math.abs on complex

Average Error: 0.6 → 0.5

Time: 6.1s

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 r2061594 = re;
        double r2061595 = r2061594 * r2061594;
        double r2061596 = im;
        double r2061597 = r2061596 * r2061596;
        double r2061598 = r2061595 + r2061597;
        double r2061599 = sqrt(r2061598);
        return r2061599;
}

↓

double f(double re, double im) {
        double r2061600 = re;
        double r2061601 = r2061600 * r2061600;
        double r2061602 = /*Error: no posit support in C */;
        double r2061603 = im;
        double r2061604 = /*Error: no posit support in C */;
        double r2061605 = /*Error: no posit support in C */;
        double r2061606 = sqrt(r2061605);
        return r2061606;
}

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 2019144 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))