Average Error: 39.1 → 19.5
Time: 4.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.463228932581569 \cdot 10^{122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -7.36720432236371972 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le -8.0555860969017378 \cdot 10^{-305}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 5.3944762495838531 \cdot 10^{143}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{2 \cdot re}\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.463228932581569 \cdot 10^{122}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\

\mathbf{elif}\;re \le -7.36720432236371972 \cdot 10^{-267}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\

\mathbf{elif}\;re \le -8.0555860969017378 \cdot 10^{-305}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\

\mathbf{elif}\;re \le 5.3944762495838531 \cdot 10^{143}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right|\right)\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{2 \cdot re}\right)}\\

\end{array}
double f(double re, double im) {
        double r20165 = 0.5;
        double r20166 = 2.0;
        double r20167 = re;
        double r20168 = r20167 * r20167;
        double r20169 = im;
        double r20170 = r20169 * r20169;
        double r20171 = r20168 + r20170;
        double r20172 = sqrt(r20171);
        double r20173 = r20172 - r20167;
        double r20174 = r20166 * r20173;
        double r20175 = sqrt(r20174);
        double r20176 = r20165 * r20175;
        return r20176;
}
↓
double f(double re, double im) {
        double r20177 = re;
        double r20178 = -2.463228932581569e+122;
        bool r20179 = r20177 <= r20178;
        double r20180 = 0.5;
        double r20181 = 2.0;
        double r20182 = -1.0;
        double r20183 = r20182 * r20177;
        double r20184 = r20183 - r20177;
        double r20185 = r20181 * r20184;
        double r20186 = sqrt(r20185);
        double r20187 = r20180 * r20186;
        double r20188 = -7.36720432236372e-267;
        bool r20189 = r20177 <= r20188;
        double r20190 = r20177 * r20177;
        double r20191 = im;
        double r20192 = r20191 * r20191;
        double r20193 = r20190 + r20192;
        double r20194 = sqrt(r20193);
        double r20195 = cbrt(r20194);
        double r20196 = r20195 * r20195;
        double r20197 = r20196 * r20195;
        double r20198 = r20197 - r20177;
        double r20199 = r20181 * r20198;
        double r20200 = sqrt(r20199);
        double r20201 = r20180 * r20200;
        double r20202 = -8.055586096901738e-305;
        bool r20203 = r20177 <= r20202;
        double r20204 = r20177 + r20191;
        double r20205 = -r20204;
        double r20206 = r20181 * r20205;
        double r20207 = sqrt(r20206);
        double r20208 = r20180 * r20207;
        double r20209 = 5.394476249583853e+143;
        bool r20210 = r20177 <= r20209;
        double r20211 = sqrt(r20181);
        double r20212 = fabs(r20191);
        double r20213 = r20194 + r20177;
        double r20214 = sqrt(r20213);
        double r20215 = r20212 / r20214;
        double r20216 = fabs(r20215);
        double r20217 = r20211 * r20216;
        double r20218 = r20180 * r20217;
        double r20219 = 2.0;
        double r20220 = r20219 * r20177;
        double r20221 = r20191 / r20220;
        double r20222 = r20191 * r20221;
        double r20223 = r20181 * r20222;
        double r20224 = sqrt(r20223);
        double r20225 = r20180 * r20224;
        double r20226 = r20210 ? r20218 : r20225;
        double r20227 = r20203 ? r20208 : r20226;
        double r20228 = r20189 ? r20201 : r20227;
        double r20229 = r20179 ? r20187 : r20228;
        return r20229;
}

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 5 regimes

  2. if re < -2.463228932581569e+122

    1. Initial program 55.7

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

    2. Taylor expanded around -inf 9.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{-1 \cdot re} - re\right)}\]

    if -2.463228932581569e+122 < re < -7.36720432236372e-267

    1. Initial program 19.7

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

    3. Applied add-cube-cbrt20.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\]

    if -7.36720432236372e-267 < re < -8.055586096901738e-305

    1. Initial program 29.8

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

    3. Applied flip--30.6

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

    4. Simplified30.6

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

    5. Taylor expanded around -inf 30.6

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

    if -8.055586096901738e-305 < re < 5.394476249583853e+143

    1. Initial program 40.2

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

    3. Applied flip--40.0

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

    4. Simplified31.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt31.5

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

    7. Applied add-sqr-sqrt31.5

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

    8. Applied times-frac31.5

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

    9. Simplified31.3

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

    10. Simplified29.2

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

    12. Applied sqrt-prod29.3

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

    13. Simplified20.6

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

    if 5.394476249583853e+143 < re

    1. Initial program 63.1

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

    3. Applied flip--63.1

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

    4. Simplified48.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity48.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}}\]

    7. Applied add-sqr-sqrt56.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{\color{blue}{\left(\sqrt{im} \cdot \sqrt{im}\right)}}^{2}}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\]

    8. Applied unpow-prod-down56.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{\left(\sqrt{im}\right)}^{2} \cdot {\left(\sqrt{im}\right)}^{2}}}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\]

    9. Applied times-frac56.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{\left(\sqrt{im}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt{im}\right)}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\]

    10. Simplified56.0

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

    11. Simplified47.7

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

    12. Taylor expanded around inf 24.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\color{blue}{2 \cdot re}}\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification19.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.463228932581569 \cdot 10^{122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -7.36720432236371972 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le -8.0555860969017378 \cdot 10^{-305}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 5.3944762495838531 \cdot 10^{143}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{2 \cdot re}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))