math.abs on complex

Average Error: 0.6 → 0.5

Time: 5.1s

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 r3166576 = re;
        double r3166577 = r3166576 * r3166576;
        double r3166578 = im;
        double r3166579 = r3166578 * r3166578;
        double r3166580 = r3166577 + r3166579;
        double r3166581 = sqrt(r3166580);
        return r3166581;
}

↓

double f(double re, double im) {
        double r3166582 = re;
        double r3166583 = r3166582 * r3166582;
        double r3166584 = /*Error: no posit support in C */;
        double r3166585 = im;
        double r3166586 = /*Error: no posit support in C */;
        double r3166587 = /*Error: no posit support in C */;
        double r3166588 = sqrt(r3166587);
        return r3166588;
}

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 2019149 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))