math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 43.4s

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 r1796980 = re;
        double r1796981 = r1796980 * r1796980;
        double r1796982 = im;
        double r1796983 = r1796982 * r1796982;
        double r1796984 = r1796981 - r1796983;
        return r1796984;
}

↓

double f(double re, double im) {
        double r1796985 = re;
        double r1796986 = im;
        double r1796987 = r1796985 - r1796986;
        double r1796988 = r1796986 + r1796985;
        double r1796989 = r1796987 * r1796988;
        return r1796989;
}

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 2019132 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))