Average Error: 37.4 → 16.8
Time: 21.0s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.534169287773102 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -1.7322803508780655 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} - re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 1.0730314248253915 \cdot 10^{-288}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 2.2094180720166524 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2.0}{\sqrt{im \cdot im + re \cdot re} + re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2.0}}{\sqrt{re + re}} \cdot \left|im\right|\right) \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.534169287773102 \cdot 10^{+132}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le -1.7322803508780655 \cdot 10^{-249}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} - re\right)} \cdot 0.5\\

\mathbf{elif}\;re \le 1.0730314248253915 \cdot 10^{-288}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im - re\right)}\\

\mathbf{elif}\;re \le 2.2094180720166524 \cdot 10^{+111}:\\
\;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2.0}{\sqrt{im \cdot im + re \cdot re} + re}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\sqrt{2.0}}{\sqrt{re + re}} \cdot \left|im\right|\right) \cdot 0.5\\

\end{array}
double f(double re, double im) {
        double r917138 = 0.5;
        double r917139 = 2.0;
        double r917140 = re;
        double r917141 = r917140 * r917140;
        double r917142 = im;
        double r917143 = r917142 * r917142;
        double r917144 = r917141 + r917143;
        double r917145 = sqrt(r917144);
        double r917146 = r917145 - r917140;
        double r917147 = r917139 * r917146;
        double r917148 = sqrt(r917147);
        double r917149 = r917138 * r917148;
        return r917149;
}
↓
double f(double re, double im) {
        double r917150 = re;
        double r917151 = -9.534169287773102e+132;
        bool r917152 = r917150 <= r917151;
        double r917153 = -2.0;
        double r917154 = r917153 * r917150;
        double r917155 = 2.0;
        double r917156 = r917154 * r917155;
        double r917157 = sqrt(r917156);
        double r917158 = 0.5;
        double r917159 = r917157 * r917158;
        double r917160 = -1.7322803508780655e-249;
        bool r917161 = r917150 <= r917160;
        double r917162 = im;
        double r917163 = r917162 * r917162;
        double r917164 = r917150 * r917150;
        double r917165 = r917163 + r917164;
        double r917166 = sqrt(r917165);
        double r917167 = sqrt(r917166);
        double r917168 = r917167 * r917167;
        double r917169 = r917168 - r917150;
        double r917170 = r917155 * r917169;
        double r917171 = sqrt(r917170);
        double r917172 = r917171 * r917158;
        double r917173 = 1.0730314248253915e-288;
        bool r917174 = r917150 <= r917173;
        double r917175 = r917162 - r917150;
        double r917176 = r917155 * r917175;
        double r917177 = sqrt(r917176);
        double r917178 = r917158 * r917177;
        double r917179 = 2.2094180720166524e+111;
        bool r917180 = r917150 <= r917179;
        double r917181 = fabs(r917162);
        double r917182 = r917166 + r917150;
        double r917183 = r917155 / r917182;
        double r917184 = sqrt(r917183);
        double r917185 = r917181 * r917184;
        double r917186 = r917158 * r917185;
        double r917187 = sqrt(r917155);
        double r917188 = r917150 + r917150;
        double r917189 = sqrt(r917188);
        double r917190 = r917187 / r917189;
        double r917191 = r917190 * r917181;
        double r917192 = r917191 * r917158;
        double r917193 = r917180 ? r917186 : r917192;
        double r917194 = r917174 ? r917178 : r917193;
        double r917195 = r917161 ? r917172 : r917194;
        double r917196 = r917152 ? r917159 : r917195;
        return r917196;
}

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 < -9.534169287773102e+132

    1. Initial program 56.0

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

    2. Taylor expanded around -inf 9.5

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(-2 \cdot re\right)}}\]

    if -9.534169287773102e+132 < re < -1.7322803508780655e-249

    1. Initial program 17.9

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt18.0

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\]

    if -1.7322803508780655e-249 < re < 1.0730314248253915e-288

    1. Initial program 29.5

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt29.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\]

    4. Taylor expanded around 0 29.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{im} - re\right)}\]

    if 1.0730314248253915e-288 < re < 2.2094180720166524e+111

    1. Initial program 38.1

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied flip--38.0

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    4. Applied associate-*r/38.0

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    5. Applied sqrt-div38.1

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    6. Simplified29.3

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(im \cdot im + 0\right) \cdot 2.0}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity29.3

      \[\leadsto 0.5 \cdot \frac{\sqrt{\left(im \cdot im + 0\right) \cdot 2.0}}{\color{blue}{1 \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    9. Applied sqrt-prod29.3

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{im \cdot im + 0} \cdot \sqrt{2.0}}}{1 \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    10. Applied times-frac29.3

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{im \cdot im + 0}}{1} \cdot \frac{\sqrt{2.0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right)}\]

    11. Simplified20.0

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left|im\right|} \cdot \frac{\sqrt{2.0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right)\]
    12. Using strategy rm

    13. Applied sqrt-undiv19.9

      \[\leadsto 0.5 \cdot \left(\left|im\right| \cdot \color{blue}{\sqrt{\frac{2.0}{\sqrt{re \cdot re + im \cdot im} + re}}}\right)\]

    if 2.2094180720166524e+111 < re

    1. Initial program 60.1

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied flip--60.2

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    4. Applied associate-*r/60.2

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    5. Applied sqrt-div60.2

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    6. Simplified44.3

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(im \cdot im + 0\right) \cdot 2.0}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity44.3

      \[\leadsto 0.5 \cdot \frac{\sqrt{\left(im \cdot im + 0\right) \cdot 2.0}}{\color{blue}{1 \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    9. Applied sqrt-prod44.3

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{im \cdot im + 0} \cdot \sqrt{2.0}}}{1 \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    10. Applied times-frac44.3

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{im \cdot im + 0}}{1} \cdot \frac{\sqrt{2.0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right)}\]

    11. Simplified42.1

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left|im\right|} \cdot \frac{\sqrt{2.0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right)\]

    12. Taylor expanded around inf 10.2

      \[\leadsto 0.5 \cdot \left(\left|im\right| \cdot \frac{\sqrt{2.0}}{\sqrt{\color{blue}{re} + re}}\right)\]
  3. Recombined 5 regimes into one program.

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.534169287773102 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -1.7322803508780655 \cdot 10^{-249}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} - re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 1.0730314248253915 \cdot 10^{-288}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 2.2094180720166524 \cdot 10^{+111}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2.0}{\sqrt{im \cdot im + re \cdot re} + re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{2.0}}{\sqrt{re + re}} \cdot \left|im\right|\right) \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))