Average Error: 38.9 → 12.1
Time: 5.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 1.666629672552327473847126428691428249131 \cdot 10^{65}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 1.666629672552327473847126428691428249131 \cdot 10^{65}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \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 r32150 = 0.5;
        double r32151 = 2.0;
        double r32152 = re;
        double r32153 = r32152 * r32152;
        double r32154 = im;
        double r32155 = r32154 * r32154;
        double r32156 = r32153 + r32155;
        double r32157 = sqrt(r32156);
        double r32158 = r32157 - r32152;
        double r32159 = r32151 * r32158;
        double r32160 = sqrt(r32159);
        double r32161 = r32150 * r32160;
        return r32161;
}
↓
double f(double re, double im) {
        double r32162 = re;
        double r32163 = 1.6666296725523275e+65;
        bool r32164 = r32162 <= r32163;
        double r32165 = 0.5;
        double r32166 = 2.0;
        double r32167 = 1.0;
        double r32168 = im;
        double r32169 = hypot(r32162, r32168);
        double r32170 = r32167 * r32169;
        double r32171 = r32170 - r32162;
        double r32172 = r32166 * r32171;
        double r32173 = sqrt(r32172);
        double r32174 = r32165 * r32173;
        double r32175 = 2.0;
        double r32176 = pow(r32168, r32175);
        double r32177 = 0.0;
        double r32178 = r32176 + r32177;
        double r32179 = r32162 + r32169;
        double r32180 = r32178 / r32179;
        double r32181 = r32166 * r32180;
        double r32182 = sqrt(r32181);
        double r32183 = r32165 * r32182;
        double r32184 = r32164 ? r32174 : r32183;
        return r32184;
}

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 < 1.6666296725523275e+65

    1. Initial program 33.5

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

    3. Applied *-un-lft-identity33.5

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

    4. Applied sqrt-prod33.5

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

    5. Simplified33.5

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

    6. Simplified7.1

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

    if 1.6666296725523275e+65 < re

    1. Initial program 59.6

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

    3. Applied flip--59.6

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

    4. Simplified44.0

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

    5. Simplified31.7

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

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 1.666629672552327473847126428691428249131 \cdot 10^{65}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))))