Average Error: 38.2 → 26.1
Time: 13.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le 7.789430045443286732009817813286096796361 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\

\mathbf{elif}\;re \le 7.789430045443286732009817813286096796361 \cdot 10^{-277}:\\
\;\;\;\;\sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)} \cdot 0.5\\

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

\end{array}
double f(double re, double im) {
        double r28308 = 0.5;
        double r28309 = 2.0;
        double r28310 = re;
        double r28311 = r28310 * r28310;
        double r28312 = im;
        double r28313 = r28312 * r28312;
        double r28314 = r28311 + r28313;
        double r28315 = sqrt(r28314);
        double r28316 = r28315 - r28310;
        double r28317 = r28309 * r28316;
        double r28318 = sqrt(r28317);
        double r28319 = r28308 * r28318;
        return r28319;
}
↓
double f(double re, double im) {
        double r28320 = re;
        double r28321 = -1.1817931832138217e+151;
        bool r28322 = r28320 <= r28321;
        double r28323 = -2.0;
        double r28324 = r28323 * r28320;
        double r28325 = 2.0;
        double r28326 = r28324 * r28325;
        double r28327 = sqrt(r28326);
        double r28328 = 0.5;
        double r28329 = r28327 * r28328;
        double r28330 = 7.789430045443287e-277;
        bool r28331 = r28320 <= r28330;
        double r28332 = im;
        double r28333 = r28332 * r28332;
        double r28334 = r28320 * r28320;
        double r28335 = r28333 + r28334;
        double r28336 = sqrt(r28335);
        double r28337 = r28336 - r28320;
        double r28338 = r28325 * r28337;
        double r28339 = sqrt(r28338);
        double r28340 = r28339 * r28328;
        double r28341 = r28325 * r28333;
        double r28342 = sqrt(r28341);
        double r28343 = r28320 + r28336;
        double r28344 = sqrt(r28343);
        double r28345 = r28342 / r28344;
        double r28346 = r28328 * r28345;
        double r28347 = r28331 ? r28340 : r28346;
        double r28348 = r28322 ? r28329 : r28347;
        return r28348;
}

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 < -1.1817931832138217e+151

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 8.4

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

    3. Simplified8.4

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

    if -1.1817931832138217e+151 < re < 7.789430045443287e-277

    1. Initial program 21.0

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

    3. Applied pow121.0

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

    if 7.789430045443287e-277 < re

    1. Initial program 46.1

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

    3. Applied pow146.1

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

    5. Applied flip--46.0

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

    6. Applied associate-*r/46.0

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

    7. Applied sqrt-div46.1

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

    8. Simplified35.1

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(im \cdot im + 0\right) \cdot 2}}}{\sqrt{\sqrt{{\left(re \cdot re + im \cdot im\right)}^{1}} + re}}\]

    9. Simplified35.1

      \[\leadsto 0.5 \cdot \frac{\sqrt{\left(im \cdot im + 0\right) \cdot 2}}{\color{blue}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification26.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.181793183213821728908776663248811693415 \cdot 10^{151}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le 7.789430045443286732009817813286096796361 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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)))))