Average Error: 38.6 → 23.2
Time: 4.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.68133238957017115 \cdot 10^{161}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{-2 \cdot re}{im}}}\\ \mathbf{elif}\;re \le 1.4937844730964523 \cdot 10^{-255}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{{im}^{1}}{\sqrt{re \cdot re + im \cdot im} - re} \cdot im\right)}\\ \mathbf{elif}\;re \le 2.37750825737911996 \cdot 10^{102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.68133238957017115 \cdot 10^{161}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{-2 \cdot re}{im}}}\\

\mathbf{elif}\;re \le 1.4937844730964523 \cdot 10^{-255}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{{im}^{1}}{\sqrt{re \cdot re + im \cdot im} - re} \cdot im\right)}\\

\mathbf{elif}\;re \le 2.37750825737911996 \cdot 10^{102}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(e^{\log \left(\sqrt{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 r235135 = 0.5;
        double r235136 = 2.0;
        double r235137 = re;
        double r235138 = r235137 * r235137;
        double r235139 = im;
        double r235140 = r235139 * r235139;
        double r235141 = r235138 + r235140;
        double r235142 = sqrt(r235141);
        double r235143 = r235142 + r235137;
        double r235144 = r235136 * r235143;
        double r235145 = sqrt(r235144);
        double r235146 = r235135 * r235145;
        return r235146;
}
↓
double f(double re, double im) {
        double r235147 = re;
        double r235148 = -1.6813323895701712e+161;
        bool r235149 = r235147 <= r235148;
        double r235150 = 0.5;
        double r235151 = 2.0;
        double r235152 = im;
        double r235153 = 1.0;
        double r235154 = pow(r235152, r235153);
        double r235155 = -2.0;
        double r235156 = r235155 * r235147;
        double r235157 = r235156 / r235152;
        double r235158 = r235154 / r235157;
        double r235159 = r235151 * r235158;
        double r235160 = sqrt(r235159);
        double r235161 = r235150 * r235160;
        double r235162 = 1.4937844730964523e-255;
        bool r235163 = r235147 <= r235162;
        double r235164 = r235147 * r235147;
        double r235165 = r235152 * r235152;
        double r235166 = r235164 + r235165;
        double r235167 = sqrt(r235166);
        double r235168 = r235167 - r235147;
        double r235169 = r235154 / r235168;
        double r235170 = r235169 * r235152;
        double r235171 = r235151 * r235170;
        double r235172 = sqrt(r235171);
        double r235173 = r235150 * r235172;
        double r235174 = 2.37750825737912e+102;
        bool r235175 = r235147 <= r235174;
        double r235176 = log(r235167);
        double r235177 = exp(r235176);
        double r235178 = r235177 + r235147;
        double r235179 = r235151 * r235178;
        double r235180 = sqrt(r235179);
        double r235181 = r235150 * r235180;
        double r235182 = 2.0;
        double r235183 = r235182 * r235147;
        double r235184 = r235151 * r235183;
        double r235185 = sqrt(r235184);
        double r235186 = r235150 * r235185;
        double r235187 = r235175 ? r235181 : r235186;
        double r235188 = r235163 ? r235173 : r235187;
        double r235189 = r235149 ? r235161 : r235188;
        return r235189;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target33.8
Herbie23.2
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -1.6813323895701712e+161

    1. Initial program 64.0

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

    3. Applied flip-+64.0

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

    4. Simplified51.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied sqr-pow51.6

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

    7. Applied associate-/l*51.2

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

    8. Simplified51.2

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

    9. Taylor expanded around -inf 23.4

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

    if -1.6813323895701712e+161 < re < 1.4937844730964523e-255

    1. Initial program 38.9

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

    3. Applied flip-+38.9

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

    4. Simplified31.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied sqr-pow31.0

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

    7. Applied associate-/l*29.3

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

    8. Simplified29.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{\left(\frac{2}{2}\right)}}{\color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{im}}}}\]
    9. Using strategy rm

    10. Applied associate-/r/29.3

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

    11. Simplified29.3

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

    if 1.4937844730964523e-255 < re < 2.37750825737912e+102

    1. Initial program 19.3

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

    3. Applied add-exp-log21.6

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

    if 2.37750825737912e+102 < re

    1. Initial program 52.7

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

    2. Taylor expanded around inf 10.4

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

  4. Final simplification23.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.68133238957017115 \cdot 10^{161}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{1}}{\frac{-2 \cdot re}{im}}}\\ \mathbf{elif}\;re \le 1.4937844730964523 \cdot 10^{-255}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{{im}^{1}}{\sqrt{re \cdot re + im \cdot im} - re} \cdot im\right)}\\ \mathbf{elif}\;re \le 2.37750825737911996 \cdot 10^{102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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)))))