math.abs on complex

Average Error: 0.6 → 0.5

Time: 5.4s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r1124767 = re;
        double r1124768 = r1124767 * r1124767;
        double r1124769 = im;
        double r1124770 = r1124769 * r1124769;
        double r1124771 = r1124768 + r1124770;
        double r1124772 = sqrt(r1124771);
        return r1124772;
}

↓

double f(double re, double im) {
        double r1124773 = re;
        double r1124774 = r1124773 * r1124773;
        double r1124775 = /*Error: no posit support in C */;
        double r1124776 = im;
        double r1124777 = /*Error: no posit support in C */;
        double r1124778 = /*Error: no posit support in C */;
        double r1124779 = sqrt(r1124778);
        return r1124779;
}

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

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))