Average Error: 2.1 → 2.1
Time: 5.7s
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 r13994 = 0.5;
        double r13995 = 2.0;
        double r13996 = re;
        double r13997 = r13996 * r13996;
        double r13998 = im;
        double r13999 = r13998 * r13998;
        double r14000 = r13997 + r13999;
        double r14001 = sqrt(r14000);
        double r14002 = r14001 + r13996;
        double r14003 = r13995 * r14002;
        double r14004 = sqrt(r14003);
        double r14005 = r13994 * r14004;
        return r14005;
}
↓
double f(double re, double im) {
        double r14006 = 0.5;
        double r14007 = 2.0;
        double r14008 = re;
        double r14009 = r14008 * r14008;
        double r14010 = im;
        double r14011 = r14010 * r14010;
        double r14012 = r14009 + r14011;
        double r14013 = sqrt(r14012);
        double r14014 = r14013 + r14008;
        double r14015 = r14007 * r14014;
        double r14016 = sqrt(r14015);
        double r14017 = r14006 * r14016;
        return r14017;
}


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 2019101 +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)))))