Average Error: 2.1 → 2.1
Time: 32.1s
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 r667673 = 0.5;
        double r667674 = /* ERROR: no posit support in C */;
        double r667675 = 2.0;
        double r667676 = /* ERROR: no posit support in C */;
        double r667677 = re;
        double r667678 = r667677 * r667677;
        double r667679 = im;
        double r667680 = r667679 * r667679;
        double r667681 = r667678 + r667680;
        double r667682 = sqrt(r667681);
        double r667683 = r667682 - r667677;
        double r667684 = r667676 * r667683;
        double r667685 = sqrt(r667684);
        double r667686 = r667674 * r667685;
        return r667686;
}
↓
double f(double re, double im) {
        double r667687 = 0.5;
        double r667688 = 2.0;
        double r667689 = re;
        double r667690 = r667689 * r667689;
        double r667691 = /*Error: no posit support in C */;
        double r667692 = im;
        double r667693 = /*Error: no posit support in C */;
        double r667694 = /*Error: no posit support in C */;
        double r667695 = sqrt(r667694);
        double r667696 = r667695 - r667689;
        double r667697 = r667688 * r667696;
        double r667698 = sqrt(r667697);
        double r667699 = r667687 * r667698;
        return r667699;
}

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 2019152 +o rules:numerics
(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)))))