Average Error: 2.1 → 2.1
Time: 5.5s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
double f(double re, double im) {
        double r13986 = 0.5;
        double r13987 = 2.0;
        double r13988 = re;
        double r13989 = r13988 * r13988;
        double r13990 = im;
        double r13991 = r13990 * r13990;
        double r13992 = r13989 + r13991;
        double r13993 = sqrt(r13992);
        double r13994 = r13993 + r13988;
        double r13995 = r13987 * r13994;
        double r13996 = sqrt(r13995);
        double r13997 = r13986 * r13996;
        return r13997;
}
↓
double f(double re, double im) {
        double r13998 = 0.5;
        double r13999 = 2.0;
        double r14000 = re;
        double r14001 = r14000 * r14000;
        double r14002 = im;
        double r14003 = r14002 * r14002;
        double r14004 = r14001 + r14003;
        double r14005 = sqrt(r14004);
        double r14006 = r14005 + r14000;
        double r14007 = r13999 * r14006;
        double r14008 = sqrt(r14007);
        double r14009 = r13998 * r14008;
        return r14009;
}


0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}

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 2019102 +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)))))