Average Error: 2.1 → 2.1
Time: 7.4s
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 r15025 = 0.5;
        double r15026 = /* ERROR: no posit support in C */;
        double r15027 = 2.0;
        double r15028 = /* ERROR: no posit support in C */;
        double r15029 = re;
        double r15030 = r15029 * r15029;
        double r15031 = im;
        double r15032 = r15031 * r15031;
        double r15033 = r15030 + r15032;
        double r15034 = sqrt(r15033);
        double r15035 = r15034 + r15029;
        double r15036 = r15028 * r15035;
        double r15037 = sqrt(r15036);
        double r15038 = r15026 * r15037;
        return r15038;
}
↓
double f(double re, double im) {
        double r15039 = 0.5;
        double r15040 = 2.0;
        double r15041 = re;
        double r15042 = r15041 * r15041;
        double r15043 = im;
        double r15044 = r15043 * r15043;
        double r15045 = r15042 + r15044;
        double r15046 = sqrt(r15045);
        double r15047 = r15046 + r15041;
        double r15048 = r15040 * r15047;
        double r15049 = sqrt(r15048);
        double r15050 = r15039 * r15049;
        return r15050;
}

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 2019107 
(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)))))