Average Error: 38.9 → 27.1
Time: 4.5s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.75406070697555614550103171070689226775 \cdot 10^{99}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -6.851668065765393813815957926512377199889 \cdot 10^{-264}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 2.821269380473622723862824940269700262969 \cdot 10^{-218}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]
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(-2 \cdot 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 r17942 = 0.5;
        double r17943 = 2.0;
        double r17944 = re;
        double r17945 = r17944 * r17944;
        double r17946 = im;
        double r17947 = r17946 * r17946;
        double r17948 = r17945 + r17947;
        double r17949 = sqrt(r17948);
        double r17950 = r17949 - r17944;
        double r17951 = r17943 * r17950;
        double r17952 = sqrt(r17951);
        double r17953 = r17942 * r17952;
        return r17953;
}
↓
double f(double re, double im) {
        double r17954 = re;
        double r17955 = -6.754060706975556e+99;
        bool r17956 = r17954 <= r17955;
        double r17957 = 0.5;
        double r17958 = 2.0;
        double r17959 = -2.0;
        double r17960 = r17959 * r17954;
        double r17961 = r17958 * r17960;
        double r17962 = sqrt(r17961);
        double r17963 = r17957 * r17962;
        double r17964 = -6.851668065765394e-264;
        bool r17965 = r17954 <= r17964;
        double r17966 = r17954 * r17954;
        double r17967 = im;
        double r17968 = r17967 * r17967;
        double r17969 = r17966 + r17968;
        double r17970 = sqrt(r17969);
        double r17971 = sqrt(r17970);
        double r17972 = r17971 * r17971;
        double r17973 = r17972 - r17954;
        double r17974 = r17958 * r17973;
        double r17975 = sqrt(r17974);
        double r17976 = r17957 * r17975;
        double r17977 = 2.8212693804736227e-218;
        bool r17978 = r17954 <= r17977;
        double r17979 = r17967 - r17954;
        double r17980 = r17958 * r17979;
        double r17981 = sqrt(r17980);
        double r17982 = r17957 * r17981;
        double r17983 = 2.0;
        double r17984 = pow(r17967, r17983);
        double r17985 = r17970 + r17954;
        double r17986 = r17984 / r17985;
        double r17987 = r17958 * r17986;
        double r17988 = sqrt(r17987);
        double r17989 = r17957 * r17988;
        double r17990 = r17978 ? r17982 : r17989;
        double r17991 = r17965 ? r17976 : r17990;
        double r17992 = r17956 ? r17963 : r17991;
        return r17992;
}

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

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

    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

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

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)))))