
        double r1135680 = im;
        double r1135681 = r1135679 - r1135680;
        double r1135682 = r1135680 + r1135679;
        double r1135683 = r1135681 * r1135682;
        return r1135683;
}

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 2019135 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))