Average Error: 38.6 → 13.9
Time: 5.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -8.1081767403708244 \cdot 10^{118} \lor \neg \left(im \le -245497705297861837000 \lor \neg \left(im \le 2.4301831339222963 \cdot 10^{-131} \lor \neg \left(im \le 9.06583591964624736 \cdot 10^{35}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -8.1081767403708244 \cdot 10^{118} \lor \neg \left(im \le -245497705297861837000 \lor \neg \left(im \le 2.4301831339222963 \cdot 10^{-131} \lor \neg \left(im \le 9.06583591964624736 \cdot 10^{35}\right)\right)\right):\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\

\end{array}
double f(double re, double im) {
        double r212882 = 0.5;
        double r212883 = 2.0;
        double r212884 = re;
        double r212885 = r212884 * r212884;
        double r212886 = im;
        double r212887 = r212886 * r212886;
        double r212888 = r212885 + r212887;
        double r212889 = sqrt(r212888);
        double r212890 = r212889 + r212884;
        double r212891 = r212883 * r212890;
        double r212892 = sqrt(r212891);
        double r212893 = r212882 * r212892;
        return r212893;
}
↓
double f(double re, double im) {
        double r212894 = im;
        double r212895 = -8.108176740370824e+118;
        bool r212896 = r212894 <= r212895;
        double r212897 = -2.4549770529786184e+20;
        bool r212898 = r212894 <= r212897;
        double r212899 = 2.4301831339222963e-131;
        bool r212900 = r212894 <= r212899;
        double r212901 = 9.065835919646247e+35;
        bool r212902 = r212894 <= r212901;
        double r212903 = !r212902;
        bool r212904 = r212900 || r212903;
        double r212905 = !r212904;
        bool r212906 = r212898 || r212905;
        double r212907 = !r212906;
        bool r212908 = r212896 || r212907;
        double r212909 = 0.5;
        double r212910 = 2.0;
        double r212911 = 1.0;
        double r212912 = re;
        double r212913 = hypot(r212912, r212894);
        double r212914 = r212912 + r212913;
        double r212915 = r212911 * r212914;
        double r212916 = r212910 * r212915;
        double r212917 = sqrt(r212916);
        double r212918 = r212909 * r212917;
        double r212919 = r212894 * r212894;
        double r212920 = r212913 - r212912;
        double r212921 = r212919 / r212920;
        double r212922 = r212910 * r212921;
        double r212923 = sqrt(r212922);
        double r212924 = r212909 * r212923;
        double r212925 = r212908 ? r212918 : r212924;
        return r212925;
}

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
Target33.7
Herbie13.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if im < -8.108176740370824e+118 or -2.4549770529786184e+20 < im < 2.4301831339222963e-131 or 9.065835919646247e+35 < im

    1. Initial program 42.8

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

    3. Applied *-un-lft-identity42.8

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

    4. Applied *-un-lft-identity42.8

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

    5. Applied distribute-lft-out42.8

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

    6. Simplified12.7

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

    if -8.108176740370824e+118 < im < -2.4549770529786184e+20 or 2.4301831339222963e-131 < im < 9.065835919646247e+35

    1. Initial program 23.7

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

    3. Applied flip-+31.0

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

    4. Simplified23.3

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

    5. Simplified18.2

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

  4. Final simplification13.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -8.1081767403708244 \cdot 10^{118} \lor \neg \left(im \le -245497705297861837000 \lor \neg \left(im \le 2.4301831339222963 \cdot 10^{-131} \lor \neg \left(im \le 9.06583591964624736 \cdot 10^{35}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(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)))))