Average Error: 2.1 → 2.1
Time: 18.9s
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 r575670 = 0.5;
        double r575671 = /* ERROR: no posit support in C */;
        double r575672 = 2.0;
        double r575673 = /* ERROR: no posit support in C */;
        double r575674 = re;
        double r575675 = r575674 * r575674;
        double r575676 = im;
        double r575677 = r575676 * r575676;
        double r575678 = r575675 + r575677;
        double r575679 = sqrt(r575678);
        double r575680 = r575679 - r575674;
        double r575681 = r575673 * r575680;
        double r575682 = sqrt(r575681);
        double r575683 = r575671 * r575682;
        return r575683;
}
↓
double f(double re, double im) {
        double r575684 = 0.5;
        double r575685 = 2.0;
        double r575686 = re;
        double r575687 = r575686 * r575686;
        double r575688 = /*Error: no posit support in C */;
        double r575689 = im;
        double r575690 = /*Error: no posit support in C */;
        double r575691 = /*Error: no posit support in C */;
        double r575692 = sqrt(r575691);
        double r575693 = r575692 - r575686;
        double r575694 = r575685 * r575693;
        double r575695 = sqrt(r575694);
        double r575696 = r575684 * r575695;
        return r575696;
}

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 2019156 
(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)))))