Average Error: 38.9 → 27.1
Time: 4.5s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.75406070697555614550103171070689226775 \cdot 10^{99}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -6.851668065765393813815957926512377199889 \cdot 10^{-264}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 2.821269380473622723862824940269700262969 \cdot 10^{-218}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.75406070697555614550103171070689226775 \cdot 10^{99}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\

\mathbf{elif}\;re \le -6.851668065765393813815957926512377199889 \cdot 10^{-264}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\

\mathbf{elif}\;re \le 2.821269380473622723862824940269700262969 \cdot 10^{-218}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\

\end{array}
double f(double re, double im) {
        double r17970 = 0.5;
        double r17971 = 2.0;
        double r17972 = re;
        double r17973 = r17972 * r17972;
        double r17974 = im;
        double r17975 = r17974 * r17974;
        double r17976 = r17973 + r17975;
        double r17977 = sqrt(r17976);
        double r17978 = r17977 - r17972;
        double r17979 = r17971 * r17978;
        double r17980 = sqrt(r17979);
        double r17981 = r17970 * r17980;
        return r17981;
}
↓
double f(double re, double im) {
        double r17982 = re;
        double r17983 = -6.754060706975556e+99;
        bool r17984 = r17982 <= r17983;
        double r17985 = 0.5;
        double r17986 = 2.0;
        double r17987 = -1.0;
        double r17988 = r17987 * r17982;
        double r17989 = r17988 - r17982;
        double r17990 = r17986 * r17989;
        double r17991 = sqrt(r17990);
        double r17992 = r17985 * r17991;
        double r17993 = -6.851668065765394e-264;
        bool r17994 = r17982 <= r17993;
        double r17995 = r17982 * r17982;
        double r17996 = im;
        double r17997 = r17996 * r17996;
        double r17998 = r17995 + r17997;
        double r17999 = sqrt(r17998);
        double r18000 = sqrt(r17999);
        double r18001 = r18000 * r18000;
        double r18002 = r18001 - r17982;
        double r18003 = r17986 * r18002;
        double r18004 = sqrt(r18003);
        double r18005 = r17985 * r18004;
        double r18006 = 2.8212693804736227e-218;
        bool r18007 = r17982 <= r18006;
        double r18008 = r17996 - r17982;
        double r18009 = r17986 * r18008;
        double r18010 = sqrt(r18009);
        double r18011 = r17985 * r18010;
        double r18012 = 2.0;
        double r18013 = pow(r17996, r18012);
        double r18014 = r17999 + r17982;
        double r18015 = r18013 / r18014;
        double r18016 = r17986 * r18015;
        double r18017 = sqrt(r18016);
        double r18018 = r17985 * r18017;
        double r18019 = r18007 ? r18011 : r18018;
        double r18020 = r17994 ? r18005 : r18019;
        double r18021 = r17984 ? r17992 : r18020;
        return r18021;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if re < -6.754060706975556e+99

    1. Initial program 50.7

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

    2. Taylor expanded around -inf 10.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{-1 \cdot re} - re\right)}\]

    if -6.754060706975556e+99 < re < -6.851668065765394e-264

    1. Initial program 19.1

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

    3. Applied add-sqr-sqrt19.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}}} - re\right)}\]

    4. Applied sqrt-prod19.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\]

    if -6.851668065765394e-264 < re < 2.8212693804736227e-218

    1. Initial program 31.4

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

    2. Taylor expanded around 0 33.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)}\]

    if 2.8212693804736227e-218 < re

    1. Initial program 49.2

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

    3. Applied flip--49.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    4. Simplified37.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification27.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.75406070697555614550103171070689226775 \cdot 10^{99}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -6.851668065765393813815957926512377199889 \cdot 10^{-264}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 2.821269380473622723862824940269700262969 \cdot 10^{-218}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))