Average Error: 38.5 → 21.9
Time: 20.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.958741127435869792223497733767457453485 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -1.998473793179742598054252273926425808671 \cdot 10^{-109}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{2 \cdot im}{\sqrt{re \cdot re + im \cdot im} + re} \cdot im}\\ \mathbf{elif}\;re \le 3.824319967948722127918572797361195693664 \cdot 10^{-251}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 5.243516123083839465290070379456236260621 \cdot 10^{57}:\\ \;\;\;\;0.5 \cdot \left(\frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \frac{\left|im\right|}{\sqrt{re + re}}\right)\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.958741127435869792223497733767457453485 \cdot 10^{-42}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;re \le -1.998473793179742598054252273926425808671 \cdot 10^{-109}:\\
\;\;\;\;0.5 \cdot \sqrt{\frac{2 \cdot im}{\sqrt{re \cdot re + im \cdot im} + re} \cdot im}\\

\mathbf{elif}\;re \le 3.824319967948722127918572797361195693664 \cdot 10^{-251}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{elif}\;re \le 5.243516123083839465290070379456236260621 \cdot 10^{57}:\\
\;\;\;\;0.5 \cdot \left(\frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\right)\\

\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 r26023 = 0.5;
        double r26024 = 2.0;
        double r26025 = re;
        double r26026 = r26025 * r26025;
        double r26027 = im;
        double r26028 = r26027 * r26027;
        double r26029 = r26026 + r26028;
        double r26030 = sqrt(r26029);
        double r26031 = r26030 - r26025;
        double r26032 = r26024 * r26031;
        double r26033 = sqrt(r26032);
        double r26034 = r26023 * r26033;
        return r26034;
}
↓
double f(double re, double im) {
        double r26035 = re;
        double r26036 = -9.95874112743587e-42;
        bool r26037 = r26035 <= r26036;
        double r26038 = 0.5;
        double r26039 = 2.0;
        double r26040 = -2.0;
        double r26041 = r26040 * r26035;
        double r26042 = r26039 * r26041;
        double r26043 = sqrt(r26042);
        double r26044 = r26038 * r26043;
        double r26045 = -1.9984737931797426e-109;
        bool r26046 = r26035 <= r26045;
        double r26047 = im;
        double r26048 = r26039 * r26047;
        double r26049 = r26035 * r26035;
        double r26050 = r26047 * r26047;
        double r26051 = r26049 + r26050;
        double r26052 = sqrt(r26051);
        double r26053 = r26052 + r26035;
        double r26054 = r26048 / r26053;
        double r26055 = r26054 * r26047;
        double r26056 = sqrt(r26055);
        double r26057 = r26038 * r26056;
        double r26058 = 3.824319967948722e-251;
        bool r26059 = r26035 <= r26058;
        double r26060 = r26047 - r26035;
        double r26061 = r26039 * r26060;
        double r26062 = sqrt(r26061);
        double r26063 = r26038 * r26062;
        double r26064 = 5.2435161230838395e+57;
        bool r26065 = r26035 <= r26064;
        double r26066 = sqrt(r26039);
        double r26067 = sqrt(r26053);
        double r26068 = sqrt(r26067);
        double r26069 = r26066 / r26068;
        double r26070 = fabs(r26047);
        double r26071 = r26070 / r26068;
        double r26072 = r26069 * r26071;
        double r26073 = r26038 * r26072;
        double r26074 = r26035 + r26035;
        double r26075 = sqrt(r26074);
        double r26076 = r26070 / r26075;
        double r26077 = r26066 * r26076;
        double r26078 = r26038 * r26077;
        double r26079 = r26065 ? r26073 : r26078;
        double r26080 = r26059 ? r26063 : r26079;
        double r26081 = r26046 ? r26057 : r26080;
        double r26082 = r26037 ? r26044 : r26081;
        return r26082;
}

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 5 regimes

  2. if re < -9.95874112743587e-42

    1. Initial program 36.7

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

    2. Taylor expanded around -inf 16.7

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

    if -9.95874112743587e-42 < re < -1.9984737931797426e-109

    1. Initial program 16.2

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

    3. Applied flip--36.1

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

    4. Applied associate-*r/36.2

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

    5. Applied sqrt-div36.4

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

    6. Simplified36.4

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

    8. Applied sqrt-undiv36.2

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

    9. Simplified35.7

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

    if -1.9984737931797426e-109 < re < 3.824319967948722e-251

    1. Initial program 27.5

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

    2. Taylor expanded around 0 35.7

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

    if 3.824319967948722e-251 < re < 5.2435161230838395e+57

    1. Initial program 38.6

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

    3. Applied flip--38.5

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

    4. Applied associate-*r/38.5

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

    5. Applied sqrt-div38.6

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

    6. Simplified31.9

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

    8. Applied add-sqr-sqrt31.9

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

    9. Applied sqrt-prod32.0

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

    10. Applied sqrt-prod32.0

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

    11. Applied times-frac32.0

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

    12. Simplified20.6

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

    if 5.2435161230838395e+57 < re

    1. Initial program 58.7

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

    3. Applied flip--58.7

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

    4. Applied associate-*r/58.7

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

    5. Applied sqrt-div58.7

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

    6. Simplified40.9

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

    8. Applied *-un-lft-identity40.9

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

    9. Applied sqrt-prod40.9

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

    10. Applied sqrt-prod40.9

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

    11. Applied times-frac40.9

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

    12. Simplified40.9

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

    13. Simplified36.4

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

    14. Taylor expanded around inf 13.1

      \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \frac{\left|im\right|}{\sqrt{\color{blue}{re} + re}}\right)\]
  3. Recombined 5 regimes into one program.

  4. Final simplification21.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.958741127435869792223497733767457453485 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -1.998473793179742598054252273926425808671 \cdot 10^{-109}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{2 \cdot im}{\sqrt{re \cdot re + im \cdot im} + re} \cdot im}\\ \mathbf{elif}\;re \le 3.824319967948722127918572797361195693664 \cdot 10^{-251}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 5.243516123083839465290070379456236260621 \cdot 10^{57}:\\ \;\;\;\;0.5 \cdot \left(\frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \frac{\left|im\right|}{\sqrt{re + re}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)))))