Average Error: 38.9 → 27.6
Time: 6.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.62652217125084502788721421106711627251 \cdot 10^{-185}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 2.110613024816268606126864582999880133873 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 1.467365952554277233358119440554894540874 \cdot 10^{121}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \sqrt{\sqrt{e^{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.62652217125084502788721421106711627251 \cdot 10^{-185}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 2.110613024816268606126864582999880133873 \cdot 10^{-296}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \le 1.467365952554277233358119440554894540874 \cdot 10^{121}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \sqrt{\sqrt{e^{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}}} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r258142 = 0.5;
        double r258143 = 2.0;
        double r258144 = re;
        double r258145 = r258144 * r258144;
        double r258146 = im;
        double r258147 = r258146 * r258146;
        double r258148 = r258145 + r258147;
        double r258149 = sqrt(r258148);
        double r258150 = r258149 + r258144;
        double r258151 = r258143 * r258150;
        double r258152 = sqrt(r258151);
        double r258153 = r258142 * r258152;
        return r258153;
}
↓
double f(double re, double im) {
        double r258154 = re;
        double r258155 = -7.626522171250845e-185;
        bool r258156 = r258154 <= r258155;
        double r258157 = 0.5;
        double r258158 = 2.0;
        double r258159 = im;
        double r258160 = r258159 * r258159;
        double r258161 = r258154 * r258154;
        double r258162 = r258161 + r258160;
        double r258163 = sqrt(r258162);
        double r258164 = r258163 - r258154;
        double r258165 = r258160 / r258164;
        double r258166 = r258158 * r258165;
        double r258167 = sqrt(r258166);
        double r258168 = r258157 * r258167;
        double r258169 = 2.1106130248162686e-296;
        bool r258170 = r258154 <= r258169;
        double r258171 = r258154 + r258159;
        double r258172 = r258158 * r258171;
        double r258173 = sqrt(r258172);
        double r258174 = r258157 * r258173;
        double r258175 = 1.4673659525542772e+121;
        bool r258176 = r258154 <= r258175;
        double r258177 = cbrt(r258162);
        double r258178 = fabs(r258177);
        double r258179 = sqrt(r258177);
        double r258180 = sqrt(r258179);
        double r258181 = r258178 * r258180;
        double r258182 = log(r258177);
        double r258183 = exp(r258182);
        double r258184 = sqrt(r258183);
        double r258185 = sqrt(r258184);
        double r258186 = r258181 * r258185;
        double r258187 = r258186 + r258154;
        double r258188 = r258158 * r258187;
        double r258189 = sqrt(r258188);
        double r258190 = r258157 * r258189;
        double r258191 = 2.0;
        double r258192 = r258191 * r258154;
        double r258193 = r258158 * r258192;
        double r258194 = sqrt(r258193);
        double r258195 = r258157 * r258194;
        double r258196 = r258176 ? r258190 : r258195;
        double r258197 = r258170 ? r258174 : r258196;
        double r258198 = r258156 ? r258168 : r258197;
        return r258198;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.9
Target33.9
Herbie27.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -7.626522171250845e-185

    1. Initial program 50.4

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

    3. Applied flip-+50.4

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

    4. Simplified38.0

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

    if -7.626522171250845e-185 < re < 2.1106130248162686e-296

    1. Initial program 31.2

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    2. Taylor expanded around 0 34.6

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

    if 2.1106130248162686e-296 < re < 1.4673659525542772e+121

    1. Initial program 20.3

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

    3. Applied add-cube-cbrt20.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\]

    4. Applied sqrt-prod20.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\]

    5. Simplified20.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} + re\right)}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt20.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\color{blue}{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]

    8. Applied sqrt-prod20.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \color{blue}{\left(\sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right)} + re\right)}\]

    9. Applied associate-*r*20.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]
    10. Using strategy rm

    11. Applied add-exp-log21.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \sqrt{\sqrt{\color{blue}{e^{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}}}} + re\right)}\]

    if 1.4673659525542772e+121 < re

    1. Initial program 55.4

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    2. Taylor expanded around inf 9.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(2 \cdot re\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification27.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.62652217125084502788721421106711627251 \cdot 10^{-185}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 2.110613024816268606126864582999880133873 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 1.467365952554277233358119440554894540874 \cdot 10^{121}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \sqrt{\sqrt{e^{\log \left(\sqrt[3]{re \cdot re + im \cdot im}\right)}}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))