math.square on complex, imaginary part

Average Error: 0.2 → 0.1

Time: 4.3s

Precision: 64

Math

\[\frac{\left(re \cdot im\right)}{\left(im \cdot re\right)}\]

↓

\[re \cdot \left(im + im\right)\]

C

double f(double re, double im) {
        double r8825 = re;
        double r8826 = im;
        double r8827 = r8825 * r8826;
        double r8828 = r8826 * r8825;
        double r8829 = r8827 + r8828;
        return r8829;
}

↓

double f(double re, double im) {
        double r8830 = re;
        double r8831 = im;
        double r8832 = r8831 + r8831;
        double r8833 = r8830 * r8832;
        return r8833;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.2

   \[\frac{\left(re \cdot im\right)}{\left(im \cdot re\right)}\]

2. Simplified 0.2

   \[\leadsto \color{blue}{\frac{\left(im \cdot re\right)}{\left(im \cdot re\right)}}\]
3. Using strategy rm

4. Applied distribute-rgt-out 0.1

   \[\leadsto \color{blue}{re \cdot \left(\frac{im}{im}\right)}\]

5. Final simplification 0.1

   \[\leadsto re \cdot \left(im + im\right)\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, imaginary part"
  (+.p16 (*.p16 re im) (*.p16 im re)))