Average Error: 39.0 → 27.2
Time: 4.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.9648958467837301 \cdot 10^{102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -5.355088044563557 \cdot 10^{-294}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left({\left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}}\right)}^{3} \cdot \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{re + \sqrt{re \cdot re + im \cdot im}}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.9648958467837301 \cdot 10^{102}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;re \le -5.355088044563557 \cdot 10^{-294}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left({\left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}}\right)}^{3} \cdot \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{re + \sqrt{re \cdot re + im \cdot im}}}\\

\end{array}
double f(double re, double im) {
        double r15987 = 0.5;
        double r15988 = 2.0;
        double r15989 = re;
        double r15990 = r15989 * r15989;
        double r15991 = im;
        double r15992 = r15991 * r15991;
        double r15993 = r15990 + r15992;
        double r15994 = sqrt(r15993);
        double r15995 = r15994 - r15989;
        double r15996 = r15988 * r15995;
        double r15997 = sqrt(r15996);
        double r15998 = r15987 * r15997;
        return r15998;
}
↓
double f(double re, double im) {
        double r15999 = re;
        double r16000 = -5.96489584678373e+102;
        bool r16001 = r15999 <= r16000;
        double r16002 = 0.5;
        double r16003 = 2.0;
        double r16004 = -2.0;
        double r16005 = r16004 * r15999;
        double r16006 = r16003 * r16005;
        double r16007 = sqrt(r16006);
        double r16008 = r16002 * r16007;
        double r16009 = -5.355088044563557e-294;
        bool r16010 = r15999 <= r16009;
        double r16011 = r15999 * r15999;
        double r16012 = im;
        double r16013 = r16012 * r16012;
        double r16014 = r16011 + r16013;
        double r16015 = sqrt(r16014);
        double r16016 = sqrt(r16015);
        double r16017 = sqrt(r16016);
        double r16018 = 3.0;
        double r16019 = pow(r16017, r16018);
        double r16020 = r16019 * r16017;
        double r16021 = r16020 - r15999;
        double r16022 = r16003 * r16021;
        double r16023 = sqrt(r16022);
        double r16024 = r16002 * r16023;
        double r16025 = 0.0;
        double r16026 = r16013 + r16025;
        double r16027 = r15999 + r16015;
        double r16028 = r16026 / r16027;
        double r16029 = r16003 * r16028;
        double r16030 = sqrt(r16029);
        double r16031 = r16002 * r16030;
        double r16032 = r16010 ? r16024 : r16031;
        double r16033 = r16001 ? r16008 : r16032;
        return r16033;
}

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 3 regimes

  2. if re < -5.96489584678373e+102

    1. Initial program 53.0

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

    2. Taylor expanded around -inf 10.1

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

    if -5.96489584678373e+102 < re < -5.355088044563557e-294

    1. Initial program 21.2

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

    3. Applied add-sqr-sqrt21.2

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

    4. Applied sqrt-prod21.3

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

    6. Applied add-sqr-sqrt21.3

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

    7. Applied sqrt-prod21.3

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

    8. Applied sqrt-prod21.4

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

    9. Applied associate-*r*21.4

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

    10. Simplified21.5

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

    if -5.355088044563557e-294 < re

    1. Initial program 45.4

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

    3. Applied add-sqr-sqrt45.4

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

    4. Applied sqrt-prod46.1

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

    6. Applied flip--45.9

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

    7. Simplified36.0

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

    8. Simplified36.0

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

  4. Final simplification27.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.9648958467837301 \cdot 10^{102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -5.355088044563557 \cdot 10^{-294}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left({\left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}}\right)}^{3} \cdot \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{re + \sqrt{re \cdot re + im \cdot im}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(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)))))