Average Error: 38.5 → 18.8
Time: 14.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.369499690488931210940314873390714060144 \cdot 10^{154}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le 2.059571469999201499073172710204654392749 \cdot 10^{-305}:\\ \;\;\;\;\left(\frac{\left|im\right|}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \cdot \frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right) \cdot 0.5\\ \mathbf{elif}\;re \le 6.557741771668357414125326525668957010105 \cdot 10^{115}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.369499690488931210940314873390714060144 \cdot 10^{154}:\\
\;\;\;\;\frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{-2 \cdot re}} \cdot 0.5\\

\mathbf{elif}\;re \le 2.059571469999201499073172710204654392749 \cdot 10^{-305}:\\
\;\;\;\;\left(\frac{\left|im\right|}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \cdot \frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right) \cdot 0.5\\

\mathbf{elif}\;re \le 6.557741771668357414125326525668957010105 \cdot 10^{115}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}\\

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

\end{array}
double f(double re, double im) {
        double r9374939 = 0.5;
        double r9374940 = 2.0;
        double r9374941 = re;
        double r9374942 = r9374941 * r9374941;
        double r9374943 = im;
        double r9374944 = r9374943 * r9374943;
        double r9374945 = r9374942 + r9374944;
        double r9374946 = sqrt(r9374945);
        double r9374947 = r9374946 + r9374941;
        double r9374948 = r9374940 * r9374947;
        double r9374949 = sqrt(r9374948);
        double r9374950 = r9374939 * r9374949;
        return r9374950;
}
↓
double f(double re, double im) {
        double r9374951 = re;
        double r9374952 = -1.3694996904889312e+154;
        bool r9374953 = r9374951 <= r9374952;
        double r9374954 = 2.0;
        double r9374955 = im;
        double r9374956 = r9374955 * r9374955;
        double r9374957 = r9374954 * r9374956;
        double r9374958 = sqrt(r9374957);
        double r9374959 = -2.0;
        double r9374960 = r9374959 * r9374951;
        double r9374961 = sqrt(r9374960);
        double r9374962 = r9374958 / r9374961;
        double r9374963 = 0.5;
        double r9374964 = r9374962 * r9374963;
        double r9374965 = 2.0595714699992015e-305;
        bool r9374966 = r9374951 <= r9374965;
        double r9374967 = fabs(r9374955);
        double r9374968 = r9374951 * r9374951;
        double r9374969 = r9374968 + r9374956;
        double r9374970 = sqrt(r9374969);
        double r9374971 = r9374970 - r9374951;
        double r9374972 = sqrt(r9374971);
        double r9374973 = sqrt(r9374972);
        double r9374974 = r9374967 / r9374973;
        double r9374975 = sqrt(r9374954);
        double r9374976 = r9374975 / r9374973;
        double r9374977 = r9374974 * r9374976;
        double r9374978 = r9374977 * r9374963;
        double r9374979 = 6.557741771668357e+115;
        bool r9374980 = r9374951 <= r9374979;
        double r9374981 = sqrt(r9374970);
        double r9374982 = r9374981 * r9374981;
        double r9374983 = r9374951 + r9374982;
        double r9374984 = r9374954 * r9374983;
        double r9374985 = sqrt(r9374984);
        double r9374986 = r9374963 * r9374985;
        double r9374987 = r9374951 + r9374951;
        double r9374988 = r9374987 * r9374954;
        double r9374989 = sqrt(r9374988);
        double r9374990 = r9374989 * r9374963;
        double r9374991 = r9374980 ? r9374986 : r9374990;
        double r9374992 = r9374966 ? r9374978 : r9374991;
        double r9374993 = r9374953 ? r9374964 : r9374992;
        return r9374993;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.5
Target33.6
Herbie18.8
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -1.3694996904889312e+154

    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. Applied associate-*r/64.0

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

    5. Applied sqrt-div64.0

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

    6. Simplified50.2

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    7. Taylor expanded around -inf 19.8

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}{\sqrt{\color{blue}{-2 \cdot re}}}\]

    if -1.3694996904889312e+154 < re < 2.0595714699992015e-305

    1. Initial program 40.4

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

    3. Applied flip-+40.2

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

    4. Applied associate-*r/40.2

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

    5. Applied sqrt-div40.3

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

    6. Simplified30.5

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt30.5

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}{\sqrt{\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}}\]

    9. Applied sqrt-prod30.6

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}{\color{blue}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}}\]

    10. Applied sqrt-prod30.6

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{im \cdot im + 0}}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    11. Applied times-frac30.6

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \cdot \frac{\sqrt{im \cdot im + 0}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)}\]

    12. Simplified21.0

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

    if 2.0595714699992015e-305 < re < 6.557741771668357e+115

    1. Initial program 20.0

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

    3. Applied add-sqr-sqrt20.0

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

    4. Applied sqrt-prod20.1

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

    if 6.557741771668357e+115 < re

    1. Initial program 54.4

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

    2. Taylor expanded around inf 10.0

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

  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.369499690488931210940314873390714060144 \cdot 10^{154}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le 2.059571469999201499073172710204654392749 \cdot 10^{-305}:\\ \;\;\;\;\left(\frac{\left|im\right|}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \cdot \frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right) \cdot 0.5\\ \mathbf{elif}\;re \le 6.557741771668357414125326525668957010105 \cdot 10^{115}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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)))))