Average Error: 39.0 → 27.2
Time: 10.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.2696195727379345 \cdot 10^{139}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -3.87971466314923503 \cdot 10^{-161}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \le 3.06622232904101925 \cdot 10^{-255}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{{im}^{2} \cdot 2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.2696195727379345 \cdot 10^{139}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;re \le -3.87971466314923503 \cdot 10^{-161}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{elif}\;re \le 3.06622232904101925 \cdot 10^{-255}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{{im}^{2} \cdot 2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\

\end{array}
double f(double re, double im) {
        double r79770 = 0.5;
        double r79771 = 2.0;
        double r79772 = re;
        double r79773 = r79772 * r79772;
        double r79774 = im;
        double r79775 = r79774 * r79774;
        double r79776 = r79773 + r79775;
        double r79777 = sqrt(r79776);
        double r79778 = r79777 - r79772;
        double r79779 = r79771 * r79778;
        double r79780 = sqrt(r79779);
        double r79781 = r79770 * r79780;
        return r79781;
}
↓
double f(double re, double im) {
        double r79782 = re;
        double r79783 = -4.2696195727379345e+139;
        bool r79784 = r79782 <= r79783;
        double r79785 = 0.5;
        double r79786 = 2.0;
        double r79787 = -2.0;
        double r79788 = r79787 * r79782;
        double r79789 = r79786 * r79788;
        double r79790 = sqrt(r79789);
        double r79791 = r79785 * r79790;
        double r79792 = -3.879714663149235e-161;
        bool r79793 = r79782 <= r79792;
        double r79794 = 1.0;
        double r79795 = sqrt(r79794);
        double r79796 = r79782 * r79782;
        double r79797 = im;
        double r79798 = r79797 * r79797;
        double r79799 = r79796 + r79798;
        double r79800 = sqrt(r79799);
        double r79801 = r79795 * r79800;
        double r79802 = r79801 - r79782;
        double r79803 = r79786 * r79802;
        double r79804 = sqrt(r79803);
        double r79805 = r79785 * r79804;
        double r79806 = 3.0662223290410192e-255;
        bool r79807 = r79782 <= r79806;
        double r79808 = r79797 - r79782;
        double r79809 = r79786 * r79808;
        double r79810 = sqrt(r79809);
        double r79811 = r79785 * r79810;
        double r79812 = 2.0;
        double r79813 = pow(r79797, r79812);
        double r79814 = r79813 * r79786;
        double r79815 = sqrt(r79814);
        double r79816 = r79800 + r79782;
        double r79817 = sqrt(r79816);
        double r79818 = r79815 / r79817;
        double r79819 = r79785 * r79818;
        double r79820 = r79807 ? r79811 : r79819;
        double r79821 = r79793 ? r79805 : r79820;
        double r79822 = r79784 ? r79791 : r79821;
        return r79822;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if re < -4.2696195727379345e+139

    1. Initial program 59.5

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

    2. Taylor expanded around -inf 8.4

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

    if -4.2696195727379345e+139 < re < -3.879714663149235e-161

    1. Initial program 16.9

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

    3. Applied add-sqr-sqrt16.9

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

    4. Applied sqrt-prod17.0

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

    6. Applied *-un-lft-identity17.0

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

    7. Applied sqrt-prod17.0

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

    8. Applied associate-*l*17.0

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

    9. Simplified16.9

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

    if -3.879714663149235e-161 < re < 3.0662223290410192e-255

    1. Initial program 31.3

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

    3. Applied add-sqr-sqrt31.3

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

    4. Applied sqrt-prod31.4

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

    6. Applied add-sqr-sqrt31.4

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

    7. Applied sqrt-prod31.4

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

    8. Applied sqrt-prod31.5

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

    9. Applied associate-*r*31.4

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

    10. Simplified31.5

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

    11. Taylor expanded around 0 33.3

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

    if 3.0662223290410192e-255 < re

    1. Initial program 47.7

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

    3. Applied flip--47.6

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

    4. Applied associate-*r/47.6

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

    5. Applied sqrt-div47.7

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

    6. Simplified36.2

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(0 + {im}^{2}\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification27.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.2696195727379345 \cdot 10^{139}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -3.87971466314923503 \cdot 10^{-161}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{1} \cdot \sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \le 3.06622232904101925 \cdot 10^{-255}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{{im}^{2} \cdot 2}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))