Average Error: 2.1 → 2.0
Time: 41.4s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\]
↓
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)\right)}\right)}{re}\right)\right)}\right)\]
\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)
↓
\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\left(\mathsf{qma}\left(\left(\left(re \cdot re\right)\right), im, im\right)\right)\right)}\right)}{re}\right)\right)}\right)
double f(double re, double im) {
        double r294671 = 0.5;
        double r294672 = /* ERROR: no posit support in C */;
        double r294673 = 2.0;
        double r294674 = /* ERROR: no posit support in C */;
        double r294675 = re;
        double r294676 = r294675 * r294675;
        double r294677 = im;
        double r294678 = r294677 * r294677;
        double r294679 = r294676 + r294678;
        double r294680 = sqrt(r294679);
        double r294681 = r294680 + r294675;
        double r294682 = r294674 * r294681;
        double r294683 = sqrt(r294682);
        double r294684 = r294672 * r294683;
        return r294684;
}
↓
double f(double re, double im) {
        double r294685 = 0.5;
        double r294686 = /* ERROR: no posit support in C */;
        double r294687 = 2.0;
        double r294688 = /* ERROR: no posit support in C */;
        double r294689 = re;
        double r294690 = r294689 * r294689;
        double r294691 = /*Error: no posit support in C */;
        double r294692 = im;
        double r294693 = /*Error: no posit support in C */;
        double r294694 = /*Error: no posit support in C */;
        double r294695 = sqrt(r294694);
        double r294696 = r294695 + r294689;
        double r294697 = r294688 * r294696;
        double r294698 = sqrt(r294697);
        double r294699 = r294686 * r294698;
        return r294699;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.1

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

  3. Applied introduce-quire2.1

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

  4. Applied insert-quire-fdp-add2.0

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

  5. Final simplification2.0

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

Reproduce

herbie shell --seed 2019164 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  (*.p16 (real->posit16 0.5) (sqrt.p16 (*.p16 (real->posit16 2.0) (+.p16 (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))) re)))))