Average Error: 38.2 → 11.9
Time: 4.3s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 2.16048530559762547 \cdot 10^{136}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 2.16048530559762547 \cdot 10^{136}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\

\end{array}
double f(double re, double im) {
        double r21146 = 0.5;
        double r21147 = 2.0;
        double r21148 = re;
        double r21149 = r21148 * r21148;
        double r21150 = im;
        double r21151 = r21150 * r21150;
        double r21152 = r21149 + r21151;
        double r21153 = sqrt(r21152);
        double r21154 = r21153 - r21148;
        double r21155 = r21147 * r21154;
        double r21156 = sqrt(r21155);
        double r21157 = r21146 * r21156;
        return r21157;
}
↓
double f(double re, double im) {
        double r21158 = re;
        double r21159 = 2.1604853055976255e+136;
        bool r21160 = r21158 <= r21159;
        double r21161 = 0.5;
        double r21162 = 2.0;
        double r21163 = im;
        double r21164 = hypot(r21158, r21163);
        double r21165 = r21164 - r21158;
        double r21166 = r21162 * r21165;
        double r21167 = sqrt(r21166);
        double r21168 = r21161 * r21167;
        double r21169 = 2.0;
        double r21170 = pow(r21163, r21169);
        double r21171 = 0.0;
        double r21172 = r21170 + r21171;
        double r21173 = r21158 + r21164;
        double r21174 = r21172 / r21173;
        double r21175 = r21162 * r21174;
        double r21176 = sqrt(r21175);
        double r21177 = r21161 * r21176;
        double r21178 = r21160 ? r21168 : r21177;
        return r21178;
}

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 2 regimes

  2. if re < 2.1604853055976255e+136

    1. Initial program 34.1

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

    3. Applied hypot-def8.7

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

    if 2.1604853055976255e+136 < re

    1. Initial program 63.0

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

    3. Applied flip--63.0

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

    4. Simplified49.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2} + 0}}{\sqrt{re \cdot re + im \cdot im} + re}}\]

    5. Simplified31.4

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

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 2.16048530559762547 \cdot 10^{136}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(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)))))