Average Error: 38.4 → 23.5
Time: 13.7s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -3.115043688932172505718341038177959053028 \cdot 10^{78}:\\ \;\;\;\;\sqrt{\left(re - im\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;im \le -6.768391877498059401608789761099910841632 \cdot 10^{-68}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2}\\ \mathbf{elif}\;im \le 1.14568583642485665409875286099617082253 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(2 \cdot re\right) \cdot 2}\\ \mathbf{elif}\;im \le 1.605778950101427917412655110579574821614 \cdot 10^{60}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{\sqrt{{im}^{2} + re \cdot re} - re} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(re + im\right)} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -3.115043688932172505718341038177959053028 \cdot 10^{78}:\\
\;\;\;\;\sqrt{\left(re - im\right) \cdot 2} \cdot 0.5\\

\mathbf{elif}\;im \le -6.768391877498059401608789761099910841632 \cdot 10^{-68}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2}\\

\mathbf{elif}\;im \le 1.14568583642485665409875286099617082253 \cdot 10^{-118}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(2 \cdot re\right) \cdot 2}\\

\mathbf{elif}\;im \le 1.605778950101427917412655110579574821614 \cdot 10^{60}:\\
\;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{\sqrt{{im}^{2} + re \cdot re} - re} \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2 \cdot \left(re + im\right)} \cdot 0.5\\

\end{array}
double f(double re, double im) {
        double r241224 = 0.5;
        double r241225 = 2.0;
        double r241226 = re;
        double r241227 = r241226 * r241226;
        double r241228 = im;
        double r241229 = r241228 * r241228;
        double r241230 = r241227 + r241229;
        double r241231 = sqrt(r241230);
        double r241232 = r241231 + r241226;
        double r241233 = r241225 * r241232;
        double r241234 = sqrt(r241233);
        double r241235 = r241224 * r241234;
        return r241235;
}
↓
double f(double re, double im) {
        double r241236 = im;
        double r241237 = -3.1150436889321725e+78;
        bool r241238 = r241236 <= r241237;
        double r241239 = re;
        double r241240 = r241239 - r241236;
        double r241241 = 2.0;
        double r241242 = r241240 * r241241;
        double r241243 = sqrt(r241242);
        double r241244 = 0.5;
        double r241245 = r241243 * r241244;
        double r241246 = -6.768391877498059e-68;
        bool r241247 = r241236 <= r241246;
        double r241248 = r241239 * r241239;
        double r241249 = r241236 * r241236;
        double r241250 = r241248 + r241249;
        double r241251 = sqrt(r241250);
        double r241252 = r241251 + r241239;
        double r241253 = r241252 * r241241;
        double r241254 = sqrt(r241253);
        double r241255 = r241244 * r241254;
        double r241256 = 1.1456858364248567e-118;
        bool r241257 = r241236 <= r241256;
        double r241258 = 2.0;
        double r241259 = r241258 * r241239;
        double r241260 = r241259 * r241241;
        double r241261 = sqrt(r241260);
        double r241262 = r241244 * r241261;
        double r241263 = 1.605778950101428e+60;
        bool r241264 = r241236 <= r241263;
        double r241265 = pow(r241236, r241258);
        double r241266 = r241265 + r241248;
        double r241267 = sqrt(r241266);
        double r241268 = r241267 - r241239;
        double r241269 = r241265 / r241268;
        double r241270 = r241269 * r241241;
        double r241271 = sqrt(r241270);
        double r241272 = r241244 * r241271;
        double r241273 = r241239 + r241236;
        double r241274 = r241241 * r241273;
        double r241275 = sqrt(r241274);
        double r241276 = r241275 * r241244;
        double r241277 = r241264 ? r241272 : r241276;
        double r241278 = r241257 ? r241262 : r241277;
        double r241279 = r241247 ? r241255 : r241278;
        double r241280 = r241238 ? r241245 : r241279;
        return r241280;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.4
Target33.3
Herbie23.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if im < -3.1150436889321725e+78

    1. Initial program 48.9

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

    3. Applied add-sqr-sqrt48.9

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

    4. Applied sqrt-prod49.0

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

    5. Simplified49.0

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

    6. Simplified49.0

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

    7. Taylor expanded around -inf 11.8

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

    if -3.1150436889321725e+78 < im < -6.768391877498059e-68

    1. Initial program 22.5

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

    if -6.768391877498059e-68 < im < 1.1456858364248567e-118

    1. Initial program 39.1

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

    3. Applied add-sqr-sqrt39.1

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

    4. Applied sqrt-prod40.0

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

    5. Simplified40.0

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

    6. Simplified40.0

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

    7. Taylor expanded around 0 36.9

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

    8. Simplified36.9

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

    if 1.1456858364248567e-118 < im < 1.605778950101428e+60

    1. Initial program 24.4

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

    3. Applied flip-+33.4

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

    4. Simplified24.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2} + 0}}{\sqrt{re \cdot re + im \cdot im} - re}}\]

    5. Simplified24.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{\color{blue}{\sqrt{{im}^{2} + re \cdot re} - re}}}\]

    if 1.605778950101428e+60 < im

    1. Initial program 47.1

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

    3. Applied add-sqr-sqrt47.1

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

    4. Applied sqrt-prod47.1

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

    5. Simplified47.1

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

    6. Simplified47.1

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

    7. Taylor expanded around inf 11.1

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

  4. Final simplification23.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -3.115043688932172505718341038177959053028 \cdot 10^{78}:\\ \;\;\;\;\sqrt{\left(re - im\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;im \le -6.768391877498059401608789761099910841632 \cdot 10^{-68}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{re \cdot re + im \cdot im} + re\right) \cdot 2}\\ \mathbf{elif}\;im \le 1.14568583642485665409875286099617082253 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(2 \cdot re\right) \cdot 2}\\ \mathbf{elif}\;im \le 1.605778950101427917412655110579574821614 \cdot 10^{60}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{{im}^{2}}{\sqrt{{im}^{2} + re \cdot re} - re} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(re + im\right)} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (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)))))