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 r453600 = re;
        double r453601 = r453600 * r453600;
        double r453602 = im;
        double r453603 = r453602 * r453602;
        double r453604 = r453601 + r453603;
        double r453605 = sqrt(r453604);
        return r453605;
}

↓

double f(double re, double im) {
        double r453606 = re;
        double r453607 = r453606 * r453606;
        double r453608 = /*Error: no posit support in C */;
        double r453609 = im;
        double r453610 = /*Error: no posit support in C */;
        double r453611 = /*Error: no posit support in C */;
        double r453612 = sqrt(r453611);
        return r453612;
}

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 2019154 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))