Average Error: 39.1 → 19.8
Time: 9.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.463228932581569 \cdot 10^{122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -7.36720432236371972 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le -8.0555860969017378 \cdot 10^{-305}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 5.3944762495838531 \cdot 10^{143}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\sqrt{\left|im\right|} \cdot \sqrt{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} + re}}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{re + re}\right)}\\ \end{array}\]
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(-2 \cdot 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|\sqrt{\left|im\right|} \cdot \sqrt{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} + re}}\right|\right)\\

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

\end{array}
double f(double re, double im) {
        double r132 = 0.5;
        double r133 = 2.0;
        double r134 = re;
        double r135 = r134 * r134;
        double r136 = im;
        double r137 = r136 * r136;
        double r138 = r135 + r137;
        double r139 = sqrt(r138);
        double r140 = r139 - r134;
        double r141 = r133 * r140;
        double r142 = sqrt(r141);
        double r143 = r132 * r142;
        return r143;
}
↓
double f(double re, double im) {
        double r144 = re;
        double r145 = -2.463228932581569e+122;
        bool r146 = r144 <= r145;
        double r147 = 0.5;
        double r148 = 2.0;
        double r149 = -2.0;
        double r150 = r149 * r144;
        double r151 = r148 * r150;
        double r152 = sqrt(r151);
        double r153 = r147 * r152;
        double r154 = -7.36720432236372e-267;
        bool r155 = r144 <= r154;
        double r156 = r144 * r144;
        double r157 = im;
        double r158 = r157 * r157;
        double r159 = r156 + r158;
        double r160 = sqrt(r159);
        double r161 = cbrt(r160);
        double r162 = r161 * r161;
        double r163 = r162 * r161;
        double r164 = r163 - r144;
        double r165 = r148 * r164;
        double r166 = sqrt(r165);
        double r167 = r147 * r166;
        double r168 = -8.055586096901738e-305;
        bool r169 = r144 <= r168;
        double r170 = r144 + r157;
        double r171 = -r170;
        double r172 = r148 * r171;
        double r173 = sqrt(r172);
        double r174 = r147 * r173;
        double r175 = 5.394476249583853e+143;
        bool r176 = r144 <= r175;
        double r177 = sqrt(r148);
        double r178 = fabs(r157);
        double r179 = sqrt(r178);
        double r180 = r160 + r144;
        double r181 = r178 / r180;
        double r182 = sqrt(r181);
        double r183 = r179 * r182;
        double r184 = fabs(r183);
        double r185 = r177 * r184;
        double r186 = r147 * r185;
        double r187 = r144 + r144;
        double r188 = r178 / r187;
        double r189 = r178 * r188;
        double r190 = r148 * r189;
        double r191 = sqrt(r190);
        double r192 = r147 * r191;
        double r193 = r176 ? r186 : r192;
        double r194 = r169 ? r174 : r193;
        double r195 = r155 ? r167 : r194;
        double r196 = r146 ? r153 : r195;
        return r196;
}

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

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

    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 *-un-lft-identity31.4

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

    7. Applied add-sqr-sqrt31.4

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

    8. Applied times-frac31.4

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

    9. Simplified31.2

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

    10. Simplified29.0

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

    12. Applied add-sqr-sqrt29.1

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

    13. Applied add-sqr-sqrt29.2

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

    14. Applied unswap-sqr29.2

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

    16. Applied sqrt-prod29.2

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

    17. Simplified21.6

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

    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-sqrt48.1

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

    8. Applied times-frac48.1

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

    9. Simplified48.1

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

    10. Simplified47.7

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

    11. Taylor expanded around inf 24.1

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

  4. Final simplification19.8

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

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)))))