Average Error: 37.5 → 25.6
Time: 18.4s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.256802803697009 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -6.395799248891705 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} - re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.256802803697009 \cdot 10^{+151}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le -6.395799248891705 \cdot 10^{-305}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} - re\right)} \cdot 0.5\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\

\end{array}
double f(double re, double im) {
        double r452045 = 0.5;
        double r452046 = 2.0;
        double r452047 = re;
        double r452048 = r452047 * r452047;
        double r452049 = im;
        double r452050 = r452049 * r452049;
        double r452051 = r452048 + r452050;
        double r452052 = sqrt(r452051);
        double r452053 = r452052 - r452047;
        double r452054 = r452046 * r452053;
        double r452055 = sqrt(r452054);
        double r452056 = r452045 * r452055;
        return r452056;
}
↓
double f(double re, double im) {
        double r452057 = re;
        double r452058 = -2.256802803697009e+151;
        bool r452059 = r452057 <= r452058;
        double r452060 = -2.0;
        double r452061 = r452060 * r452057;
        double r452062 = 2.0;
        double r452063 = r452061 * r452062;
        double r452064 = sqrt(r452063);
        double r452065 = 0.5;
        double r452066 = r452064 * r452065;
        double r452067 = -6.395799248891705e-305;
        bool r452068 = r452057 <= r452067;
        double r452069 = im;
        double r452070 = r452069 * r452069;
        double r452071 = r452057 * r452057;
        double r452072 = r452070 + r452071;
        double r452073 = sqrt(r452072);
        double r452074 = sqrt(r452073);
        double r452075 = r452074 * r452074;
        double r452076 = r452075 - r452057;
        double r452077 = r452062 * r452076;
        double r452078 = sqrt(r452077);
        double r452079 = r452078 * r452065;
        double r452080 = r452070 * r452062;
        double r452081 = sqrt(r452080);
        double r452082 = r452073 + r452057;
        double r452083 = sqrt(r452082);
        double r452084 = r452081 / r452083;
        double r452085 = r452065 * r452084;
        double r452086 = r452068 ? r452079 : r452085;
        double r452087 = r452059 ? r452066 : r452086;
        return r452087;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if re < -2.256802803697009e+151

    1. Initial program 60.5

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

    2. Taylor expanded around -inf 6.7

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(-2 \cdot re\right)}}\]

    if -2.256802803697009e+151 < re < -6.395799248891705e-305

    1. Initial program 19.0

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt19.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\]

    if -6.395799248891705e-305 < re

    1. Initial program 45.0

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    2. Using strategy rm

    3. Applied flip--45.0

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    4. Applied associate-*r/45.0

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    5. Applied sqrt-div45.1

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    6. Simplified35.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(im \cdot im\right) \cdot 2.0 + 0}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.256802803697009 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -6.395799248891705 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} - re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))