Average Error: 2.0 → 0.8
Time: 8.8s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -0.014678955078125:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \end{array}\]
\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)
↓
\begin{array}{l}
\mathbf{if}\;re \le -0.014678955078125:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\

\end{array}
double f(double re, double im) {
        double r1018771 = 0.5;
        double r1018772 = /* ERROR: no posit support in C */;
        double r1018773 = 2.0;
        double r1018774 = /* ERROR: no posit support in C */;
        double r1018775 = re;
        double r1018776 = r1018775 * r1018775;
        double r1018777 = im;
        double r1018778 = r1018777 * r1018777;
        double r1018779 = r1018776 + r1018778;
        double r1018780 = sqrt(r1018779);
        double r1018781 = r1018780 - r1018775;
        double r1018782 = r1018774 * r1018781;
        double r1018783 = sqrt(r1018782);
        double r1018784 = r1018772 * r1018783;
        return r1018784;
}
↓
double f(double re, double im) {
        double r1018785 = re;
        double r1018786 = -0.014678955078125;
        bool r1018787 = r1018785 <= r1018786;
        double r1018788 = 0.5;
        double r1018789 = 2.0;
        double r1018790 = r1018785 * r1018785;
        double r1018791 = im;
        double r1018792 = r1018791 * r1018791;
        double r1018793 = r1018790 + r1018792;
        double r1018794 = sqrt(r1018793);
        double r1018795 = r1018794 - r1018785;
        double r1018796 = r1018789 * r1018795;
        double r1018797 = sqrt(r1018796);
        double r1018798 = r1018788 * r1018797;
        double r1018799 = r1018794 + r1018785;
        double r1018800 = r1018799 / r1018791;
        double r1018801 = r1018791 / r1018800;
        double r1018802 = r1018789 * r1018801;
        double r1018803 = sqrt(r1018802);
        double r1018804 = r1018788 * r1018803;
        double r1018805 = r1018787 ? r1018798 : r1018804;
        return r1018805;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 2 regimes

  2. if re < -0.014678955078125

    1. Initial program 0.7

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

    if -0.014678955078125 < re

    1. Initial program 3.3

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

    3. Applied p16-flip--3.0

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

    4. Simplified0.9

      \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\color{blue}{\left(im \cdot im\right)}}{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}\right)\right)}\right)\]
    5. Using strategy rm

    6. Applied associate-/l*0.8

      \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \color{blue}{\left(\frac{im}{\left(\frac{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}{im}\right)}\right)}\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -0.014678955078125:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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)))))