Average Error: 37.3 → 25.2
Time: 18.4s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.6008532152452964 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 2.3724251741656645 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) - re\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.6008532152452964 \cdot 10^{+111}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le 2.3724251741656645 \cdot 10^{-300}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) - re\right) \cdot 2.0}\\

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

\end{array}
double f(double re, double im) {
        double r916158 = 0.5;
        double r916159 = 2.0;
        double r916160 = re;
        double r916161 = r916160 * r916160;
        double r916162 = im;
        double r916163 = r916162 * r916162;
        double r916164 = r916161 + r916163;
        double r916165 = sqrt(r916164);
        double r916166 = r916165 - r916160;
        double r916167 = r916159 * r916166;
        double r916168 = sqrt(r916167);
        double r916169 = r916158 * r916168;
        return r916169;
}
↓
double f(double re, double im) {
        double r916170 = re;
        double r916171 = -2.6008532152452964e+111;
        bool r916172 = r916170 <= r916171;
        double r916173 = -2.0;
        double r916174 = r916173 * r916170;
        double r916175 = 2.0;
        double r916176 = r916174 * r916175;
        double r916177 = sqrt(r916176);
        double r916178 = 0.5;
        double r916179 = r916177 * r916178;
        double r916180 = 2.3724251741656645e-300;
        bool r916181 = r916170 <= r916180;
        double r916182 = im;
        double r916183 = r916182 * r916182;
        double r916184 = r916170 * r916170;
        double r916185 = r916183 + r916184;
        double r916186 = sqrt(r916185);
        double r916187 = sqrt(r916186);
        double r916188 = sqrt(r916187);
        double r916189 = r916188 * r916187;
        double r916190 = r916188 * r916189;
        double r916191 = r916190 - r916170;
        double r916192 = r916191 * r916175;
        double r916193 = sqrt(r916192);
        double r916194 = r916178 * r916193;
        double r916195 = r916175 * r916183;
        double r916196 = sqrt(r916195);
        double r916197 = r916186 + r916170;
        double r916198 = sqrt(r916197);
        double r916199 = r916196 / r916198;
        double r916200 = r916178 * r916199;
        double r916201 = r916181 ? r916194 : r916200;
        double r916202 = r916172 ? r916179 : r916201;
        return r916202;
}

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 < -2.6008532152452964e+111

    1. Initial program 50.9

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

    2. Taylor expanded around -inf 9.0

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

    if -2.6008532152452964e+111 < re < 2.3724251741656645e-300

    1. Initial program 21.0

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

    3. Applied add-sqr-sqrt21.0

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

    4. Applied sqrt-prod21.1

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

    6. Applied add-sqr-sqrt21.1

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

    7. Applied sqrt-prod21.1

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

    8. Applied sqrt-prod21.2

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

    9. Applied associate-*r*21.2

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

    if 2.3724251741656645e-300 < re

    1. Initial program 44.3

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

    3. Applied flip--44.3

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

    4. Applied associate-*r/44.3

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

    5. Applied sqrt-div44.4

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

    6. Simplified33.4

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

  4. Final simplification25.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.6008532152452964 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 2.3724251741656645 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) - re\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))))