Average Error: 38.6 → 21.8
Time: 17.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.615973159790100816135112956922113830758 \cdot 10^{-204}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le 2.88234410399163811959624616338485820472 \cdot 10^{143}:\\ \;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \frac{\left|im\right|}{\sqrt{re + re}}\right)\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.615973159790100816135112956922113830758 \cdot 10^{-204}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;re \le 2.88234410399163811959624616338485820472 \cdot 10^{143}:\\
\;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \frac{\left|im\right|}{\sqrt{re + re}}\right)\\

\end{array}
double f(double re, double im) {
        double r25886 = 0.5;
        double r25887 = 2.0;
        double r25888 = re;
        double r25889 = r25888 * r25888;
        double r25890 = im;
        double r25891 = r25890 * r25890;
        double r25892 = r25889 + r25891;
        double r25893 = sqrt(r25892);
        double r25894 = r25893 - r25888;
        double r25895 = r25887 * r25894;
        double r25896 = sqrt(r25895);
        double r25897 = r25886 * r25896;
        return r25897;
}
↓
double f(double re, double im) {
        double r25898 = re;
        double r25899 = -8.615973159790101e-204;
        bool r25900 = r25898 <= r25899;
        double r25901 = 0.5;
        double r25902 = 2.0;
        double r25903 = -2.0;
        double r25904 = r25903 * r25898;
        double r25905 = r25902 * r25904;
        double r25906 = sqrt(r25905);
        double r25907 = r25901 * r25906;
        double r25908 = 2.882344103991638e+143;
        bool r25909 = r25898 <= r25908;
        double r25910 = im;
        double r25911 = fabs(r25910);
        double r25912 = sqrt(r25902);
        double r25913 = r25911 * r25912;
        double r25914 = r25898 * r25898;
        double r25915 = r25910 * r25910;
        double r25916 = r25914 + r25915;
        double r25917 = sqrt(r25916);
        double r25918 = r25917 + r25898;
        double r25919 = sqrt(r25918);
        double r25920 = r25913 / r25919;
        double r25921 = r25901 * r25920;
        double r25922 = r25898 + r25898;
        double r25923 = sqrt(r25922);
        double r25924 = r25911 / r25923;
        double r25925 = r25912 * r25924;
        double r25926 = r25901 * r25925;
        double r25927 = r25909 ? r25921 : r25926;
        double r25928 = r25900 ? r25907 : r25927;
        return r25928;
}

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 3 regimes

  2. if re < -8.615973159790101e-204

    1. Initial program 31.9

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

    2. Taylor expanded around -inf 25.2

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

    if -8.615973159790101e-204 < re < 2.882344103991638e+143

    1. Initial program 37.5

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

    3. Applied flip--37.6

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

    4. Applied associate-*r/37.6

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

    5. Applied sqrt-div37.9

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

    6. Simplified29.5

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity29.5

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

    9. Applied sqrt-prod29.5

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

    10. Applied sqrt-prod29.6

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{im \cdot im}}}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    11. Applied times-frac29.6

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

    12. Simplified29.6

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

    13. Simplified22.4

      \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \color{blue}{\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\right)\]
    14. Using strategy rm

    15. Applied associate-*r/22.4

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

    16. Simplified22.4

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

    if 2.882344103991638e+143 < re

    1. Initial program 63.1

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

    3. Applied flip--63.1

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

    4. Applied associate-*r/63.1

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

    5. Applied sqrt-div63.1

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

    6. Simplified49.3

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity49.3

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

    9. Applied sqrt-prod49.3

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

    10. Applied sqrt-prod49.3

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{im \cdot im}}}{\sqrt{1} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    11. Applied times-frac49.3

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

    12. Simplified49.3

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

    13. Simplified48.2

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

    14. Taylor expanded around inf 9.0

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

  4. Final simplification21.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.615973159790100816135112956922113830758 \cdot 10^{-204}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le 2.88234410399163811959624616338485820472 \cdot 10^{143}:\\ \;\;\;\;0.5 \cdot \frac{\left|im\right| \cdot \sqrt{2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \frac{\left|im\right|}{\sqrt{re + re}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))))