Average Error: 37.0 → 26.4
Time: 18.9s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.471563953414917 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le -8.308754597855719 \cdot 10^{-213}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 9.946851897479392 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} + re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.471563953414917 \cdot 10^{-82}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\

\mathbf{elif}\;re \le -8.308754597855719 \cdot 10^{-213}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\

\mathbf{elif}\;re \le 9.946851897479392 \cdot 10^{+107}:\\
\;\;\;\;\sqrt{\left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} + re\right) \cdot 2.0} \cdot 0.5\\

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

\end{array}
double f(double re, double im) {
        double r8152744 = 0.5;
        double r8152745 = 2.0;
        double r8152746 = re;
        double r8152747 = r8152746 * r8152746;
        double r8152748 = im;
        double r8152749 = r8152748 * r8152748;
        double r8152750 = r8152747 + r8152749;
        double r8152751 = sqrt(r8152750);
        double r8152752 = r8152751 + r8152746;
        double r8152753 = r8152745 * r8152752;
        double r8152754 = sqrt(r8152753);
        double r8152755 = r8152744 * r8152754;
        return r8152755;
}
↓
double f(double re, double im) {
        double r8152756 = re;
        double r8152757 = -3.471563953414917e-82;
        bool r8152758 = r8152756 <= r8152757;
        double r8152759 = 0.5;
        double r8152760 = 2.0;
        double r8152761 = im;
        double r8152762 = r8152761 * r8152761;
        double r8152763 = r8152760 * r8152762;
        double r8152764 = sqrt(r8152763);
        double r8152765 = r8152756 * r8152756;
        double r8152766 = r8152762 + r8152765;
        double r8152767 = sqrt(r8152766);
        double r8152768 = r8152767 - r8152756;
        double r8152769 = sqrt(r8152768);
        double r8152770 = r8152764 / r8152769;
        double r8152771 = r8152759 * r8152770;
        double r8152772 = -8.308754597855719e-213;
        bool r8152773 = r8152756 <= r8152772;
        double r8152774 = r8152761 + r8152756;
        double r8152775 = r8152760 * r8152774;
        double r8152776 = sqrt(r8152775);
        double r8152777 = r8152759 * r8152776;
        double r8152778 = 9.946851897479392e+107;
        bool r8152779 = r8152756 <= r8152778;
        double r8152780 = sqrt(r8152767);
        double r8152781 = r8152780 * r8152780;
        double r8152782 = r8152781 + r8152756;
        double r8152783 = r8152782 * r8152760;
        double r8152784 = sqrt(r8152783);
        double r8152785 = r8152784 * r8152759;
        double r8152786 = r8152756 + r8152756;
        double r8152787 = r8152786 * r8152760;
        double r8152788 = sqrt(r8152787);
        double r8152789 = r8152788 * r8152759;
        double r8152790 = r8152779 ? r8152785 : r8152789;
        double r8152791 = r8152773 ? r8152777 : r8152790;
        double r8152792 = r8152758 ? r8152771 : r8152791;
        return r8152792;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target32.4
Herbie26.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -3.471563953414917e-82

    1. Initial program 53.2

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

    3. Applied flip-+53.2

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

    4. Applied associate-*r/53.2

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

    5. Applied sqrt-div53.3

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

    6. Simplified37.4

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

    if -3.471563953414917e-82 < re < -8.308754597855719e-213

    1. Initial program 33.8

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

    3. Applied add-sqr-sqrt33.8

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

    4. Applied sqrt-prod33.9

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

    6. Applied *-un-lft-identity33.9

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

    7. Applied sqrt-prod33.9

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

    8. Applied associate-*r*33.9

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

    9. Simplified33.9

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

    11. Applied add-sqr-sqrt33.9

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

    12. Applied sqrt-prod34.0

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

    13. Applied associate-*r*34.0

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

    14. Taylor expanded around 0 39.7

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

    if -8.308754597855719e-213 < re < 9.946851897479392e+107

    1. Initial program 20.9

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

    3. Applied add-sqr-sqrt20.9

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

    4. Applied sqrt-prod21.0

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

    6. Applied *-un-lft-identity21.0

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

    7. Applied sqrt-prod21.0

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

    8. Applied associate-*r*21.0

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

    9. Simplified21.0

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

    if 9.946851897479392e+107 < re

    1. Initial program 50.1

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

    3. Applied add-sqr-sqrt50.1

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

    4. Applied sqrt-prod50.1

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

    5. Taylor expanded around inf 9.8

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.471563953414917 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le -8.308754597855719 \cdot 10^{-213}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\ \mathbf{elif}\;re \le 9.946851897479392 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{\left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} + re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(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)))))