Average Error: 38.3 → 19.2
Time: 43.4s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.4169040822526505 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -3.862148986123479 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} - re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 3.3402689576194064 \cdot 10^{+144}:\\ \;\;\;\;\frac{\sqrt{2.0}}{\frac{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}{\left|im\right|}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{re + re}} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.4169040822526505 \cdot 10^{+151}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le -3.862148986123479 \cdot 10^{-264}:\\
\;\;\;\;\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{elif}\;re \le 3.3402689576194064 \cdot 10^{+144}:\\
\;\;\;\;\frac{\sqrt{2.0}}{\frac{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}{\left|im\right|}} \cdot 0.5\\

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

\end{array}
double f(double re, double im) {
        double r938086 = 0.5;
        double r938087 = 2.0;
        double r938088 = re;
        double r938089 = r938088 * r938088;
        double r938090 = im;
        double r938091 = r938090 * r938090;
        double r938092 = r938089 + r938091;
        double r938093 = sqrt(r938092);
        double r938094 = r938093 - r938088;
        double r938095 = r938087 * r938094;
        double r938096 = sqrt(r938095);
        double r938097 = r938086 * r938096;
        return r938097;
}
↓
double f(double re, double im) {
        double r938098 = re;
        double r938099 = -3.4169040822526505e+151;
        bool r938100 = r938098 <= r938099;
        double r938101 = -2.0;
        double r938102 = r938101 * r938098;
        double r938103 = 2.0;
        double r938104 = r938102 * r938103;
        double r938105 = sqrt(r938104);
        double r938106 = 0.5;
        double r938107 = r938105 * r938106;
        double r938108 = -3.862148986123479e-264;
        bool r938109 = r938098 <= r938108;
        double r938110 = im;
        double r938111 = r938110 * r938110;
        double r938112 = r938098 * r938098;
        double r938113 = r938111 + r938112;
        double r938114 = sqrt(r938113);
        double r938115 = sqrt(r938114);
        double r938116 = r938115 * r938115;
        double r938117 = r938116 - r938098;
        double r938118 = r938103 * r938117;
        double r938119 = sqrt(r938118);
        double r938120 = r938119 * r938106;
        double r938121 = 3.3402689576194064e+144;
        bool r938122 = r938098 <= r938121;
        double r938123 = sqrt(r938103);
        double r938124 = r938114 + r938098;
        double r938125 = sqrt(r938124);
        double r938126 = fabs(r938110);
        double r938127 = r938125 / r938126;
        double r938128 = r938123 / r938127;
        double r938129 = r938128 * r938106;
        double r938130 = r938111 * r938103;
        double r938131 = sqrt(r938130);
        double r938132 = r938098 + r938098;
        double r938133 = sqrt(r938132);
        double r938134 = r938131 / r938133;
        double r938135 = r938134 * r938106;
        double r938136 = r938122 ? r938129 : r938135;
        double r938137 = r938109 ? r938120 : r938136;
        double r938138 = r938100 ? r938107 : r938137;
        return r938138;
}

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 < -3.4169040822526505e+151

    1. Initial program 63.2

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

    2. Taylor expanded around -inf 8.8

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

    if -3.4169040822526505e+151 < re < -3.862148986123479e-264

    1. Initial program 19.3

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

    3. Applied add-sqr-sqrt19.3

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

    4. Applied sqrt-prod19.4

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

    if -3.862148986123479e-264 < re < 3.3402689576194064e+144

    1. Initial program 38.1

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

    3. Applied flip--38.0

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

    4. Applied associate-*r/38.0

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

    5. Applied sqrt-div38.2

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

    6. Simplified29.4

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(0 + im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
    7. Using strategy rm

    8. Applied sqrt-prod29.4

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0} \cdot \sqrt{0 + im \cdot im}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    9. Applied associate-/l*29.4

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}{\sqrt{0 + im \cdot im}}}}\]

    10. Simplified21.3

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

    if 3.3402689576194064e+144 < re

    1. Initial program 63.6

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

    3. Applied flip--63.6

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

    4. Applied associate-*r/63.6

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

    5. Applied sqrt-div63.6

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

    6. Simplified49.3

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(0 + im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    7. Taylor expanded around inf 22.6

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0 \cdot \left(0 + im \cdot im\right)}}{\sqrt{\color{blue}{re} + re}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification19.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.4169040822526505 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -3.862148986123479 \cdot 10^{-264}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} - re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 3.3402689576194064 \cdot 10^{+144}:\\ \;\;\;\;\frac{\sqrt{2.0}}{\frac{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}{\left|im\right|}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{re + re}} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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)))))