Average Error: 38.5 → 26.1
Time: 15.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.273448994496035253530762983989520870583 \cdot 10^{87}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le -6.316428651001183134615642765647102049934 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} - re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.273448994496035253530762983989520870583 \cdot 10^{87}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\

\mathbf{elif}\;re \le -6.316428651001183134615642765647102049934 \cdot 10^{-302}:\\
\;\;\;\;\sqrt{2 \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{else}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\

\end{array}
double f(double re, double im) {
        double r948104 = 0.5;
        double r948105 = 2.0;
        double r948106 = re;
        double r948107 = r948106 * r948106;
        double r948108 = im;
        double r948109 = r948108 * r948108;
        double r948110 = r948107 + r948109;
        double r948111 = sqrt(r948110);
        double r948112 = r948111 - r948106;
        double r948113 = r948105 * r948112;
        double r948114 = sqrt(r948113);
        double r948115 = r948104 * r948114;
        return r948115;
}
↓
double f(double re, double im) {
        double r948116 = re;
        double r948117 = -2.2734489944960353e+87;
        bool r948118 = r948116 <= r948117;
        double r948119 = -2.0;
        double r948120 = r948119 * r948116;
        double r948121 = 2.0;
        double r948122 = r948120 * r948121;
        double r948123 = sqrt(r948122);
        double r948124 = 0.5;
        double r948125 = r948123 * r948124;
        double r948126 = -6.316428651001183e-302;
        bool r948127 = r948116 <= r948126;
        double r948128 = im;
        double r948129 = r948128 * r948128;
        double r948130 = r948116 * r948116;
        double r948131 = r948129 + r948130;
        double r948132 = sqrt(r948131);
        double r948133 = sqrt(r948132);
        double r948134 = r948133 * r948133;
        double r948135 = r948134 - r948116;
        double r948136 = r948121 * r948135;
        double r948137 = sqrt(r948136);
        double r948138 = r948137 * r948124;
        double r948139 = r948121 * r948129;
        double r948140 = sqrt(r948139);
        double r948141 = r948132 + r948116;
        double r948142 = sqrt(r948141);
        double r948143 = r948140 / r948142;
        double r948144 = r948124 * r948143;
        double r948145 = r948127 ? r948138 : r948144;
        double r948146 = r948118 ? r948125 : r948145;
        return r948146;
}

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 < -2.2734489944960353e+87

    1. Initial program 49.1

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

    2. Taylor expanded around -inf 11.1

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

    if -2.2734489944960353e+87 < re < -6.316428651001183e-302

    1. Initial program 21.6

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

    3. Applied add-sqr-sqrt21.6

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

    4. Applied sqrt-prod21.7

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

    if -6.316428651001183e-302 < re

    1. Initial program 45.4

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

    3. Applied flip--45.3

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

    4. Applied associate-*r/45.3

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

    5. Applied sqrt-div45.4

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

    6. Simplified34.2

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

  4. Final simplification26.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.273448994496035253530762983989520870583 \cdot 10^{87}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le -6.316428651001183134615642765647102049934 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} - re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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)))))