Average Error: 38.1 → 25.9
Time: 17.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.953163933293596454341424469878526728026 \cdot 10^{119}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le 1.416987785013479855894798143638187490549 \cdot 10^{-250}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.953163933293596454341424469878526728026 \cdot 10^{119}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;re \le 1.416987785013479855894798143638187490549 \cdot 10^{-250}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} - re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r22946 = 0.5;
        double r22947 = 2.0;
        double r22948 = re;
        double r22949 = r22948 * r22948;
        double r22950 = im;
        double r22951 = r22950 * r22950;
        double r22952 = r22949 + r22951;
        double r22953 = sqrt(r22952);
        double r22954 = r22953 - r22948;
        double r22955 = r22947 * r22954;
        double r22956 = sqrt(r22955);
        double r22957 = r22946 * r22956;
        return r22957;
}
↓
double f(double re, double im) {
        double r22958 = re;
        double r22959 = -8.953163933293596e+119;
        bool r22960 = r22958 <= r22959;
        double r22961 = 0.5;
        double r22962 = 2.0;
        double r22963 = -2.0;
        double r22964 = r22963 * r22958;
        double r22965 = r22962 * r22964;
        double r22966 = sqrt(r22965);
        double r22967 = r22961 * r22966;
        double r22968 = 1.41698778501348e-250;
        bool r22969 = r22958 <= r22968;
        double r22970 = r22958 * r22958;
        double r22971 = im;
        double r22972 = r22971 * r22971;
        double r22973 = r22970 + r22972;
        double r22974 = 0.5;
        double r22975 = pow(r22973, r22974);
        double r22976 = r22975 - r22958;
        double r22977 = r22962 * r22976;
        double r22978 = sqrt(r22977);
        double r22979 = r22961 * r22978;
        double r22980 = r22962 * r22972;
        double r22981 = sqrt(r22980);
        double r22982 = sqrt(r22973);
        double r22983 = r22982 + r22958;
        double r22984 = sqrt(r22983);
        double r22985 = r22981 / r22984;
        double r22986 = r22961 * r22985;
        double r22987 = r22969 ? r22979 : r22986;
        double r22988 = r22960 ? r22967 : r22987;
        return r22988;
}

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 < -8.953163933293596e+119

    1. Initial program 55.7

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

    2. Taylor expanded around -inf 9.4

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

    if -8.953163933293596e+119 < re < 1.41698778501348e-250

    1. Initial program 21.3

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

    3. Applied add-sqr-sqrt21.3

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

    4. Applied sqrt-prod21.4

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

    6. Applied pow1/221.4

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

    7. Applied sqrt-pow121.4

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

    8. Applied pow1/221.4

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

    9. Applied sqrt-pow121.4

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

    10. Applied pow-prod-up21.3

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

    11. Simplified21.3

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

    if 1.41698778501348e-250 < re

    1. Initial program 47.4

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

    3. Applied flip--47.3

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

    4. Applied associate-*r/47.4

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

    5. Applied sqrt-div47.4

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

    6. Simplified35.7

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

  4. Final simplification25.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.953163933293596454341424469878526728026 \cdot 10^{119}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le 1.416987785013479855894798143638187490549 \cdot 10^{-250}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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)))))