math.sqrt on complex, real part

Average Error: 38.1 → 24.3

Time: 15.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -2.036766738766978654329181944509117917379 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(-im\right)\right)}\\ \mathbf{elif}\;im \le -5.289548942669333189149718055604630252585 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{2 \cdot \left(re + re\right)} \cdot 0.5\\ \mathbf{elif}\;im \le 3.606820219200127121090676194295429685619 \cdot 10^{-171}:\\ \;\;\;\;0.5 \cdot \left(\frac{\left|im\right|}{\sqrt{re \cdot -2}} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;im \le 1.471945877265292806314351903468873946637 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;im \le 1.11277121648235116127597920377105142481 \cdot 10^{127}:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \frac{\left|im\right| \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right) \cdot 0.5\\ \mathbf{elif}\;im \le 7.340779959888026082952331047777691515948 \cdot 10^{187}:\\ \;\;\;\;\left(\frac{\left|im\right|}{\sqrt{im - re}} \cdot \sqrt{2}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array}\]

C

double f(double re, double im) {
        double r181944 = 0.5;
        double r181945 = 2.0;
        double r181946 = re;
        double r181947 = r181946 * r181946;
        double r181948 = im;
        double r181949 = r181948 * r181948;
        double r181950 = r181947 + r181949;
        double r181951 = sqrt(r181950);
        double r181952 = r181951 + r181946;
        double r181953 = r181945 * r181952;
        double r181954 = sqrt(r181953);
        double r181955 = r181944 * r181954;
        return r181955;
}

↓

double f(double re, double im) {
        double r181956 = im;
        double r181957 = -2.0367667387669787e-24;
        bool r181958 = r181956 <= r181957;
        double r181959 = 0.5;
        double r181960 = 2.0;
        double r181961 = re;
        double r181962 = -r181956;
        double r181963 = r181961 + r181962;
        double r181964 = r181960 * r181963;
        double r181965 = sqrt(r181964);
        double r181966 = r181959 * r181965;
        double r181967 = -5.289548942669333e-269;
        bool r181968 = r181956 <= r181967;
        double r181969 = r181961 + r181961;
        double r181970 = r181960 * r181969;
        double r181971 = sqrt(r181970);
        double r181972 = r181971 * r181959;
        double r181973 = 3.606820219200127e-171;
        bool r181974 = r181956 <= r181973;
        double r181975 = fabs(r181956);
        double r181976 = -2.0;
        double r181977 = r181961 * r181976;
        double r181978 = sqrt(r181977);
        double r181979 = r181975 / r181978;
        double r181980 = sqrt(r181960);
        double r181981 = r181979 * r181980;
        double r181982 = r181959 * r181981;
        double r181983 = 1.4719458772652928e-143;
        bool r181984 = r181956 <= r181983;
        double r181985 = r181961 + r181956;
        double r181986 = r181960 * r181985;
        double r181987 = sqrt(r181986);
        double r181988 = r181959 * r181987;
        double r181989 = 1.1127712164823512e+127;
        bool r181990 = r181956 <= r181989;
        double r181991 = cbrt(r181980);
        double r181992 = r181991 * r181991;
        double r181993 = r181975 * r181991;
        double r181994 = r181961 * r181961;
        double r181995 = r181956 * r181956;
        double r181996 = r181994 + r181995;
        double r181997 = sqrt(r181996);
        double r181998 = r181997 - r181961;
        double r181999 = sqrt(r181998);
        double r182000 = r181993 / r181999;
        double r182001 = r181992 * r182000;
        double r182002 = r182001 * r181959;
        double r182003 = 7.340779959888026e+187;
        bool r182004 = r181956 <= r182003;
        double r182005 = r181956 - r181961;
        double r182006 = sqrt(r182005);
        double r182007 = r181975 / r182006;
        double r182008 = r182007 * r181980;
        double r182009 = r182008 * r181959;
        double r182010 = r182004 ? r182009 : r181988;
        double r182011 = r181990 ? r182002 : r182010;
        double r182012 = r181984 ? r181988 : r182011;
        double r182013 = r181974 ? r181982 : r182012;
        double r182014 = r181968 ? r181972 : r182013;
        double r182015 = r181958 ? r181966 : r182014;
        return r182015;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.1
Target	33.2
Herbie	24.3

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

Derivation

1. Split input into 6 regimes

2. if im < -2.0367667387669787e-24

   1. Initial program 39.8

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

   2. Simplified 39.8

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

   4. Applied add-cube-cbrt 40.0

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

   5. Simplified 40.0

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

   6. Simplified 40.0

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

   7. Taylor expanded around -inf 15.5

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

   8. Simplified 15.5

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

   if -2.0367667387669787e-24 < im < -5.289548942669333e-269

   1. Initial program 34.8

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

   2. Simplified 34.8

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

   4. Applied add-cube-cbrt 35.4

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

   5. Simplified 35.4

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

   6. Simplified 35.4

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

   7. Taylor expanded around 0 40.0

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

   if -5.289548942669333e-269 < im < 3.606820219200127e-171

   1. Initial program 42.4

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

   2. Simplified 42.4

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

   4. Applied flip-+ 59.4

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

   5. Applied associate-*l/ 59.4

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

   6. Applied sqrt-div 59.7

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

   7. Simplified 52.8

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

   8. Simplified 52.8

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

   10. Applied *-un-lft-identity 52.8

       \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}{\sqrt{\color{blue}{1 \cdot \left(\sqrt{im \cdot im + {re}^{2}} - re\right)}}}\]

   11. Applied sqrt-prod 52.8

       \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}{\color{blue}{\sqrt{1} \cdot \sqrt{\sqrt{im \cdot im + {re}^{2}} - re}}}\]

   12. Applied sqrt-prod 52.8

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{im \cdot im + 0}}}{\sqrt{1} \cdot \sqrt{\sqrt{im \cdot im + {re}^{2}} - re}}\]

   13. Applied times-frac 52.8

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{1}} \cdot \frac{\sqrt{im \cdot im + 0}}{\sqrt{\sqrt{im \cdot im + {re}^{2}} - re}}\right)}\]

   14. Simplified 52.8

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

   15. Simplified 39.1

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

   16. Taylor expanded around -inf 34.4

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

   17. Simplified 34.4

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

   if 3.606820219200127e-171 < im < 1.4719458772652928e-143 or 7.340779959888026e+187 < im

   1. Initial program 59.1

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

   2. Simplified 59.1

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

   3. Taylor expanded around 0 12.3

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

   if 1.4719458772652928e-143 < im < 1.1127712164823512e+127

   1. Initial program 24.0

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

   2. Simplified 24.0

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

   4. Applied flip-+ 31.7

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

   5. Applied associate-*l/ 31.7

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

   6. Applied sqrt-div 31.8

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

   7. Simplified 23.5

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

   8. Simplified 23.5

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

   10. Applied *-un-lft-identity 23.5

       \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}{\sqrt{\color{blue}{1 \cdot \left(\sqrt{im \cdot im + {re}^{2}} - re\right)}}}\]

   11. Applied sqrt-prod 23.5

       \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}{\color{blue}{\sqrt{1} \cdot \sqrt{\sqrt{im \cdot im + {re}^{2}} - re}}}\]

   12. Applied sqrt-prod 23.6

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{im \cdot im + 0}}}{\sqrt{1} \cdot \sqrt{\sqrt{im \cdot im + {re}^{2}} - re}}\]

   13. Applied times-frac 23.6

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{1}} \cdot \frac{\sqrt{im \cdot im + 0}}{\sqrt{\sqrt{im \cdot im + {re}^{2}} - re}}\right)}\]

   14. Simplified 23.6

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

   15. Simplified 23.6

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

   17. Applied add-cube-cbrt 23.6

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

   18. Applied associate-*l* 23.6

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

   19. Simplified 23.5

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

   if 1.1127712164823512e+127 < im < 7.340779959888026e+187

   1. Initial program 39.8

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

   2. Simplified 39.8

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

   4. Applied flip-+ 40.1

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

   5. Applied associate-*l/ 40.2

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

   6. Applied sqrt-div 40.4

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

   7. Simplified 39.5

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

   8. Simplified 39.5

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

   10. Applied *-un-lft-identity 39.5

       \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}{\sqrt{\color{blue}{1 \cdot \left(\sqrt{im \cdot im + {re}^{2}} - re\right)}}}\]

   11. Applied sqrt-prod 39.5

       \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}{\color{blue}{\sqrt{1} \cdot \sqrt{\sqrt{im \cdot im + {re}^{2}} - re}}}\]

   12. Applied sqrt-prod 39.4

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{im \cdot im + 0}}}{\sqrt{1} \cdot \sqrt{\sqrt{im \cdot im + {re}^{2}} - re}}\]

   13. Applied times-frac 39.4

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{1}} \cdot \frac{\sqrt{im \cdot im + 0}}{\sqrt{\sqrt{im \cdot im + {re}^{2}} - re}}\right)}\]

   14. Simplified 39.4

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

   15. Simplified 38.6

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

   16. Taylor expanded around 0 15.6

       \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \frac{\left|im\right|}{\sqrt{\color{blue}{im} - re}}\right)\]
3. Recombined 6 regimes into one program.

4. Final simplification 24.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -2.036766738766978654329181944509117917379 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(-im\right)\right)}\\ \mathbf{elif}\;im \le -5.289548942669333189149718055604630252585 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{2 \cdot \left(re + re\right)} \cdot 0.5\\ \mathbf{elif}\;im \le 3.606820219200127121090676194295429685619 \cdot 10^{-171}:\\ \;\;\;\;0.5 \cdot \left(\frac{\left|im\right|}{\sqrt{re \cdot -2}} \cdot \sqrt{2}\right)\\ \mathbf{elif}\;im \le 1.471945877265292806314351903468873946637 \cdot 10^{-143}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;im \le 1.11277121648235116127597920377105142481 \cdot 10^{127}:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \frac{\left|im\right| \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right) \cdot 0.5\\ \mathbf{elif}\;im \le 7.340779959888026082952331047777691515948 \cdot 10^{187}:\\ \;\;\;\;\left(\frac{\left|im\right|}{\sqrt{im - re}} \cdot \sqrt{2}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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)))))