math.abs on complex

Average Error: 0.6 → 0.5

Time: 5.0s

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 r2581986 = re;
        double r2581987 = r2581986 * r2581986;
        double r2581988 = im;
        double r2581989 = r2581988 * r2581988;
        double r2581990 = r2581987 + r2581989;
        double r2581991 = sqrt(r2581990);
        return r2581991;
}

↓

double f(double re, double im) {
        double r2581992 = re;
        double r2581993 = r2581992 * r2581992;
        double r2581994 = /*Error: no posit support in C */;
        double r2581995 = im;
        double r2581996 = /*Error: no posit support in C */;
        double r2581997 = /*Error: no posit support in C */;
        double r2581998 = sqrt(r2581997);
        return r2581998;
}

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 2019142 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))