Average Error: 2.1 → 2.1
Time: 34.5s
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 r2014579 = 0.5;
        double r2014580 = /* ERROR: no posit support in C */;
        double r2014581 = 2.0;
        double r2014582 = /* ERROR: no posit support in C */;
        double r2014583 = re;
        double r2014584 = r2014583 * r2014583;
        double r2014585 = im;
        double r2014586 = r2014585 * r2014585;
        double r2014587 = r2014584 + r2014586;
        double r2014588 = sqrt(r2014587);
        double r2014589 = r2014588 - r2014583;
        double r2014590 = r2014582 * r2014589;
        double r2014591 = sqrt(r2014590);
        double r2014592 = r2014580 * r2014591;
        return r2014592;
}
↓
double f(double re, double im) {
        double r2014593 = 0.5;
        double r2014594 = 2.0;
        double r2014595 = re;
        double r2014596 = r2014595 * r2014595;
        double r2014597 = /*Error: no posit support in C */;
        double r2014598 = im;
        double r2014599 = /*Error: no posit support in C */;
        double r2014600 = /*Error: no posit support in C */;
        double r2014601 = sqrt(r2014600);
        double r2014602 = r2014601 - r2014595;
        double r2014603 = r2014594 * r2014602;
        double r2014604 = sqrt(r2014603);
        double r2014605 = r2014593 * r2014604;
        return r2014605;
}

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 +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)))))