Average Error: 38.9 → 28.1
Time: 5.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.2223700059947939 \cdot 10^{102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{-1 \cdot re - re}\right)}\\ \mathbf{elif}\;re \le -1.5528913295942832 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\left|im\right| \cdot \left(\sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\right) \cdot \sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}}\right)}\\ \mathbf{elif}\;re \le 2.730492367188375 \cdot 10^{28}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.2223700059947939 \cdot 10^{102}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{-1 \cdot re - re}\right)}\\

\mathbf{elif}\;re \le -1.5528913295942832 \cdot 10^{-276}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\left|im\right| \cdot \left(\sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\right) \cdot \sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}}\right)}\\

\mathbf{elif}\;re \le 2.730492367188375 \cdot 10^{28}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

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

\end{array}
double f(double re, double im) {
        double r138893 = 0.5;
        double r138894 = 2.0;
        double r138895 = re;
        double r138896 = r138895 * r138895;
        double r138897 = im;
        double r138898 = r138897 * r138897;
        double r138899 = r138896 + r138898;
        double r138900 = sqrt(r138899);
        double r138901 = r138900 + r138895;
        double r138902 = r138894 * r138901;
        double r138903 = sqrt(r138902);
        double r138904 = r138893 * r138903;
        return r138904;
}
↓
double f(double re, double im) {
        double r138905 = re;
        double r138906 = -1.2223700059947939e+102;
        bool r138907 = r138905 <= r138906;
        double r138908 = 0.5;
        double r138909 = 2.0;
        double r138910 = im;
        double r138911 = fabs(r138910);
        double r138912 = -1.0;
        double r138913 = r138912 * r138905;
        double r138914 = r138913 - r138905;
        double r138915 = r138911 / r138914;
        double r138916 = r138911 * r138915;
        double r138917 = r138909 * r138916;
        double r138918 = sqrt(r138917);
        double r138919 = r138908 * r138918;
        double r138920 = -1.5528913295942832e-276;
        bool r138921 = r138905 <= r138920;
        double r138922 = r138905 * r138905;
        double r138923 = r138910 * r138910;
        double r138924 = r138922 + r138923;
        double r138925 = sqrt(r138924);
        double r138926 = r138925 - r138905;
        double r138927 = r138911 / r138926;
        double r138928 = cbrt(r138927);
        double r138929 = r138928 * r138928;
        double r138930 = r138911 * r138929;
        double r138931 = r138930 * r138928;
        double r138932 = r138909 * r138931;
        double r138933 = sqrt(r138932);
        double r138934 = r138908 * r138933;
        double r138935 = 2.730492367188375e+28;
        bool r138936 = r138905 <= r138935;
        double r138937 = r138905 + r138910;
        double r138938 = r138909 * r138937;
        double r138939 = sqrt(r138938);
        double r138940 = r138908 * r138939;
        double r138941 = 2.0;
        double r138942 = r138941 * r138905;
        double r138943 = r138909 * r138942;
        double r138944 = sqrt(r138943);
        double r138945 = r138908 * r138944;
        double r138946 = r138936 ? r138940 : r138945;
        double r138947 = r138921 ? r138934 : r138946;
        double r138948 = r138907 ? r138919 : r138947;
        return r138948;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.9
Target34.0
Herbie28.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -1.2223700059947939e+102

    1. Initial program 61.8

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

    3. Applied flip-+61.8

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

    4. Simplified46.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity46.7

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

    7. Applied add-sqr-sqrt46.7

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

    8. Applied times-frac46.7

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

    9. Simplified46.7

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

    10. Simplified46.3

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

    11. Taylor expanded around -inf 26.6

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

    if -1.2223700059947939e+102 < re < -1.5528913295942832e-276

    1. Initial program 39.9

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

    3. Applied flip-+39.7

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

    4. Simplified31.8

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity31.8

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

    7. Applied add-sqr-sqrt31.8

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

    8. Applied times-frac31.8

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

    9. Simplified31.7

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

    10. Simplified29.3

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

    12. Applied add-cube-cbrt29.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}}\right) \cdot \sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}}\right)}\right)}\]

    13. Applied associate-*r*29.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\left|im\right| \cdot \left(\sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\right) \cdot \sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}}\right)}}\]

    if -1.5528913295942832e-276 < re < 2.730492367188375e+28

    1. Initial program 23.3

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

    2. Taylor expanded around 0 37.2

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

    if 2.730492367188375e+28 < re

    1. Initial program 42.4

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

    2. Taylor expanded around inf 14.8

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

  4. Final simplification28.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.2223700059947939 \cdot 10^{102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{-1 \cdot re - re}\right)}\\ \mathbf{elif}\;re \le -1.5528913295942832 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\left|im\right| \cdot \left(\sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\right) \cdot \sqrt[3]{\frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} - re}}\right)}\\ \mathbf{elif}\;re \le 2.730492367188375 \cdot 10^{28}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(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)))))