Average Error: 2.0 → 2.0
Time: 6.0s
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 r13811 = 0.5;
        double r13812 = /* ERROR: no posit support in C */;
        double r13813 = 2.0;
        double r13814 = /* ERROR: no posit support in C */;
        double r13815 = re;
        double r13816 = r13815 * r13815;
        double r13817 = im;
        double r13818 = r13817 * r13817;
        double r13819 = r13816 + r13818;
        double r13820 = sqrt(r13819);
        double r13821 = r13820 + r13815;
        double r13822 = r13814 * r13821;
        double r13823 = sqrt(r13822);
        double r13824 = r13812 * r13823;
        return r13824;
}
↓
double f(double re, double im) {
        double r13825 = 0.5;
        double r13826 = 2.0;
        double r13827 = re;
        double r13828 = r13827 * r13827;
        double r13829 = im;
        double r13830 = r13829 * r13829;
        double r13831 = r13828 + r13830;
        double r13832 = sqrt(r13831);
        double r13833 = r13832 + r13827;
        double r13834 = r13826 * r13833;
        double r13835 = sqrt(r13834);
        double r13836 = r13825 * r13835;
        return r13836;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.0

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

  2. Final simplification2.0

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

Reproduce

herbie shell --seed 2019119 
(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)))))