Average Error: 38.1 → 20.4
Time: 15.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.05252701517827769261902701918403618702 \cdot 10^{150}:\\ \;\;\;\;\left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{re \cdot -2}}\right) \cdot 0.5\\ \mathbf{elif}\;re \le 1.859645186058447714528239489016906729797 \cdot 10^{-128}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{2}{\sqrt{im \cdot im + re \cdot re} - re}} \cdot \left|im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.05252701517827769261902701918403618702 \cdot 10^{150}:\\
\;\;\;\;\left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{re \cdot -2}}\right) \cdot 0.5\\

\mathbf{elif}\;re \le 1.859645186058447714528239489016906729797 \cdot 10^{-128}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{\frac{2}{\sqrt{im \cdot im + re \cdot re} - re}} \cdot \left|im\right|\right)\\

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

\end{array}
double f(double re, double im) {
        double r7597062 = 0.5;
        double r7597063 = 2.0;
        double r7597064 = re;
        double r7597065 = r7597064 * r7597064;
        double r7597066 = im;
        double r7597067 = r7597066 * r7597066;
        double r7597068 = r7597065 + r7597067;
        double r7597069 = sqrt(r7597068);
        double r7597070 = r7597069 + r7597064;
        double r7597071 = r7597063 * r7597070;
        double r7597072 = sqrt(r7597071);
        double r7597073 = r7597062 * r7597072;
        return r7597073;
}
↓
double f(double re, double im) {
        double r7597074 = re;
        double r7597075 = -6.052527015178278e+150;
        bool r7597076 = r7597074 <= r7597075;
        double r7597077 = im;
        double r7597078 = fabs(r7597077);
        double r7597079 = 2.0;
        double r7597080 = sqrt(r7597079);
        double r7597081 = -2.0;
        double r7597082 = r7597074 * r7597081;
        double r7597083 = sqrt(r7597082);
        double r7597084 = r7597080 / r7597083;
        double r7597085 = r7597078 * r7597084;
        double r7597086 = 0.5;
        double r7597087 = r7597085 * r7597086;
        double r7597088 = 1.8596451860584477e-128;
        bool r7597089 = r7597074 <= r7597088;
        double r7597090 = r7597077 * r7597077;
        double r7597091 = r7597074 * r7597074;
        double r7597092 = r7597090 + r7597091;
        double r7597093 = sqrt(r7597092);
        double r7597094 = r7597093 - r7597074;
        double r7597095 = r7597079 / r7597094;
        double r7597096 = sqrt(r7597095);
        double r7597097 = r7597096 * r7597078;
        double r7597098 = r7597086 * r7597097;
        double r7597099 = r7597074 + r7597074;
        double r7597100 = r7597079 * r7597099;
        double r7597101 = sqrt(r7597100);
        double r7597102 = r7597086 * r7597101;
        double r7597103 = r7597089 ? r7597098 : r7597102;
        double r7597104 = r7597076 ? r7597087 : r7597103;
        return r7597104;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.1
Target33.2
Herbie20.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -6.052527015178278e+150

    1. Initial program 63.7

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

    3. Applied flip-+63.7

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

    4. Applied associate-*r/63.7

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

    5. Applied sqrt-div63.7

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

    6. Simplified50.6

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

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

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

    9. Applied sqrt-prod50.6

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

    10. Applied sqrt-prod50.6

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

    11. Applied times-frac50.6

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

    12. Simplified50.1

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

    13. Taylor expanded around -inf 8.7

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

    if -6.052527015178278e+150 < re < 1.8596451860584477e-128

    1. Initial program 35.4

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

    3. Applied flip-+36.4

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

    4. Applied associate-*r/36.4

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

    5. Applied sqrt-div36.7

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

    6. Simplified29.6

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

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

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

    9. Applied sqrt-prod29.6

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

    10. Applied sqrt-prod29.7

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

    11. Applied times-frac29.7

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

    12. Simplified23.1

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

    14. Applied sqrt-undiv22.9

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

    if 1.8596451860584477e-128 < re

    1. Initial program 33.0

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

    2. Taylor expanded around inf 20.9

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

  4. Final simplification20.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.05252701517827769261902701918403618702 \cdot 10^{150}:\\ \;\;\;\;\left(\left|im\right| \cdot \frac{\sqrt{2}}{\sqrt{re \cdot -2}}\right) \cdot 0.5\\ \mathbf{elif}\;re \le 1.859645186058447714528239489016906729797 \cdot 10^{-128}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{2}{\sqrt{im \cdot im + re \cdot re} - re}} \cdot \left|im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

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)))))