Average Error: 2.1 → 2.1
Time: 37.1s
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 r294688 = 0.5;
        double r294689 = /* ERROR: no posit support in C */;
        double r294690 = 2.0;
        double r294691 = /* ERROR: no posit support in C */;
        double r294692 = re;
        double r294693 = r294692 * r294692;
        double r294694 = im;
        double r294695 = r294694 * r294694;
        double r294696 = r294693 + r294695;
        double r294697 = sqrt(r294696);
        double r294698 = r294697 + r294692;
        double r294699 = r294691 * r294698;
        double r294700 = sqrt(r294699);
        double r294701 = r294689 * r294700;
        return r294701;
}
↓
double f(double re, double im) {
        double r294702 = 0.5;
        double r294703 = /* ERROR: no posit support in C */;
        double r294704 = 2.0;
        double r294705 = /* ERROR: no posit support in C */;
        double r294706 = re;
        double r294707 = r294706 * r294706;
        double r294708 = /*Error: no posit support in C */;
        double r294709 = im;
        double r294710 = /*Error: no posit support in C */;
        double r294711 = /*Error: no posit support in C */;
        double r294712 = sqrt(r294711);
        double r294713 = r294712 + r294706;
        double r294714 = r294705 * r294713;
        double r294715 = sqrt(r294714);
        double r294716 = r294703 * r294715;
        return r294716;
}

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-add2.1

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

  5. Final simplification2.1

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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)))))