Average Error: 38.5 → 21.9
Time: 19.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.958741127435869792223497733767457453485 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -1.998473793179742598054252273926425808671 \cdot 10^{-109}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{2 \cdot im}{\sqrt{re \cdot re + im \cdot im} + re} \cdot im}\\ \mathbf{elif}\;re \le 3.824319967948722127918572797361195693664 \cdot 10^{-251}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 5.243516123083839465290070379456236260621 \cdot 10^{57}:\\ \;\;\;\;0.5 \cdot \left(\frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \frac{\left|im\right|}{\sqrt{re + re}}\right)\\ \end{array}\]
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 r26089 = 0.5;
        double r26090 = 2.0;
        double r26091 = re;
        double r26092 = r26091 * r26091;
        double r26093 = im;
        double r26094 = r26093 * r26093;
        double r26095 = r26092 + r26094;
        double r26096 = sqrt(r26095);
        double r26097 = r26096 - r26091;
        double r26098 = r26090 * r26097;
        double r26099 = sqrt(r26098);
        double r26100 = r26089 * r26099;
        return r26100;
}
↓
double f(double re, double im) {
        double r26101 = re;
        double r26102 = -9.95874112743587e-42;
        bool r26103 = r26101 <= r26102;
        double r26104 = 0.5;
        double r26105 = 2.0;
        double r26106 = -2.0;
        double r26107 = r26106 * r26101;
        double r26108 = r26105 * r26107;
        double r26109 = sqrt(r26108);
        double r26110 = r26104 * r26109;
        double r26111 = -1.9984737931797426e-109;
        bool r26112 = r26101 <= r26111;
        double r26113 = im;
        double r26114 = r26105 * r26113;
        double r26115 = r26101 * r26101;
        double r26116 = r26113 * r26113;
        double r26117 = r26115 + r26116;
        double r26118 = sqrt(r26117);
        double r26119 = r26118 + r26101;
        double r26120 = r26114 / r26119;
        double r26121 = r26120 * r26113;
        double r26122 = sqrt(r26121);
        double r26123 = r26104 * r26122;
        double r26124 = 3.824319967948722e-251;
        bool r26125 = r26101 <= r26124;
        double r26126 = r26113 - r26101;
        double r26127 = r26105 * r26126;
        double r26128 = sqrt(r26127);
        double r26129 = r26104 * r26128;
        double r26130 = 5.2435161230838395e+57;
        bool r26131 = r26101 <= r26130;
        double r26132 = sqrt(r26105);
        double r26133 = sqrt(r26119);
        double r26134 = sqrt(r26133);
        double r26135 = r26132 / r26134;
        double r26136 = fabs(r26113);
        double r26137 = r26136 / r26134;
        double r26138 = r26135 * r26137;
        double r26139 = r26104 * r26138;
        double r26140 = r26101 + r26101;
        double r26141 = sqrt(r26140);
        double r26142 = r26136 / r26141;
        double r26143 = r26132 * r26142;
        double r26144 = r26104 * r26143;
        double r26145 = r26131 ? r26139 : r26144;
        double r26146 = r26125 ? r26129 : r26145;
        double r26147 = r26112 ? r26123 : r26146;
        double r26148 = r26103 ? r26110 : r26147;
        return r26148;
}

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)))))