Average Error: 38.9 → 26.7
Time: 7.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.528167954050153861894555232350182828574 \cdot 10^{-301}:\\ \;\;\;\;0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{elif}\;re \le 4.941794989955851304909697931543184438921 \cdot 10^{75}:\\ \;\;\;\;0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\right)\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.528167954050153861894555232350182828574 \cdot 10^{-301}:\\
\;\;\;\;0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\

\mathbf{elif}\;re \le 4.941794989955851304909697931543184438921 \cdot 10^{75}:\\
\;\;\;\;0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\right)\\

\end{array}
double f(double re, double im) {
        double r135183 = 0.5;
        double r135184 = 2.0;
        double r135185 = re;
        double r135186 = r135185 * r135185;
        double r135187 = im;
        double r135188 = r135187 * r135187;
        double r135189 = r135186 + r135188;
        double r135190 = sqrt(r135189);
        double r135191 = r135190 + r135185;
        double r135192 = r135184 * r135191;
        double r135193 = sqrt(r135192);
        double r135194 = r135183 * r135193;
        return r135194;
}
↓
double f(double re, double im) {
        double r135195 = re;
        double r135196 = -3.528167954050154e-301;
        bool r135197 = r135195 <= r135196;
        double r135198 = 0.5;
        double r135199 = 1.0;
        double r135200 = 2.0;
        double r135201 = im;
        double r135202 = r135201 * r135201;
        double r135203 = r135195 * r135195;
        double r135204 = r135203 + r135202;
        double r135205 = sqrt(r135204);
        double r135206 = r135205 - r135195;
        double r135207 = r135202 / r135206;
        double r135208 = r135200 * r135207;
        double r135209 = sqrt(r135208);
        double r135210 = r135199 * r135209;
        double r135211 = r135198 * r135210;
        double r135212 = 4.941794989955851e+75;
        bool r135213 = r135195 <= r135212;
        double r135214 = r135205 + r135195;
        double r135215 = r135200 * r135214;
        double r135216 = sqrt(r135215);
        double r135217 = r135199 * r135216;
        double r135218 = r135198 * r135217;
        double r135219 = 2.0;
        double r135220 = r135219 * r135195;
        double r135221 = r135200 * r135220;
        double r135222 = sqrt(r135221);
        double r135223 = r135199 * r135222;
        double r135224 = r135198 * r135223;
        double r135225 = r135213 ? r135218 : r135224;
        double r135226 = r135197 ? r135211 : r135225;
        return r135226;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.9
Target33.6
Herbie26.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -3.528167954050154e-301

    1. Initial program 46.8

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

    3. Applied *-un-lft-identity46.8

      \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}\]
    4. Using strategy rm

    5. Applied flip-+46.6

      \[\leadsto 0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\right)\]

    6. Simplified35.9

      \[\leadsto 0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\]

    if -3.528167954050154e-301 < re < 4.941794989955851e+75

    1. Initial program 21.7

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

    3. Applied *-un-lft-identity21.7

      \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}\]

    if 4.941794989955851e+75 < re

    1. Initial program 48.0

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

    3. Applied *-un-lft-identity48.0

      \[\leadsto 0.5 \cdot \color{blue}{\left(1 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt48.0

      \[\leadsto 0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}}} + re\right)}\right)\]

    6. Applied sqrt-prod48.0

      \[\leadsto 0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\right)\]

    7. Taylor expanded around inf 11.1

      \[\leadsto 0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \color{blue}{\left(2 \cdot re\right)}}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification26.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.528167954050153861894555232350182828574 \cdot 10^{-301}:\\ \;\;\;\;0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{elif}\;re \le 4.941794989955851304909697931543184438921 \cdot 10^{75}:\\ \;\;\;\;0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))