Average Error: 38.6 → 26.7
Time: 4.7s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.018010836528902491605763563734697486774 \cdot 10^{-289}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.918788687392367223139377332243671629337 \cdot 10^{56}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.018010836528902491605763563734697486774 \cdot 10^{-289}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 1.918788687392367223139377332243671629337 \cdot 10^{56}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r402951 = 0.5;
        double r402952 = 2.0;
        double r402953 = re;
        double r402954 = r402953 * r402953;
        double r402955 = im;
        double r402956 = r402955 * r402955;
        double r402957 = r402954 + r402956;
        double r402958 = sqrt(r402957);
        double r402959 = r402958 + r402953;
        double r402960 = r402952 * r402959;
        double r402961 = sqrt(r402960);
        double r402962 = r402951 * r402961;
        return r402962;
}
↓
double f(double re, double im) {
        double r402963 = re;
        double r402964 = -8.0180108365289025e-289;
        bool r402965 = r402963 <= r402964;
        double r402966 = 0.5;
        double r402967 = 2.0;
        double r402968 = im;
        double r402969 = r402968 * r402968;
        double r402970 = r402963 * r402963;
        double r402971 = r402970 + r402969;
        double r402972 = sqrt(r402971);
        double r402973 = r402972 - r402963;
        double r402974 = r402969 / r402973;
        double r402975 = r402967 * r402974;
        double r402976 = sqrt(r402975);
        double r402977 = r402966 * r402976;
        double r402978 = 1.9187886873923672e+56;
        bool r402979 = r402963 <= r402978;
        double r402980 = r402972 + r402963;
        double r402981 = r402967 * r402980;
        double r402982 = sqrt(r402981);
        double r402983 = r402966 * r402982;
        double r402984 = 2.0;
        double r402985 = r402984 * r402963;
        double r402986 = r402967 * r402985;
        double r402987 = sqrt(r402986);
        double r402988 = r402966 * r402987;
        double r402989 = r402979 ? r402983 : r402988;
        double r402990 = r402965 ? r402977 : r402989;
        return r402990;
}

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
Target33.5
Herbie26.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -8.0180108365289025e-289

    1. Initial program 46.4

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

    3. Applied flip-+46.3

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

    4. Simplified35.7

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

    if -8.0180108365289025e-289 < re < 1.9187886873923672e+56

    1. Initial program 22.3

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

    3. Applied add-exp-log24.6

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

    5. Applied rem-exp-log22.3

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

    if 1.9187886873923672e+56 < re

    1. Initial program 45.6

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

    2. Taylor expanded around inf 12.2

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

  4. Final simplification26.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.018010836528902491605763563734697486774 \cdot 10^{-289}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.918788687392367223139377332243671629337 \cdot 10^{56}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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)))))