Average Error: 2.0 → 2.0
Time: 5.4s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
\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 r13978 = 0.5;
        double r13979 = /* ERROR: no posit support in C */;
        double r13980 = 2.0;
        double r13981 = /* ERROR: no posit support in C */;
        double r13982 = re;
        double r13983 = r13982 * r13982;
        double r13984 = im;
        double r13985 = r13984 * r13984;
        double r13986 = r13983 + r13985;
        double r13987 = sqrt(r13986);
        double r13988 = r13987 + r13982;
        double r13989 = r13981 * r13988;
        double r13990 = sqrt(r13989);
        double r13991 = r13979 * r13990;
        return r13991;
}
↓
double f(double re, double im) {
        double r13992 = 0.5;
        double r13993 = 2.0;
        double r13994 = re;
        double r13995 = r13994 * r13994;
        double r13996 = im;
        double r13997 = r13996 * r13996;
        double r13998 = r13995 + r13997;
        double r13999 = sqrt(r13998);
        double r14000 = r13999 + r13994;
        double r14001 = r13993 * r14000;
        double r14002 = sqrt(r14001);
        double r14003 = r13992 * r14002;
        return r14003;
}

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 2019120 
(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)))))