math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 6.0s

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 r74338 = re;
        double r74339 = r74338 * r74338;
        double r74340 = im;
        double r74341 = r74340 * r74340;
        double r74342 = r74339 - r74341;
        return r74342;
}

↓

double f(double re, double im) {
        double r74343 = re;
        double r74344 = im;
        double r74345 = r74343 - r74344;
        double r74346 = r74344 + r74343;
        double r74347 = r74345 * r74346;
        return r74347;
}

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 2019124 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))