Average Error: 37.5 → 16.5
Time: 17.2s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.2601941440528353 \cdot 10^{+122}:\\ \;\;\;\;\frac{\left|im\right| \cdot \sqrt{2.0}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le -3.1198242249434154 \cdot 10^{-257}:\\ \;\;\;\;0.5 \cdot \left(\frac{1}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \left(\sqrt{\sqrt{2.0}} \cdot \left(\left|im\right| \cdot \sqrt{\sqrt{2.0}}\right)\right)\right)\\ \mathbf{elif}\;re \le 9.299865300702547 \cdot 10^{+104}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2.0}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.2601941440528353 \cdot 10^{+122}:\\
\;\;\;\;\frac{\left|im\right| \cdot \sqrt{2.0}}{\sqrt{-2 \cdot re}} \cdot 0.5\\

\mathbf{elif}\;re \le -3.1198242249434154 \cdot 10^{-257}:\\
\;\;\;\;0.5 \cdot \left(\frac{1}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \left(\sqrt{\sqrt{2.0}} \cdot \left(\left|im\right| \cdot \sqrt{\sqrt{2.0}}\right)\right)\right)\\

\mathbf{elif}\;re \le 9.299865300702547 \cdot 10^{+104}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r4632143 = 0.5;
        double r4632144 = 2.0;
        double r4632145 = re;
        double r4632146 = r4632145 * r4632145;
        double r4632147 = im;
        double r4632148 = r4632147 * r4632147;
        double r4632149 = r4632146 + r4632148;
        double r4632150 = sqrt(r4632149);
        double r4632151 = r4632150 + r4632145;
        double r4632152 = r4632144 * r4632151;
        double r4632153 = sqrt(r4632152);
        double r4632154 = r4632143 * r4632153;
        return r4632154;
}
↓
double f(double re, double im) {
        double r4632155 = re;
        double r4632156 = -3.2601941440528353e+122;
        bool r4632157 = r4632155 <= r4632156;
        double r4632158 = im;
        double r4632159 = fabs(r4632158);
        double r4632160 = 2.0;
        double r4632161 = sqrt(r4632160);
        double r4632162 = r4632159 * r4632161;
        double r4632163 = -2.0;
        double r4632164 = r4632163 * r4632155;
        double r4632165 = sqrt(r4632164);
        double r4632166 = r4632162 / r4632165;
        double r4632167 = 0.5;
        double r4632168 = r4632166 * r4632167;
        double r4632169 = -3.1198242249434154e-257;
        bool r4632170 = r4632155 <= r4632169;
        double r4632171 = 1.0;
        double r4632172 = r4632155 * r4632155;
        double r4632173 = r4632158 * r4632158;
        double r4632174 = r4632172 + r4632173;
        double r4632175 = sqrt(r4632174);
        double r4632176 = r4632175 - r4632155;
        double r4632177 = sqrt(r4632176);
        double r4632178 = r4632171 / r4632177;
        double r4632179 = sqrt(r4632161);
        double r4632180 = r4632159 * r4632179;
        double r4632181 = r4632179 * r4632180;
        double r4632182 = r4632178 * r4632181;
        double r4632183 = r4632167 * r4632182;
        double r4632184 = 9.299865300702547e+104;
        bool r4632185 = r4632155 <= r4632184;
        double r4632186 = r4632175 + r4632155;
        double r4632187 = r4632160 * r4632186;
        double r4632188 = sqrt(r4632187);
        double r4632189 = r4632167 * r4632188;
        double r4632190 = r4632155 + r4632155;
        double r4632191 = r4632190 * r4632160;
        double r4632192 = sqrt(r4632191);
        double r4632193 = r4632167 * r4632192;
        double r4632194 = r4632185 ? r4632189 : r4632193;
        double r4632195 = r4632170 ? r4632183 : r4632194;
        double r4632196 = r4632157 ? r4632168 : r4632195;
        return r4632196;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target32.4
Herbie16.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -3.2601941440528353e+122

    1. Initial program 61.2

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

    3. Applied flip-+61.2

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

    4. Applied associate-*r/61.2

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

    5. Applied sqrt-div61.2

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

    6. Simplified45.3

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

    8. Applied sqrt-prod45.4

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

    9. Simplified44.2

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

    10. Taylor expanded around -inf 8.8

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0} \cdot \left|im\right|}{\sqrt{\color{blue}{-2 \cdot re}}}\]

    if -3.2601941440528353e+122 < re < -3.1198242249434154e-257

    1. Initial program 39.1

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

    3. Applied flip-+39.1

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

    4. Applied associate-*r/39.1

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

    5. Applied sqrt-div39.2

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

    6. Simplified28.7

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

    8. Applied sqrt-prod28.8

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

    9. Simplified18.9

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

    11. Applied add-sqr-sqrt18.9

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

    12. Applied sqrt-prod19.0

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

    13. Applied associate-*l*18.9

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

    15. Applied div-inv19.0

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

    if -3.1198242249434154e-257 < re < 9.299865300702547e+104

    1. Initial program 20.4

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

    if 9.299865300702547e+104 < re

    1. Initial program 51.4

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

    2. Taylor expanded around inf 9.7

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.2601941440528353 \cdot 10^{+122}:\\ \;\;\;\;\frac{\left|im\right| \cdot \sqrt{2.0}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le -3.1198242249434154 \cdot 10^{-257}:\\ \;\;\;\;0.5 \cdot \left(\frac{1}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \left(\sqrt{\sqrt{2.0}} \cdot \left(\left|im\right| \cdot \sqrt{\sqrt{2.0}}\right)\right)\right)\\ \mathbf{elif}\;re \le 9.299865300702547 \cdot 10^{+104}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))