math.square on complex, real part

Average Error: 0.3 → 0.1

Time: 11.7s

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 r253314 = re;
        double r253315 = r253314 * r253314;
        double r253316 = im;
        double r253317 = r253316 * r253316;
        double r253318 = r253315 - r253317;
        return r253318;
}

↓

double f(double re, double im) {
        double r253319 = re;
        double r253320 = im;
        double r253321 = r253319 - r253320;
        double r253322 = r253320 + r253319;
        double r253323 = r253321 * r253322;
        return r253323;
}

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 2019151 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, real part"
  (-.p16 (*.p16 re re) (*.p16 im im)))