Average Error: 2.0 → 2.0
Time: 21.2s
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 r733090 = 0.5;
        double r733091 = /* ERROR: no posit support in C */;
        double r733092 = 2.0;
        double r733093 = /* ERROR: no posit support in C */;
        double r733094 = re;
        double r733095 = r733094 * r733094;
        double r733096 = im;
        double r733097 = r733096 * r733096;
        double r733098 = r733095 + r733097;
        double r733099 = sqrt(r733098);
        double r733100 = r733099 - r733094;
        double r733101 = r733093 * r733100;
        double r733102 = sqrt(r733101);
        double r733103 = r733091 * r733102;
        return r733103;
}
↓
double f(double re, double im) {
        double r733104 = 0.5;
        double r733105 = 2.0;
        double r733106 = re;
        double r733107 = r733106 * r733106;
        double r733108 = im;
        double r733109 = r733108 * r733108;
        double r733110 = r733107 + r733109;
        double r733111 = sqrt(r733110);
        double r733112 = r733111 - r733106;
        double r733113 = r733105 * r733112;
        double r733114 = sqrt(r733113);
        double r733115 = r733104 * r733114;
        return r733115;
}

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)))))