Average Error: 37.5 → 16.5
Time: 19.1s
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 r5013183 = 0.5;
        double r5013184 = 2.0;
        double r5013185 = re;
        double r5013186 = r5013185 * r5013185;
        double r5013187 = im;
        double r5013188 = r5013187 * r5013187;
        double r5013189 = r5013186 + r5013188;
        double r5013190 = sqrt(r5013189);
        double r5013191 = r5013190 + r5013185;
        double r5013192 = r5013184 * r5013191;
        double r5013193 = sqrt(r5013192);
        double r5013194 = r5013183 * r5013193;
        return r5013194;
}
↓
double f(double re, double im) {
        double r5013195 = re;
        double r5013196 = -3.2601941440528353e+122;
        bool r5013197 = r5013195 <= r5013196;
        double r5013198 = im;
        double r5013199 = fabs(r5013198);
        double r5013200 = 2.0;
        double r5013201 = sqrt(r5013200);
        double r5013202 = r5013199 * r5013201;
        double r5013203 = -2.0;
        double r5013204 = r5013203 * r5013195;
        double r5013205 = sqrt(r5013204);
        double r5013206 = r5013202 / r5013205;
        double r5013207 = 0.5;
        double r5013208 = r5013206 * r5013207;
        double r5013209 = -3.1198242249434154e-257;
        bool r5013210 = r5013195 <= r5013209;
        double r5013211 = 1.0;
        double r5013212 = r5013195 * r5013195;
        double r5013213 = r5013198 * r5013198;
        double r5013214 = r5013212 + r5013213;
        double r5013215 = sqrt(r5013214);
        double r5013216 = r5013215 - r5013195;
        double r5013217 = sqrt(r5013216);
        double r5013218 = r5013211 / r5013217;
        double r5013219 = sqrt(r5013201);
        double r5013220 = r5013199 * r5013219;
        double r5013221 = r5013219 * r5013220;
        double r5013222 = r5013218 * r5013221;
        double r5013223 = r5013207 * r5013222;
        double r5013224 = 9.299865300702547e+104;
        bool r5013225 = r5013195 <= r5013224;
        double r5013226 = r5013215 + r5013195;
        double r5013227 = r5013200 * r5013226;
        double r5013228 = sqrt(r5013227);
        double r5013229 = r5013207 * r5013228;
        double r5013230 = r5013195 + r5013195;
        double r5013231 = r5013230 * r5013200;
        double r5013232 = sqrt(r5013231);
        double r5013233 = r5013207 * r5013232;
        double r5013234 = r5013225 ? r5013229 : r5013233;
        double r5013235 = r5013210 ? r5013223 : r5013234;
        double r5013236 = r5013197 ? r5013208 : r5013235;
        return r5013236;
}

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)))))