Average Error: 38.6 → 20.1
Time: 4.7s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.5187168670293087 \cdot 10^{168}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{-2 \cdot re}}\\ \mathbf{elif}\;re \le 4.41651637794876776 \cdot 10^{-304}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right|\right)\\ \mathbf{elif}\;re \le 2.7618747866440439 \cdot 10^{122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.5187168670293087 \cdot 10^{168}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{-2 \cdot re}}\\

\mathbf{elif}\;re \le 4.41651637794876776 \cdot 10^{-304}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right|\right)\\

\mathbf{elif}\;re \le 2.7618747866440439 \cdot 10^{122}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r320032 = 0.5;
        double r320033 = 2.0;
        double r320034 = re;
        double r320035 = r320034 * r320034;
        double r320036 = im;
        double r320037 = r320036 * r320036;
        double r320038 = r320035 + r320037;
        double r320039 = sqrt(r320038);
        double r320040 = r320039 + r320034;
        double r320041 = r320033 * r320040;
        double r320042 = sqrt(r320041);
        double r320043 = r320032 * r320042;
        return r320043;
}
↓
double f(double re, double im) {
        double r320044 = re;
        double r320045 = -3.5187168670293087e+168;
        bool r320046 = r320044 <= r320045;
        double r320047 = 0.5;
        double r320048 = 2.0;
        double r320049 = im;
        double r320050 = 2.0;
        double r320051 = pow(r320049, r320050);
        double r320052 = -2.0;
        double r320053 = r320052 * r320044;
        double r320054 = r320051 / r320053;
        double r320055 = r320048 * r320054;
        double r320056 = sqrt(r320055);
        double r320057 = r320047 * r320056;
        double r320058 = 4.416516377948768e-304;
        bool r320059 = r320044 <= r320058;
        double r320060 = sqrt(r320048);
        double r320061 = fabs(r320049);
        double r320062 = r320044 * r320044;
        double r320063 = r320049 * r320049;
        double r320064 = r320062 + r320063;
        double r320065 = sqrt(r320064);
        double r320066 = r320065 - r320044;
        double r320067 = sqrt(r320066);
        double r320068 = r320061 / r320067;
        double r320069 = fabs(r320068);
        double r320070 = r320060 * r320069;
        double r320071 = r320047 * r320070;
        double r320072 = 2.761874786644044e+122;
        bool r320073 = r320044 <= r320072;
        double r320074 = r320065 * r320065;
        double r320075 = sqrt(r320074);
        double r320076 = r320075 + r320044;
        double r320077 = r320048 * r320076;
        double r320078 = sqrt(r320077);
        double r320079 = r320047 * r320078;
        double r320080 = r320050 * r320044;
        double r320081 = r320048 * r320080;
        double r320082 = sqrt(r320081);
        double r320083 = r320047 * r320082;
        double r320084 = r320073 ? r320079 : r320083;
        double r320085 = r320059 ? r320071 : r320084;
        double r320086 = r320046 ? r320057 : r320085;
        return r320086;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target33.1
Herbie20.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -3.5187168670293087e+168

    1. Initial program 64.0

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

    3. Applied flip-+64.0

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

    4. Simplified49.8

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

    5. Taylor expanded around -inf 30.8

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

    if -3.5187168670293087e+168 < re < 4.416516377948768e-304

    1. Initial program 40.6

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

    3. Applied flip-+40.4

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

    4. Simplified30.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt30.5

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

    7. Applied add-sqr-sqrt30.5

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

    8. Applied times-frac30.6

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

    9. Simplified30.5

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

    10. Simplified28.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \color{blue}{\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)}\]
    11. Using strategy rm

    12. Applied sqrt-prod28.7

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

    13. Simplified20.6

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

    if 4.416516377948768e-304 < re < 2.761874786644044e+122

    1. Initial program 20.4

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

    3. Applied add-sqr-sqrt20.4

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

    if 2.761874786644044e+122 < re

    1. Initial program 57.1

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

    3. Applied flip-+63.7

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

    4. Simplified62.7

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

    5. Taylor expanded around 0 9.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(2 \cdot re\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification20.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.5187168670293087 \cdot 10^{168}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{-2 \cdot re}}\\ \mathbf{elif}\;re \le 4.41651637794876776 \cdot 10^{-304}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right|\right)\\ \mathbf{elif}\;re \le 2.7618747866440439 \cdot 10^{122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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)))))