Average Error: 38.8 → 23.0
Time: 20.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.0368624680251324307491276130279703668 \cdot 10^{161}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-2 \cdot re}\right)}\\ \mathbf{elif}\;re \le -2.180790474222033898637506221995594051551 \cdot 10^{-229}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} - re}\right)}\\ \mathbf{elif}\;re \le -5.338971335212211404004695596415328758017 \cdot 10^{-289}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 2.076008095509726135706521303221876391822 \cdot 10^{76}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.0368624680251324307491276130279703668 \cdot 10^{161}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-2 \cdot re}\right)}\\

\mathbf{elif}\;re \le -2.180790474222033898637506221995594051551 \cdot 10^{-229}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} - re}\right)}\\

\mathbf{elif}\;re \le -5.338971335212211404004695596415328758017 \cdot 10^{-289}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \le 2.076008095509726135706521303221876391822 \cdot 10^{76}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\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 r168894 = 0.5;
        double r168895 = 2.0;
        double r168896 = re;
        double r168897 = r168896 * r168896;
        double r168898 = im;
        double r168899 = r168898 * r168898;
        double r168900 = r168897 + r168899;
        double r168901 = sqrt(r168900);
        double r168902 = r168901 + r168896;
        double r168903 = r168895 * r168902;
        double r168904 = sqrt(r168903);
        double r168905 = r168894 * r168904;
        return r168905;
}
↓
double f(double re, double im) {
        double r168906 = re;
        double r168907 = -3.0368624680251324e+161;
        bool r168908 = r168906 <= r168907;
        double r168909 = 0.5;
        double r168910 = 2.0;
        double r168911 = im;
        double r168912 = -2.0;
        double r168913 = r168912 * r168906;
        double r168914 = r168911 / r168913;
        double r168915 = r168911 * r168914;
        double r168916 = r168910 * r168915;
        double r168917 = sqrt(r168916);
        double r168918 = r168909 * r168917;
        double r168919 = -2.180790474222034e-229;
        bool r168920 = r168906 <= r168919;
        double r168921 = r168906 * r168906;
        double r168922 = r168911 * r168911;
        double r168923 = r168921 + r168922;
        double r168924 = sqrt(r168923);
        double r168925 = r168924 - r168906;
        double r168926 = r168911 / r168925;
        double r168927 = r168911 * r168926;
        double r168928 = r168910 * r168927;
        double r168929 = sqrt(r168928);
        double r168930 = r168909 * r168929;
        double r168931 = -5.338971335212211e-289;
        bool r168932 = r168906 <= r168931;
        double r168933 = r168906 + r168911;
        double r168934 = r168910 * r168933;
        double r168935 = sqrt(r168934);
        double r168936 = r168909 * r168935;
        double r168937 = 2.076008095509726e+76;
        bool r168938 = r168906 <= r168937;
        double r168939 = cbrt(r168924);
        double r168940 = r168939 * r168939;
        double r168941 = r168940 * r168939;
        double r168942 = r168941 + r168906;
        double r168943 = r168910 * r168942;
        double r168944 = sqrt(r168943);
        double r168945 = r168909 * r168944;
        double r168946 = 2.0;
        double r168947 = r168946 * r168906;
        double r168948 = r168910 * r168947;
        double r168949 = sqrt(r168948);
        double r168950 = r168909 * r168949;
        double r168951 = r168938 ? r168945 : r168950;
        double r168952 = r168932 ? r168936 : r168951;
        double r168953 = r168920 ? r168930 : r168952;
        double r168954 = r168908 ? r168918 : r168953;
        return r168954;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.8
Target33.8
Herbie23.0
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -3.0368624680251324e+161

    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. Simplified50.3

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

    6. Applied *-un-lft-identity50.3

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

    7. Applied add-sqr-sqrt57.3

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

    8. Applied unpow-prod-down57.3

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

    9. Applied times-frac57.1

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

    10. Simplified57.1

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

    11. Simplified49.9

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

    12. Taylor expanded around -inf 22.3

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

    if -3.0368624680251324e+161 < re < -2.180790474222034e-229

    1. Initial program 42.3

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

    3. Applied flip-+42.2

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

    4. Simplified31.5

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

    6. Applied *-un-lft-identity31.5

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

    7. Applied add-sqr-sqrt48.0

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

    8. Applied unpow-prod-down48.0

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

    9. Applied times-frac46.6

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

    10. Simplified46.6

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

    11. Simplified29.3

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

    if -2.180790474222034e-229 < re < -5.338971335212211e-289

    1. Initial program 29.5

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

    2. Taylor expanded around 0 32.0

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

    if -5.338971335212211e-289 < re < 2.076008095509726e+76

    1. Initial program 21.8

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

    3. Applied add-cube-cbrt22.2

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

    if 2.076008095509726e+76 < re

    1. Initial program 48.1

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

    3. Applied add-cube-cbrt48.2

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

    4. Taylor expanded around inf 12.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(2 \cdot re\right)}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification23.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.0368624680251324307491276130279703668 \cdot 10^{161}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{-2 \cdot re}\right)}\\ \mathbf{elif}\;re \le -2.180790474222033898637506221995594051551 \cdot 10^{-229}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} - re}\right)}\\ \mathbf{elif}\;re \le -5.338971335212211404004695596415328758017 \cdot 10^{-289}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 2.076008095509726135706521303221876391822 \cdot 10^{76}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(\sqrt[3]{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(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)))))