Average Error: 37.6 → 18.6
Time: 20.6s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.1292868428778451 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le -2.053769551615154 \cdot 10^{-273}:\\ \;\;\;\;\left(\frac{\sqrt{\left|im\right|}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot \left(\sqrt{\left|im\right|} \cdot \sqrt{2.0}\right)\right) \cdot 0.5\\ \mathbf{elif}\;re \le 7.239807700907349 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 1.2290590931535932 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + \sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.1292868428778451 \cdot 10^{+139}:\\
\;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{-2 \cdot re}} \cdot 0.5\\

\mathbf{elif}\;re \le -2.053769551615154 \cdot 10^{-273}:\\
\;\;\;\;\left(\frac{\sqrt{\left|im\right|}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot \left(\sqrt{\left|im\right|} \cdot \sqrt{2.0}\right)\right) \cdot 0.5\\

\mathbf{elif}\;re \le 7.239807700907349 \cdot 10^{-222}:\\
\;\;\;\;\sqrt{\left(im + re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le 1.2290590931535932 \cdot 10^{+68}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(re + \sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) \cdot 2.0}\\

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

\end{array}
double f(double re, double im) {
        double r6324945 = 0.5;
        double r6324946 = 2.0;
        double r6324947 = re;
        double r6324948 = r6324947 * r6324947;
        double r6324949 = im;
        double r6324950 = r6324949 * r6324949;
        double r6324951 = r6324948 + r6324950;
        double r6324952 = sqrt(r6324951);
        double r6324953 = r6324952 + r6324947;
        double r6324954 = r6324946 * r6324953;
        double r6324955 = sqrt(r6324954);
        double r6324956 = r6324945 * r6324955;
        return r6324956;
}
↓
double f(double re, double im) {
        double r6324957 = re;
        double r6324958 = -1.1292868428778451e+139;
        bool r6324959 = r6324957 <= r6324958;
        double r6324960 = 2.0;
        double r6324961 = im;
        double r6324962 = r6324961 * r6324961;
        double r6324963 = r6324960 * r6324962;
        double r6324964 = sqrt(r6324963);
        double r6324965 = -2.0;
        double r6324966 = r6324965 * r6324957;
        double r6324967 = sqrt(r6324966);
        double r6324968 = r6324964 / r6324967;
        double r6324969 = 0.5;
        double r6324970 = r6324968 * r6324969;
        double r6324971 = -2.053769551615154e-273;
        bool r6324972 = r6324957 <= r6324971;
        double r6324973 = fabs(r6324961);
        double r6324974 = sqrt(r6324973);
        double r6324975 = r6324957 * r6324957;
        double r6324976 = r6324962 + r6324975;
        double r6324977 = sqrt(r6324976);
        double r6324978 = r6324977 - r6324957;
        double r6324979 = sqrt(r6324978);
        double r6324980 = r6324974 / r6324979;
        double r6324981 = sqrt(r6324960);
        double r6324982 = r6324974 * r6324981;
        double r6324983 = r6324980 * r6324982;
        double r6324984 = r6324983 * r6324969;
        double r6324985 = 7.239807700907349e-222;
        bool r6324986 = r6324957 <= r6324985;
        double r6324987 = r6324961 + r6324957;
        double r6324988 = r6324987 * r6324960;
        double r6324989 = sqrt(r6324988);
        double r6324990 = r6324989 * r6324969;
        double r6324991 = 1.2290590931535932e+68;
        bool r6324992 = r6324957 <= r6324991;
        double r6324993 = sqrt(r6324977);
        double r6324994 = r6324993 * r6324993;
        double r6324995 = r6324957 + r6324994;
        double r6324996 = r6324995 * r6324960;
        double r6324997 = sqrt(r6324996);
        double r6324998 = r6324969 * r6324997;
        double r6324999 = r6324957 + r6324957;
        double r6325000 = r6324960 * r6324999;
        double r6325001 = sqrt(r6325000);
        double r6325002 = r6324969 * r6325001;
        double r6325003 = r6324992 ? r6324998 : r6325002;
        double r6325004 = r6324986 ? r6324990 : r6325003;
        double r6325005 = r6324972 ? r6324984 : r6325004;
        double r6325006 = r6324959 ? r6324970 : r6325005;
        return r6325006;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target32.6
Herbie18.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -1.1292868428778451e+139

    1. Initial program 61.7

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

    3. Applied flip-+61.7

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

    4. Applied associate-*r/61.7

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

    5. Applied sqrt-div61.7

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

    6. Simplified47.4

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    7. Taylor expanded around -inf 18.8

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}{\sqrt{\color{blue}{-2 \cdot re}}}\]

    if -1.1292868428778451e+139 < re < -2.053769551615154e-273

    1. Initial program 39.8

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

    3. Applied flip-+39.7

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

    4. Applied associate-*r/39.7

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

    5. Applied sqrt-div39.8

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

    6. Simplified29.6

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity29.6

      \[\leadsto 0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}{\color{blue}{1 \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    9. Applied sqrt-prod29.7

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0} \cdot \sqrt{im \cdot im + 0}}}{1 \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    10. Applied times-frac29.7

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{2.0}}{1} \cdot \frac{\sqrt{im \cdot im + 0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)}\]

    11. Simplified29.7

      \[\leadsto 0.5 \cdot \left(\color{blue}{\sqrt{2.0}} \cdot \frac{\sqrt{im \cdot im + 0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\]

    12. Simplified19.7

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

    14. Applied *-un-lft-identity19.7

      \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \frac{\left|im\right|}{\color{blue}{1 \cdot \sqrt{\sqrt{im \cdot im + re \cdot re} - re}}}\right)\]

    15. Applied add-sqr-sqrt19.7

      \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \frac{\color{blue}{\sqrt{\left|im\right|} \cdot \sqrt{\left|im\right|}}}{1 \cdot \sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\right)\]

    16. Applied times-frac19.7

      \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \color{blue}{\left(\frac{\sqrt{\left|im\right|}}{1} \cdot \frac{\sqrt{\left|im\right|}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\right)}\right)\]

    17. Applied associate-*r*19.7

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt{2.0} \cdot \frac{\sqrt{\left|im\right|}}{1}\right) \cdot \frac{\sqrt{\left|im\right|}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\right)}\]

    18. Simplified19.7

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

    if -2.053769551615154e-273 < re < 7.239807700907349e-222

    1. Initial program 30.6

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

    2. Taylor expanded around 0 32.1

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

    if 7.239807700907349e-222 < re < 1.2290590931535932e+68

    1. Initial program 18.4

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

    3. Applied add-sqr-sqrt18.5

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

    if 1.2290590931535932e+68 < re

    1. Initial program 43.1

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

    2. Taylor expanded around inf 10.5

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification18.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1292868428778451 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{-2 \cdot re}} \cdot 0.5\\ \mathbf{elif}\;re \le -2.053769551615154 \cdot 10^{-273}:\\ \;\;\;\;\left(\frac{\sqrt{\left|im\right|}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot \left(\sqrt{\left|im\right|} \cdot \sqrt{2.0}\right)\right) \cdot 0.5\\ \mathbf{elif}\;re \le 7.239807700907349 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{\left(im + re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 1.2290590931535932 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + \sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (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)))))