Average Error: 37.4 → 24.1
Time: 13.6s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -4.3633837281117996 \cdot 10^{+105}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-\left(re + im\right)\right) \cdot 2.0}\\ \mathbf{elif}\;im \le -3.3981436272511237 \cdot 10^{-181}:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{2.0} \cdot im}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}\\ \mathbf{elif}\;im \le -1.330280636298453 \cdot 10^{-261}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-2 \cdot re\right) \cdot 2.0}\\ \mathbf{elif}\;im \le 2.9021426245006 \cdot 10^{-309}:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{2.0} \cdot im}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}\\ \mathbf{elif}\;im \le 1.6408918126130706 \cdot 10^{-203}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-2 \cdot re\right) \cdot 2.0}\\ \mathbf{elif}\;im \le 6.1294351368168115 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0} \cdot im}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}\\ \mathbf{elif}\;im \le 1523.943947555021:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-2 \cdot re\right) \cdot 2.0}\\ \mathbf{elif}\;im \le 1.1910552598904152 \cdot 10^{+142}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im}{\frac{re + \sqrt{im \cdot im + re \cdot re}}{im}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(im - re\right)} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -4.3633837281117996 \cdot 10^{+105}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(-\left(re + im\right)\right) \cdot 2.0}\\

\mathbf{elif}\;im \le -3.3981436272511237 \cdot 10^{-181}:\\
\;\;\;\;-0.5 \cdot \frac{\sqrt{2.0} \cdot im}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}\\

\mathbf{elif}\;im \le -1.330280636298453 \cdot 10^{-261}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(-2 \cdot re\right) \cdot 2.0}\\

\mathbf{elif}\;im \le 2.9021426245006 \cdot 10^{-309}:\\
\;\;\;\;-0.5 \cdot \frac{\sqrt{2.0} \cdot im}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}\\

\mathbf{elif}\;im \le 1.6408918126130706 \cdot 10^{-203}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(-2 \cdot re\right) \cdot 2.0}\\

\mathbf{elif}\;im \le 6.1294351368168115 \cdot 10^{-31}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2.0} \cdot im}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}\\

\mathbf{elif}\;im \le 1523.943947555021:\\
\;\;\;\;0.5 \cdot \sqrt{\left(-2 \cdot re\right) \cdot 2.0}\\

\mathbf{elif}\;im \le 1.1910552598904152 \cdot 10^{+142}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im}{\frac{re + \sqrt{im \cdot im + re \cdot re}}{im}}}\\

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

\end{array}
double f(double re, double im) {
        double r297001 = 0.5;
        double r297002 = 2.0;
        double r297003 = re;
        double r297004 = r297003 * r297003;
        double r297005 = im;
        double r297006 = r297005 * r297005;
        double r297007 = r297004 + r297006;
        double r297008 = sqrt(r297007);
        double r297009 = r297008 - r297003;
        double r297010 = r297002 * r297009;
        double r297011 = sqrt(r297010);
        double r297012 = r297001 * r297011;
        return r297012;
}
↓
double f(double re, double im) {
        double r297013 = im;
        double r297014 = -4.3633837281117996e+105;
        bool r297015 = r297013 <= r297014;
        double r297016 = 0.5;
        double r297017 = re;
        double r297018 = r297017 + r297013;
        double r297019 = -r297018;
        double r297020 = 2.0;
        double r297021 = r297019 * r297020;
        double r297022 = sqrt(r297021);
        double r297023 = r297016 * r297022;
        double r297024 = -3.3981436272511237e-181;
        bool r297025 = r297013 <= r297024;
        double r297026 = sqrt(r297020);
        double r297027 = r297026 * r297013;
        double r297028 = r297013 * r297013;
        double r297029 = r297017 * r297017;
        double r297030 = r297028 + r297029;
        double r297031 = sqrt(r297030);
        double r297032 = r297017 + r297031;
        double r297033 = sqrt(r297032);
        double r297034 = r297027 / r297033;
        double r297035 = r297016 * r297034;
        double r297036 = -r297035;
        double r297037 = -1.330280636298453e-261;
        bool r297038 = r297013 <= r297037;
        double r297039 = -2.0;
        double r297040 = r297039 * r297017;
        double r297041 = r297040 * r297020;
        double r297042 = sqrt(r297041);
        double r297043 = r297016 * r297042;
        double r297044 = 2.9021426245006e-309;
        bool r297045 = r297013 <= r297044;
        double r297046 = 1.6408918126130706e-203;
        bool r297047 = r297013 <= r297046;
        double r297048 = 6.1294351368168115e-31;
        bool r297049 = r297013 <= r297048;
        double r297050 = 1523.943947555021;
        bool r297051 = r297013 <= r297050;
        double r297052 = 1.1910552598904152e+142;
        bool r297053 = r297013 <= r297052;
        double r297054 = r297032 / r297013;
        double r297055 = r297013 / r297054;
        double r297056 = r297020 * r297055;
        double r297057 = sqrt(r297056);
        double r297058 = r297016 * r297057;
        double r297059 = r297013 - r297017;
        double r297060 = r297020 * r297059;
        double r297061 = sqrt(r297060);
        double r297062 = r297061 * r297016;
        double r297063 = r297053 ? r297058 : r297062;
        double r297064 = r297051 ? r297043 : r297063;
        double r297065 = r297049 ? r297035 : r297064;
        double r297066 = r297047 ? r297043 : r297065;
        double r297067 = r297045 ? r297036 : r297066;
        double r297068 = r297038 ? r297043 : r297067;
        double r297069 = r297025 ? r297036 : r297068;
        double r297070 = r297015 ? r297023 : r297069;
        return r297070;
}

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 6 regimes

  2. if im < -4.3633837281117996e+105

    1. Initial program 51.1

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

    3. Applied flip--51.2

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

    4. Simplified51.1

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

    5. Taylor expanded around -inf 9.5

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

    if -4.3633837281117996e+105 < im < -3.3981436272511237e-181 or -1.330280636298453e-261 < im < 2.9021426245006e-309

    1. Initial program 27.7

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

    3. Applied flip--38.7

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

    4. Simplified31.8

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
    5. Using strategy rm

    6. Applied associate-*r/31.8

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

    7. Applied sqrt-div31.3

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

    8. Taylor expanded around -inf 28.6

      \[\leadsto 0.5 \cdot \frac{\color{blue}{-1 \cdot \left(\sqrt{2.0} \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    9. Simplified28.6

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

    if -3.3981436272511237e-181 < im < -1.330280636298453e-261 or 2.9021426245006e-309 < im < 1.6408918126130706e-203 or 6.1294351368168115e-31 < im < 1523.943947555021

    1. Initial program 40.6

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

    2. Taylor expanded around -inf 36.1

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

    if 1.6408918126130706e-203 < im < 6.1294351368168115e-31

    1. Initial program 31.5

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

    3. Applied flip--46.4

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

    4. Simplified38.7

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
    5. Using strategy rm

    6. Applied associate-*r/38.7

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

    7. Applied sqrt-div37.3

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

    8. Taylor expanded around inf 33.8

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

    if 1523.943947555021 < im < 1.1910552598904152e+142

    1. Initial program 20.0

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

    3. Applied flip--23.9

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

    4. Simplified21.3

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
    5. Using strategy rm

    6. Applied associate-/l*21.3

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

    if 1.1910552598904152e+142 < im

    1. Initial program 58.1

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

    2. Taylor expanded around 0 7.9

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{im} - re\right)}\]
  3. Recombined 6 regimes into one program.

  4. Final simplification24.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -4.3633837281117996 \cdot 10^{+105}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-\left(re + im\right)\right) \cdot 2.0}\\ \mathbf{elif}\;im \le -3.3981436272511237 \cdot 10^{-181}:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{2.0} \cdot im}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}\\ \mathbf{elif}\;im \le -1.330280636298453 \cdot 10^{-261}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-2 \cdot re\right) \cdot 2.0}\\ \mathbf{elif}\;im \le 2.9021426245006 \cdot 10^{-309}:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{2.0} \cdot im}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}\\ \mathbf{elif}\;im \le 1.6408918126130706 \cdot 10^{-203}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-2 \cdot re\right) \cdot 2.0}\\ \mathbf{elif}\;im \le 6.1294351368168115 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0} \cdot im}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}\\ \mathbf{elif}\;im \le 1523.943947555021:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-2 \cdot re\right) \cdot 2.0}\\ \mathbf{elif}\;im \le 1.1910552598904152 \cdot 10^{+142}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im}{\frac{re + \sqrt{im \cdot im + re \cdot re}}{im}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(im - re\right)} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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)))))