Average Error: 38.3 → 25.7
Time: 37.8s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.0154571595336588 \cdot 10^{-301}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 3.161818111636812 \cdot 10^{+106}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\left|\sqrt[3]{im \cdot im + re \cdot re}\right| \cdot \sqrt{\sqrt[3]{im \cdot im + re \cdot re}}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.0154571595336588 \cdot 10^{-301}:\\
\;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\

\mathbf{elif}\;re \le 3.161818111636812 \cdot 10^{+106}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\left|\sqrt[3]{im \cdot im + re \cdot re}\right| \cdot \sqrt{\sqrt[3]{im \cdot im + re \cdot re}}} + re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + re\right)}\\

\end{array}
double f(double re, double im) {
        double r4748125 = 0.5;
        double r4748126 = 2.0;
        double r4748127 = re;
        double r4748128 = r4748127 * r4748127;
        double r4748129 = im;
        double r4748130 = r4748129 * r4748129;
        double r4748131 = r4748128 + r4748130;
        double r4748132 = sqrt(r4748131);
        double r4748133 = r4748132 + r4748127;
        double r4748134 = r4748126 * r4748133;
        double r4748135 = sqrt(r4748134);
        double r4748136 = r4748125 * r4748135;
        return r4748136;
}
↓
double f(double re, double im) {
        double r4748137 = re;
        double r4748138 = -3.0154571595336588e-301;
        bool r4748139 = r4748137 <= r4748138;
        double r4748140 = im;
        double r4748141 = r4748140 * r4748140;
        double r4748142 = 2.0;
        double r4748143 = r4748141 * r4748142;
        double r4748144 = sqrt(r4748143);
        double r4748145 = r4748137 * r4748137;
        double r4748146 = r4748141 + r4748145;
        double r4748147 = sqrt(r4748146);
        double r4748148 = r4748147 - r4748137;
        double r4748149 = sqrt(r4748148);
        double r4748150 = r4748144 / r4748149;
        double r4748151 = 0.5;
        double r4748152 = r4748150 * r4748151;
        double r4748153 = 3.161818111636812e+106;
        bool r4748154 = r4748137 <= r4748153;
        double r4748155 = sqrt(r4748147);
        double r4748156 = cbrt(r4748146);
        double r4748157 = fabs(r4748156);
        double r4748158 = sqrt(r4748156);
        double r4748159 = r4748157 * r4748158;
        double r4748160 = sqrt(r4748159);
        double r4748161 = r4748155 * r4748160;
        double r4748162 = r4748161 + r4748137;
        double r4748163 = r4748142 * r4748162;
        double r4748164 = sqrt(r4748163);
        double r4748165 = r4748151 * r4748164;
        double r4748166 = r4748137 + r4748137;
        double r4748167 = r4748142 * r4748166;
        double r4748168 = sqrt(r4748167);
        double r4748169 = r4748151 * r4748168;
        double r4748170 = r4748154 ? r4748165 : r4748169;
        double r4748171 = r4748139 ? r4748152 : r4748170;
        return r4748171;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.3
Target33.0
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2.0} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -3.0154571595336588e-301

    1. Initial program 45.7

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

    3. Applied flip-+45.6

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

    4. Applied associate-*r/45.6

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

    5. Applied sqrt-div45.7

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

    6. Simplified34.2

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

    if -3.0154571595336588e-301 < re < 3.161818111636812e+106

    1. Initial program 20.4

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

    3. Applied add-sqr-sqrt20.4

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

    4. Applied sqrt-prod20.5

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt20.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}\]

    7. Applied sqrt-prod20.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}\]

    8. Simplified20.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}\]

    if 3.161818111636812e+106 < re

    1. Initial program 52.7

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

    2. Taylor expanded around inf 10.5

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

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.0154571595336588 \cdot 10^{-301}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le 3.161818111636812 \cdot 10^{+106}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\left|\sqrt[3]{im \cdot im + re \cdot re}\right| \cdot \sqrt{\sqrt[3]{im \cdot im + re \cdot re}}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))