math.abs on complex

Average Error: 0.6 → 0.5

Time: 8.6s

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 r397210 = re;
        double r397211 = r397210 * r397210;
        double r397212 = im;
        double r397213 = r397212 * r397212;
        double r397214 = r397211 + r397213;
        double r397215 = sqrt(r397214);
        return r397215;
}

↓

double f(double re, double im) {
        double r397216 = re;
        double r397217 = r397216 * r397216;
        double r397218 = /*Error: no posit support in C */;
        double r397219 = im;
        double r397220 = /*Error: no posit support in C */;
        double r397221 = /*Error: no posit support in C */;
        double r397222 = sqrt(r397221);
        return r397222;
}

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