math.abs on complex

Average Error: 0.6 → 0.5

Time: 5.3s

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 r1142019 = re;
        double r1142020 = r1142019 * r1142019;
        double r1142021 = im;
        double r1142022 = r1142021 * r1142021;
        double r1142023 = r1142020 + r1142022;
        double r1142024 = sqrt(r1142023);
        return r1142024;
}

↓

double f(double re, double im) {
        double r1142025 = re;
        double r1142026 = r1142025 * r1142025;
        double r1142027 = /*Error: no posit support in C */;
        double r1142028 = im;
        double r1142029 = /*Error: no posit support in C */;
        double r1142030 = /*Error: no posit support in C */;
        double r1142031 = sqrt(r1142030);
        return r1142031;
}

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 2019158 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))