Average Error: 38.3 → 11.2
Time: 3.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 7.84862278851990408 \cdot 10^{68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 7.84862278851990408 \cdot 10^{68}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\

\end{array}
double f(double re, double im) {
        double r20911 = 0.5;
        double r20912 = 2.0;
        double r20913 = re;
        double r20914 = r20913 * r20913;
        double r20915 = im;
        double r20916 = r20915 * r20915;
        double r20917 = r20914 + r20916;
        double r20918 = sqrt(r20917);
        double r20919 = r20918 - r20913;
        double r20920 = r20912 * r20919;
        double r20921 = sqrt(r20920);
        double r20922 = r20911 * r20921;
        return r20922;
}
↓
double f(double re, double im) {
        double r20923 = re;
        double r20924 = 7.848622788519904e+68;
        bool r20925 = r20923 <= r20924;
        double r20926 = 0.5;
        double r20927 = 2.0;
        double r20928 = im;
        double r20929 = hypot(r20923, r20928);
        double r20930 = r20929 - r20923;
        double r20931 = 0.0;
        double r20932 = r20930 + r20931;
        double r20933 = r20927 * r20932;
        double r20934 = sqrt(r20933);
        double r20935 = r20926 * r20934;
        double r20936 = 2.0;
        double r20937 = pow(r20928, r20936);
        double r20938 = r20937 + r20931;
        double r20939 = r20923 + r20929;
        double r20940 = r20938 / r20939;
        double r20941 = r20927 * r20940;
        double r20942 = sqrt(r20941);
        double r20943 = r20926 * r20942;
        double r20944 = r20925 ? r20935 : r20943;
        return r20944;
}

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 2 regimes

  2. if re < 7.848622788519904e+68

    1. Initial program 33.3

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

    3. Applied add-cube-cbrt33.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - \color{blue}{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}}\right)}\]

    4. Applied add-sqr-sqrt33.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}}} - \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}\right)}\]

    5. Applied sqrt-prod33.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} - \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}\right)}\]

    6. Applied prod-diff33.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{\sqrt{re \cdot re + im \cdot im}}, \sqrt{\sqrt{re \cdot re + im \cdot im}}, -\sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{re}, \sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)\right)\right)}}\]

    7. Simplified6.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\mathsf{hypot}\left(re, im\right) - re\right)} + \mathsf{fma}\left(-\sqrt[3]{re}, \sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re} \cdot \left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right)\right)\right)}\]

    8. Simplified6.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + \color{blue}{0}\right)}\]

    if 7.848622788519904e+68 < re

    1. Initial program 59.6

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

    3. Applied flip--59.6

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

    4. Simplified43.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2} + 0}}{\sqrt{re \cdot re + im \cdot im} + re}}\]

    5. Simplified30.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{\color{blue}{re + \mathsf{hypot}\left(re, im\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 7.84862278851990408 \cdot 10^{68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(re, im\right) - re\right) + 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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)))))