Average Error: 38.8 → 26.8
Time: 18.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.688721359903120564949235071844216306814 \cdot 10^{100}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le -9.064518896973367303560175417863365194412 \cdot 10^{-262}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) - re\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} + re}} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.688721359903120564949235071844216306814 \cdot 10^{100}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\

\mathbf{elif}\;re \le -9.064518896973367303560175417863365194412 \cdot 10^{-262}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) - re\right) \cdot 2}\\

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

\end{array}
double f(double re, double im) {
        double r1089906 = 0.5;
        double r1089907 = 2.0;
        double r1089908 = re;
        double r1089909 = r1089908 * r1089908;
        double r1089910 = im;
        double r1089911 = r1089910 * r1089910;
        double r1089912 = r1089909 + r1089911;
        double r1089913 = sqrt(r1089912);
        double r1089914 = r1089913 - r1089908;
        double r1089915 = r1089907 * r1089914;
        double r1089916 = sqrt(r1089915);
        double r1089917 = r1089906 * r1089916;
        return r1089917;
}
↓
double f(double re, double im) {
        double r1089918 = re;
        double r1089919 = -1.6887213599031206e+100;
        bool r1089920 = r1089918 <= r1089919;
        double r1089921 = -2.0;
        double r1089922 = r1089921 * r1089918;
        double r1089923 = 2.0;
        double r1089924 = r1089922 * r1089923;
        double r1089925 = sqrt(r1089924);
        double r1089926 = 0.5;
        double r1089927 = r1089925 * r1089926;
        double r1089928 = -9.064518896973367e-262;
        bool r1089929 = r1089918 <= r1089928;
        double r1089930 = im;
        double r1089931 = r1089930 * r1089930;
        double r1089932 = r1089918 * r1089918;
        double r1089933 = r1089931 + r1089932;
        double r1089934 = sqrt(r1089933);
        double r1089935 = sqrt(r1089934);
        double r1089936 = sqrt(r1089935);
        double r1089937 = r1089936 * r1089935;
        double r1089938 = r1089936 * r1089937;
        double r1089939 = r1089938 - r1089918;
        double r1089940 = r1089939 * r1089923;
        double r1089941 = sqrt(r1089940);
        double r1089942 = r1089926 * r1089941;
        double r1089943 = r1089934 + r1089918;
        double r1089944 = r1089931 / r1089943;
        double r1089945 = r1089923 * r1089944;
        double r1089946 = sqrt(r1089945);
        double r1089947 = r1089946 * r1089926;
        double r1089948 = r1089929 ? r1089942 : r1089947;
        double r1089949 = r1089920 ? r1089927 : r1089948;
        return r1089949;
}

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 < -1.6887213599031206e+100

    1. Initial program 51.8

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

    2. Taylor expanded around -inf 10.5

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

    if -1.6887213599031206e+100 < re < -9.064518896973367e-262

    1. Initial program 20.3

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

    3. Applied add-sqr-sqrt20.3

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

    4. Applied sqrt-prod20.4

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

    6. Applied add-sqr-sqrt20.4

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

    7. Applied sqrt-prod20.4

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

    8. Applied sqrt-prod20.4

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

    9. Applied associate-*r*20.4

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

    if -9.064518896973367e-262 < re

    1. Initial program 44.8

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

    3. Applied add-sqr-sqrt44.8

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

    4. Applied sqrt-prod45.6

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

    6. Applied flip--45.6

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

    7. Simplified35.7

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

    8. Simplified35.6

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

  4. Final simplification26.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.688721359903120564949235071844216306814 \cdot 10^{100}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le -9.064518896973367303560175417863365194412 \cdot 10^{-262}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) - re\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} + re}} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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)))))