math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 17.1s

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 r1305755 = re;
        double r1305756 = r1305755 * r1305755;
        double r1305757 = im;
        double r1305758 = r1305757 * r1305757;
        double r1305759 = r1305756 - r1305758;
        return r1305759;
}

↓

double f(double re, double im) {
        double r1305760 = re;
        double r1305761 = im;
        double r1305762 = r1305760 - r1305761;
        double r1305763 = r1305761 + r1305760;
        double r1305764 = r1305762 * r1305763;
        return r1305764;
}

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 2019134 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))