Average Error: 38.3 → 23.2
Time: 4.3s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.980267520528827579962452667551640625594 \cdot 10^{108}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -9.566365453995885312129846183313491609438 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \le 2.488831011052814504029106273604114956723 \cdot 10^{128}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{re + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.980267520528827579962452667551640625594 \cdot 10^{108}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\

\mathbf{elif}\;re \le -9.566365453995885312129846183313491609438 \cdot 10^{-295}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{elif}\;re \le 2.488831011052814504029106273604114956723 \cdot 10^{128}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{re + re}}\\

\end{array}
double f(double re, double im) {
        double r17333 = 0.5;
        double r17334 = 2.0;
        double r17335 = re;
        double r17336 = r17335 * r17335;
        double r17337 = im;
        double r17338 = r17337 * r17337;
        double r17339 = r17336 + r17338;
        double r17340 = sqrt(r17339);
        double r17341 = r17340 - r17335;
        double r17342 = r17334 * r17341;
        double r17343 = sqrt(r17342);
        double r17344 = r17333 * r17343;
        return r17344;
}
↓
double f(double re, double im) {
        double r17345 = re;
        double r17346 = -1.9802675205288276e+108;
        bool r17347 = r17345 <= r17346;
        double r17348 = 0.5;
        double r17349 = 2.0;
        double r17350 = -1.0;
        double r17351 = r17350 * r17345;
        double r17352 = r17351 - r17345;
        double r17353 = r17349 * r17352;
        double r17354 = sqrt(r17353);
        double r17355 = r17348 * r17354;
        double r17356 = -9.566365453995885e-295;
        bool r17357 = r17345 <= r17356;
        double r17358 = r17345 * r17345;
        double r17359 = im;
        double r17360 = r17359 * r17359;
        double r17361 = r17358 + r17360;
        double r17362 = sqrt(r17361);
        double r17363 = r17362 - r17345;
        double r17364 = r17349 * r17363;
        double r17365 = sqrt(r17364);
        double r17366 = r17348 * r17365;
        double r17367 = 2.4888310110528145e+128;
        bool r17368 = r17345 <= r17367;
        double r17369 = r17362 + r17345;
        double r17370 = r17369 / r17359;
        double r17371 = r17359 / r17370;
        double r17372 = r17349 * r17371;
        double r17373 = sqrt(r17372);
        double r17374 = r17348 * r17373;
        double r17375 = 2.0;
        double r17376 = pow(r17359, r17375);
        double r17377 = r17345 + r17345;
        double r17378 = r17376 / r17377;
        double r17379 = r17349 * r17378;
        double r17380 = sqrt(r17379);
        double r17381 = r17348 * r17380;
        double r17382 = r17368 ? r17374 : r17381;
        double r17383 = r17357 ? r17366 : r17382;
        double r17384 = r17347 ? r17355 : r17383;
        return r17384;
}

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 4 regimes

  2. if re < -1.9802675205288276e+108

    1. Initial program 52.1

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

    2. Taylor expanded around -inf 9.9

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

    if -1.9802675205288276e+108 < re < -9.566365453995885e-295

    1. Initial program 20.0

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

    if -9.566365453995885e-295 < re < 2.4888310110528145e+128

    1. Initial program 38.4

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

    3. Applied flip--38.3

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

    4. Simplified30.7

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

    6. Applied unpow230.7

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

    7. Applied associate-/l*28.5

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

    if 2.4888310110528145e+128 < re

    1. Initial program 62.5

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

    3. Applied flip--62.5

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

    4. Simplified47.8

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

    5. Taylor expanded around inf 31.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{re} + re}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification23.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.980267520528827579962452667551640625594 \cdot 10^{108}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -9.566365453995885312129846183313491609438 \cdot 10^{-295}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \le 2.488831011052814504029106273604114956723 \cdot 10^{128}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{re + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))