Average Error: 37.2 → 24.0
Time: 9.6s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -1.3770145852390023 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \sqrt{-\left(re + im\right) \cdot 2.0}\\ \mathbf{elif}\;im \le 9.324165927345474 \cdot 10^{-188}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re \cdot -2\right) \cdot 2.0}\\ \mathbf{elif}\;im \le 2.5714137250259334 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im - re\right)}\\ \mathbf{elif}\;im \le 1.3866396419306994 \cdot 10^{-139}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re \cdot -2\right) \cdot 2.0}\\ \mathbf{elif}\;im \le 2.0068865055559888 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\frac{im}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}} \cdot \frac{im}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im - re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -1.3770145852390023 \cdot 10^{-143}:\\
\;\;\;\;0.5 \cdot \sqrt{-\left(re + im\right) \cdot 2.0}\\

\mathbf{elif}\;im \le 9.324165927345474 \cdot 10^{-188}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(re \cdot -2\right) \cdot 2.0}\\

\mathbf{elif}\;im \le 2.5714137250259334 \cdot 10^{-143}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im - re\right)}\\

\mathbf{elif}\;im \le 1.3866396419306994 \cdot 10^{-139}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(re \cdot -2\right) \cdot 2.0}\\

\mathbf{elif}\;im \le 2.0068865055559888 \cdot 10^{+60}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(\frac{im}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}} \cdot \frac{im}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\right) \cdot 2.0}\\

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

\end{array}
double f(double re, double im) {
        double r273101 = 0.5;
        double r273102 = 2.0;
        double r273103 = re;
        double r273104 = r273103 * r273103;
        double r273105 = im;
        double r273106 = r273105 * r273105;
        double r273107 = r273104 + r273106;
        double r273108 = sqrt(r273107);
        double r273109 = r273108 - r273103;
        double r273110 = r273102 * r273109;
        double r273111 = sqrt(r273110);
        double r273112 = r273101 * r273111;
        return r273112;
}
↓
double f(double re, double im) {
        double r273113 = im;
        double r273114 = -1.3770145852390023e-143;
        bool r273115 = r273113 <= r273114;
        double r273116 = 0.5;
        double r273117 = re;
        double r273118 = r273117 + r273113;
        double r273119 = 2.0;
        double r273120 = r273118 * r273119;
        double r273121 = -r273120;
        double r273122 = sqrt(r273121);
        double r273123 = r273116 * r273122;
        double r273124 = 9.324165927345474e-188;
        bool r273125 = r273113 <= r273124;
        double r273126 = -2.0;
        double r273127 = r273117 * r273126;
        double r273128 = r273127 * r273119;
        double r273129 = sqrt(r273128);
        double r273130 = r273116 * r273129;
        double r273131 = 2.5714137250259334e-143;
        bool r273132 = r273113 <= r273131;
        double r273133 = r273113 - r273117;
        double r273134 = r273119 * r273133;
        double r273135 = sqrt(r273134);
        double r273136 = r273116 * r273135;
        double r273137 = 1.3866396419306994e-139;
        bool r273138 = r273113 <= r273137;
        double r273139 = 2.0068865055559888e+60;
        bool r273140 = r273113 <= r273139;
        double r273141 = r273113 * r273113;
        double r273142 = r273117 * r273117;
        double r273143 = r273141 + r273142;
        double r273144 = sqrt(r273143);
        double r273145 = r273144 + r273117;
        double r273146 = sqrt(r273145);
        double r273147 = r273113 / r273146;
        double r273148 = r273147 * r273147;
        double r273149 = r273148 * r273119;
        double r273150 = sqrt(r273149);
        double r273151 = r273116 * r273150;
        double r273152 = r273140 ? r273151 : r273136;
        double r273153 = r273138 ? r273130 : r273152;
        double r273154 = r273132 ? r273136 : r273153;
        double r273155 = r273125 ? r273130 : r273154;
        double r273156 = r273115 ? r273123 : r273155;
        return r273156;
}

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 im < -1.3770145852390023e-143

    1. Initial program 35.1

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

    3. Applied flip--39.8

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

    4. Simplified35.6

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

    5. Taylor expanded around -inf 20.4

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

    if -1.3770145852390023e-143 < im < 9.324165927345474e-188 or 2.5714137250259334e-143 < im < 1.3866396419306994e-139

    1. Initial program 42.0

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

    2. Taylor expanded around -inf 35.1

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

    if 9.324165927345474e-188 < im < 2.5714137250259334e-143 or 2.0068865055559888e+60 < im

    1. Initial program 44.5

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

    2. Taylor expanded around 0 17.0

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

    if 1.3866396419306994e-139 < im < 2.0068865055559888e+60

    1. Initial program 25.0

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

    3. Applied flip--34.3

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

    4. Simplified25.3

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

    6. Applied add-sqr-sqrt25.5

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

    7. Applied times-frac25.5

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

  4. Final simplification24.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -1.3770145852390023 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \sqrt{-\left(re + im\right) \cdot 2.0}\\ \mathbf{elif}\;im \le 9.324165927345474 \cdot 10^{-188}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re \cdot -2\right) \cdot 2.0}\\ \mathbf{elif}\;im \le 2.5714137250259334 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im - re\right)}\\ \mathbf{elif}\;im \le 1.3866396419306994 \cdot 10^{-139}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re \cdot -2\right) \cdot 2.0}\\ \mathbf{elif}\;im \le 2.0068865055559888 \cdot 10^{+60}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\frac{im}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}} \cdot \frac{im}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im - re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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)))))