Average Error: 2.1 → 2.1
Time: 38.8s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)} - re\right)}\]
\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)
↓
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)} - re\right)}
double f(double re, double im) {
        double r1508616 = 0.5;
        double r1508617 = /* ERROR: no posit support in C */;
        double r1508618 = 2.0;
        double r1508619 = /* ERROR: no posit support in C */;
        double r1508620 = re;
        double r1508621 = r1508620 * r1508620;
        double r1508622 = im;
        double r1508623 = r1508622 * r1508622;
        double r1508624 = r1508621 + r1508623;
        double r1508625 = sqrt(r1508624);
        double r1508626 = r1508625 - r1508620;
        double r1508627 = r1508619 * r1508626;
        double r1508628 = sqrt(r1508627);
        double r1508629 = r1508617 * r1508628;
        return r1508629;
}
↓
double f(double re, double im) {
        double r1508630 = 0.5;
        double r1508631 = 2.0;
        double r1508632 = re;
        double r1508633 = r1508632 * r1508632;
        double r1508634 = /*Error: no posit support in C */;
        double r1508635 = im;
        double r1508636 = /*Error: no posit support in C */;
        double r1508637 = /*Error: no posit support in C */;
        double r1508638 = sqrt(r1508637);
        double r1508639 = r1508638 - r1508632;
        double r1508640 = r1508631 * r1508639;
        double r1508641 = sqrt(r1508640);
        double r1508642 = r1508630 * r1508641;
        return r1508642;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.1

    \[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
  2. Using strategy rm

  3. Applied introduce-quire2.1

    \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\color{blue}{\left(\left(\left(re \cdot re\right)\right)\right)}}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]

  4. Applied insert-quire-fdp-add2.1

    \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\color{blue}{\left(\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)\right)}}\right) - re\right)\right)}\right)\]

  5. Final simplification2.1

    \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)} - re\right)}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (*.p16 (real->posit16 0.5) (sqrt.p16 (*.p16 (real->posit16 2.0) (-.p16 (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))) re)))))