Average Error: 37.3 → 26.9
Time: 18.9s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.0974932438808633 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le -4.4945327826415316 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(im + re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le -7.961223836723572 \cdot 10^{-96}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le -2.538815066158378 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(im + re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 1.8791426213625292 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + re\right)} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.0974932438808633 \cdot 10^{+26}:\\
\;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\

\mathbf{elif}\;re \le -4.4945327826415316 \cdot 10^{-20}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(im + re\right)} \cdot 0.5\\

\mathbf{elif}\;re \le -7.961223836723572 \cdot 10^{-96}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re}}\\

\mathbf{elif}\;re \le -2.538815066158378 \cdot 10^{-267}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(im + re\right)} \cdot 0.5\\

\mathbf{elif}\;re \le 1.8791426213625292 \cdot 10^{+66}:\\
\;\;\;\;\sqrt{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot 2.0} \cdot 0.5\\

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

\end{array}
double f(double re, double im) {
        double r6645803 = 0.5;
        double r6645804 = 2.0;
        double r6645805 = re;
        double r6645806 = r6645805 * r6645805;
        double r6645807 = im;
        double r6645808 = r6645807 * r6645807;
        double r6645809 = r6645806 + r6645808;
        double r6645810 = sqrt(r6645809);
        double r6645811 = r6645810 + r6645805;
        double r6645812 = r6645804 * r6645811;
        double r6645813 = sqrt(r6645812);
        double r6645814 = r6645803 * r6645813;
        return r6645814;
}
↓
double f(double re, double im) {
        double r6645815 = re;
        double r6645816 = -1.0974932438808633e+26;
        bool r6645817 = r6645815 <= r6645816;
        double r6645818 = im;
        double r6645819 = r6645818 * r6645818;
        double r6645820 = 2.0;
        double r6645821 = r6645819 * r6645820;
        double r6645822 = sqrt(r6645821);
        double r6645823 = r6645815 * r6645815;
        double r6645824 = r6645819 + r6645823;
        double r6645825 = sqrt(r6645824);
        double r6645826 = r6645825 - r6645815;
        double r6645827 = sqrt(r6645826);
        double r6645828 = r6645822 / r6645827;
        double r6645829 = 0.5;
        double r6645830 = r6645828 * r6645829;
        double r6645831 = -4.4945327826415316e-20;
        bool r6645832 = r6645815 <= r6645831;
        double r6645833 = r6645818 + r6645815;
        double r6645834 = r6645820 * r6645833;
        double r6645835 = sqrt(r6645834);
        double r6645836 = r6645835 * r6645829;
        double r6645837 = -7.961223836723572e-96;
        bool r6645838 = r6645815 <= r6645837;
        double r6645839 = r6645819 / r6645826;
        double r6645840 = r6645820 * r6645839;
        double r6645841 = sqrt(r6645840);
        double r6645842 = r6645829 * r6645841;
        double r6645843 = -2.538815066158378e-267;
        bool r6645844 = r6645815 <= r6645843;
        double r6645845 = 1.8791426213625292e+66;
        bool r6645846 = r6645815 <= r6645845;
        double r6645847 = r6645815 + r6645825;
        double r6645848 = r6645847 * r6645820;
        double r6645849 = sqrt(r6645848);
        double r6645850 = r6645849 * r6645829;
        double r6645851 = r6645815 + r6645815;
        double r6645852 = r6645820 * r6645851;
        double r6645853 = sqrt(r6645852);
        double r6645854 = r6645853 * r6645829;
        double r6645855 = r6645846 ? r6645850 : r6645854;
        double r6645856 = r6645844 ? r6645836 : r6645855;
        double r6645857 = r6645838 ? r6645842 : r6645856;
        double r6645858 = r6645832 ? r6645836 : r6645857;
        double r6645859 = r6645817 ? r6645830 : r6645858;
        return r6645859;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target32.5
Herbie26.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -1.0974932438808633e+26

    1. Initial program 56.7

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

    3. Applied flip-+56.7

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

    4. Applied associate-*r/56.7

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

    5. Applied sqrt-div56.7

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

    6. Simplified38.8

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

    if -1.0974932438808633e+26 < re < -4.4945327826415316e-20 or -7.961223836723572e-96 < re < -2.538815066158378e-267

    1. Initial program 33.8

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

    2. Taylor expanded around 0 38.9

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

    if -4.4945327826415316e-20 < re < -7.961223836723572e-96

    1. Initial program 38.9

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

    3. Applied flip-+38.9

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

    4. Simplified28.1

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

    if -2.538815066158378e-267 < re < 1.8791426213625292e+66

    1. Initial program 21.6

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

    if 1.8791426213625292e+66 < re

    1. Initial program 45.6

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

    2. Taylor expanded around inf 11.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification26.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.0974932438808633 \cdot 10^{+26}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}} \cdot 0.5\\ \mathbf{elif}\;re \le -4.4945327826415316 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(im + re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le -7.961223836723572 \cdot 10^{-96}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le -2.538815066158378 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(im + re\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 1.8791426213625292 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{\left(re + \sqrt{im \cdot im + re \cdot re}\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + re\right)} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))))