Average Error: 38.3 → 27.1
Time: 11.7s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 2.009174386030369778869788105096174124076 \cdot 10^{-187}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 371427111337525632:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt[3]{{e}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}} \cdot \sqrt[3]{{e}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}}\right) \cdot \sqrt[3]{{e}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 2.009174386030369778869788105096174124076 \cdot 10^{-187}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 371427111337525632:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt[3]{{e}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}} \cdot \sqrt[3]{{e}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}}\right) \cdot \sqrt[3]{{e}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r328086 = 0.5;
        double r328087 = 2.0;
        double r328088 = re;
        double r328089 = r328088 * r328088;
        double r328090 = im;
        double r328091 = r328090 * r328090;
        double r328092 = r328089 + r328091;
        double r328093 = sqrt(r328092);
        double r328094 = r328093 + r328088;
        double r328095 = r328087 * r328094;
        double r328096 = sqrt(r328095);
        double r328097 = r328086 * r328096;
        return r328097;
}
↓
double f(double re, double im) {
        double r328098 = re;
        double r328099 = 2.0091743860303698e-187;
        bool r328100 = r328098 <= r328099;
        double r328101 = 0.5;
        double r328102 = im;
        double r328103 = r328102 * r328102;
        double r328104 = 2.0;
        double r328105 = r328103 * r328104;
        double r328106 = sqrt(r328105);
        double r328107 = r328098 * r328098;
        double r328108 = r328107 + r328103;
        double r328109 = sqrt(r328108);
        double r328110 = r328109 - r328098;
        double r328111 = sqrt(r328110);
        double r328112 = r328106 / r328111;
        double r328113 = r328101 * r328112;
        double r328114 = 3.714271113375256e+17;
        bool r328115 = r328098 <= r328114;
        double r328116 = exp(1.0);
        double r328117 = log(r328109);
        double r328118 = pow(r328116, r328117);
        double r328119 = cbrt(r328118);
        double r328120 = r328119 * r328119;
        double r328121 = r328120 * r328119;
        double r328122 = r328121 + r328098;
        double r328123 = r328104 * r328122;
        double r328124 = sqrt(r328123);
        double r328125 = r328101 * r328124;
        double r328126 = r328098 + r328098;
        double r328127 = r328104 * r328126;
        double r328128 = sqrt(r328127);
        double r328129 = r328101 * r328128;
        double r328130 = r328115 ? r328125 : r328129;
        double r328131 = r328100 ? r328113 : r328130;
        return r328131;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.3
Target33.3
Herbie27.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < 2.0091743860303698e-187

    1. Initial program 42.9

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

    3. Applied flip-+43.0

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

    4. Applied associate-*r/43.1

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

    5. Applied sqrt-div43.3

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

    6. Simplified34.2

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

    if 2.0091743860303698e-187 < re < 3.714271113375256e+17

    1. Initial program 17.5

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

    3. Applied add-exp-log19.9

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

    5. Applied pow119.9

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

    6. Applied log-pow19.9

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

    7. Applied exp-prod20.1

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

    8. Simplified20.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left({\color{blue}{e}}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)} + re\right)}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt20.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\sqrt[3]{{e}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}} \cdot \sqrt[3]{{e}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}}\right) \cdot \sqrt[3]{{e}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}}} + re\right)}\]

    if 3.714271113375256e+17 < re

    1. Initial program 41.2

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

    2. Taylor expanded around inf 13.5

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

  4. Final simplification27.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 2.009174386030369778869788105096174124076 \cdot 10^{-187}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 371427111337525632:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt[3]{{e}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}} \cdot \sqrt[3]{{e}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}}\right) \cdot \sqrt[3]{{e}^{\left(\log \left(\sqrt{re \cdot re + im \cdot im}\right)\right)}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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)))))