Average Error: 37.2 → 25.1
Time: 15.0s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.2601941440528353 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 3.3104042253656975 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.2601941440528353 \cdot 10^{+122}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le 3.3104042253656975 \cdot 10^{-296}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\

\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 r363203 = 0.5;
        double r363204 = 2.0;
        double r363205 = re;
        double r363206 = r363205 * r363205;
        double r363207 = im;
        double r363208 = r363207 * r363207;
        double r363209 = r363206 + r363208;
        double r363210 = sqrt(r363209);
        double r363211 = r363210 - r363205;
        double r363212 = r363204 * r363211;
        double r363213 = sqrt(r363212);
        double r363214 = r363203 * r363213;
        return r363214;
}
↓
double f(double re, double im) {
        double r363215 = re;
        double r363216 = -3.2601941440528353e+122;
        bool r363217 = r363215 <= r363216;
        double r363218 = -2.0;
        double r363219 = r363218 * r363215;
        double r363220 = 2.0;
        double r363221 = r363219 * r363220;
        double r363222 = sqrt(r363221);
        double r363223 = 0.5;
        double r363224 = r363222 * r363223;
        double r363225 = 3.3104042253656975e-296;
        bool r363226 = r363215 <= r363225;
        double r363227 = im;
        double r363228 = r363227 * r363227;
        double r363229 = r363215 * r363215;
        double r363230 = r363228 + r363229;
        double r363231 = sqrt(r363230);
        double r363232 = r363231 - r363215;
        double r363233 = r363220 * r363232;
        double r363234 = sqrt(r363233);
        double r363235 = r363223 * r363234;
        double r363236 = r363220 * r363228;
        double r363237 = sqrt(r363236);
        double r363238 = r363231 + r363215;
        double r363239 = sqrt(r363238);
        double r363240 = r363237 / r363239;
        double r363241 = r363223 * r363240;
        double r363242 = r363226 ? r363235 : r363241;
        double r363243 = r363217 ? r363224 : r363242;
        return r363243;
}

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 < -3.2601941440528353e+122

    1. Initial program 53.7

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

    2. Taylor expanded around -inf 8.7

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

    if -3.2601941440528353e+122 < re < 3.3104042253656975e-296

    1. Initial program 19.6

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

    if 3.3104042253656975e-296 < re

    1. Initial program 45.1

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

    3. Applied flip--45.0

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

    4. Applied associate-*r/45.0

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

    5. Applied sqrt-div45.1

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

    6. Simplified34.4

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

  4. Final simplification25.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.2601941440528353 \cdot 10^{+122}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 3.3104042253656975 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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)))))