Average Error: 2.0 → 2.0
Time: 20.7s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\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 r641024 = 0.5;
        double r641025 = /* ERROR: no posit support in C */;
        double r641026 = 2.0;
        double r641027 = /* ERROR: no posit support in C */;
        double r641028 = re;
        double r641029 = r641028 * r641028;
        double r641030 = im;
        double r641031 = r641030 * r641030;
        double r641032 = r641029 + r641031;
        double r641033 = sqrt(r641032);
        double r641034 = r641033 - r641028;
        double r641035 = r641027 * r641034;
        double r641036 = sqrt(r641035);
        double r641037 = r641025 * r641036;
        return r641037;
}
↓
double f(double re, double im) {
        double r641038 = 0.5;
        double r641039 = 2.0;
        double r641040 = re;
        double r641041 = r641040 * r641040;
        double r641042 = im;
        double r641043 = r641042 * r641042;
        double r641044 = r641041 + r641043;
        double r641045 = sqrt(r641044);
        double r641046 = r641045 - r641040;
        double r641047 = r641039 * r641046;
        double r641048 = sqrt(r641047);
        double r641049 = r641038 * r641048;
        return r641049;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.0

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

  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 2019153 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (*.p16 (real->posit16 0.5) (sqrt.p16 (*.p16 (real->posit16 2.0) (-.p16 (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))) re)))))