Average Error: 2.1 → 2.2
Time: 18.4s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \sqrt{re \cdot re + im \cdot im} + 2.0 \cdot \left(-re\right)}\]
\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 \sqrt{re \cdot re + im \cdot im} + 2.0 \cdot \left(-re\right)}
double f(double re, double im) {
        double r852784 = 0.5;
        double r852785 = /* ERROR: no posit support in C */;
        double r852786 = 2.0;
        double r852787 = /* ERROR: no posit support in C */;
        double r852788 = re;
        double r852789 = r852788 * r852788;
        double r852790 = im;
        double r852791 = r852790 * r852790;
        double r852792 = r852789 + r852791;
        double r852793 = sqrt(r852792);
        double r852794 = r852793 - r852788;
        double r852795 = r852787 * r852794;
        double r852796 = sqrt(r852795);
        double r852797 = r852785 * r852796;
        return r852797;
}
↓
double f(double re, double im) {
        double r852798 = 0.5;
        double r852799 = 2.0;
        double r852800 = re;
        double r852801 = r852800 * r852800;
        double r852802 = im;
        double r852803 = r852802 * r852802;
        double r852804 = r852801 + r852803;
        double r852805 = sqrt(r852804);
        double r852806 = r852799 * r852805;
        double r852807 = -r852800;
        double r852808 = r852799 * r852807;
        double r852809 = r852806 + r852808;
        double r852810 = sqrt(r852809);
        double r852811 = r852798 * r852810;
        return r852811;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 2.1

    \[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
  2. Using strategy rm

  3. Applied sub-neg2.1

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

  4. Applied distribute-lft-in2.2

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

  5. Final simplification2.2

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

Reproduce

herbie shell --seed 2019104 
(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)))))