Average Error: 2.1 → 2.1
Time: 5.7s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
\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)
↓
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
double f(double re, double im) {
        double r13831 = 0.5;
        double r13832 = /* ERROR: no posit support in C */;
        double r13833 = 2.0;
        double r13834 = /* ERROR: no posit support in C */;
        double r13835 = re;
        double r13836 = r13835 * r13835;
        double r13837 = im;
        double r13838 = r13837 * r13837;
        double r13839 = r13836 + r13838;
        double r13840 = sqrt(r13839);
        double r13841 = r13840 + r13835;
        double r13842 = r13834 * r13841;
        double r13843 = sqrt(r13842);
        double r13844 = r13832 * r13843;
        return r13844;
}
↓
double f(double re, double im) {
        double r13845 = 0.5;
        double r13846 = 2.0;
        double r13847 = re;
        double r13848 = r13847 * r13847;
        double r13849 = im;
        double r13850 = r13849 * r13849;
        double r13851 = r13848 + r13850;
        double r13852 = sqrt(r13851);
        double r13853 = r13852 + r13847;
        double r13854 = r13846 * r13853;
        double r13855 = sqrt(r13854);
        double r13856 = r13845 * r13855;
        return r13856;
}

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. Final simplification2.1

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

Reproduce

herbie shell --seed 2019104 
(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)))))