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 r1127500 = re;
        double r1127501 = r1127500 * r1127500;
        double r1127502 = im;
        double r1127503 = r1127502 * r1127502;
        double r1127504 = r1127501 + r1127503;
        double r1127505 = sqrt(r1127504);
        return r1127505;
}

↓

double f(double re, double im) {
        double r1127506 = re;
        double r1127507 = r1127506 * r1127506;
        double r1127508 = /*Error: no posit support in C */;
        double r1127509 = im;
        double r1127510 = /*Error: no posit support in C */;
        double r1127511 = /*Error: no posit support in C */;
        double r1127512 = sqrt(r1127511);
        return r1127512;
}

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 2019164 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))