math.abs on complex

Average Error: 0.6 → 0.5

Time: 6.4s

Precision: 64

Math

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

↓

\[\sqrt{\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)}\]

C

double f(double re, double im) {
        double r551853 = re;
        double r551854 = r551853 * r551853;
        double r551855 = im;
        double r551856 = r551855 * r551855;
        double r551857 = r551854 + r551856;
        double r551858 = sqrt(r551857);
        return r551858;
}

↓

double f(double re, double im) {
        double r551859 = re;
        double r551860 = r551859 * r551859;
        double r551861 = /*Error: no posit support in C */;
        double r551862 = im;
        double r551863 = /*Error: no posit support in C */;
        double r551864 = /*Error: no posit support in C */;
        double r551865 = sqrt(r551864);
        return r551865;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 0.6

   \[\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\]
2. Using strategy rm

3. Applied introduce-quire 0.6

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

4. Applied insert-quire-fdp-add 0.5

   \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)\right)}}\]

5. Final simplification 0.5

   \[\leadsto \sqrt{\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))