Average Error: 38.4 → 25.8
Time: 34.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.60085321524529635506480666887989434939 \cdot 10^{111}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le 1.569656924578216529544264149613932691613 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) - re\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.60085321524529635506480666887989434939 \cdot 10^{111}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\

\mathbf{elif}\;re \le 1.569656924578216529544264149613932691613 \cdot 10^{-300}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) - re\right) \cdot 2}\\

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

\end{array}
double f(double re, double im) {
        double r934059 = 0.5;
        double r934060 = 2.0;
        double r934061 = re;
        double r934062 = r934061 * r934061;
        double r934063 = im;
        double r934064 = r934063 * r934063;
        double r934065 = r934062 + r934064;
        double r934066 = sqrt(r934065);
        double r934067 = r934066 - r934061;
        double r934068 = r934060 * r934067;
        double r934069 = sqrt(r934068);
        double r934070 = r934059 * r934069;
        return r934070;
}
↓
double f(double re, double im) {
        double r934071 = re;
        double r934072 = -2.6008532152452964e+111;
        bool r934073 = r934071 <= r934072;
        double r934074 = -2.0;
        double r934075 = r934074 * r934071;
        double r934076 = 2.0;
        double r934077 = r934075 * r934076;
        double r934078 = sqrt(r934077);
        double r934079 = 0.5;
        double r934080 = r934078 * r934079;
        double r934081 = 1.5696569245782165e-300;
        bool r934082 = r934071 <= r934081;
        double r934083 = im;
        double r934084 = r934083 * r934083;
        double r934085 = r934071 * r934071;
        double r934086 = r934084 + r934085;
        double r934087 = sqrt(r934086);
        double r934088 = sqrt(r934087);
        double r934089 = sqrt(r934088);
        double r934090 = r934089 * r934088;
        double r934091 = r934089 * r934090;
        double r934092 = r934091 - r934071;
        double r934093 = r934092 * r934076;
        double r934094 = sqrt(r934093);
        double r934095 = r934079 * r934094;
        double r934096 = r934076 * r934084;
        double r934097 = sqrt(r934096);
        double r934098 = r934087 + r934071;
        double r934099 = sqrt(r934098);
        double r934100 = r934097 / r934099;
        double r934101 = r934079 * r934100;
        double r934102 = r934082 ? r934095 : r934101;
        double r934103 = r934073 ? r934080 : r934102;
        return r934103;
}

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 < -2.6008532152452964e+111

    1. Initial program 53.2

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

    2. Taylor expanded around -inf 9.0

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

    if -2.6008532152452964e+111 < re < 1.5696569245782165e-300

    1. Initial program 21.8

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

    3. Applied add-sqr-sqrt21.8

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

    4. Applied sqrt-prod21.8

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

    6. Applied add-sqr-sqrt21.8

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

    7. Applied sqrt-prod21.8

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

    8. Applied sqrt-prod21.9

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

    9. Applied associate-*r*21.9

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

    if 1.5696569245782165e-300 < re

    1. Initial program 45.3

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

    3. Applied flip--45.2

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

    4. Applied associate-*r/45.2

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

    5. Applied sqrt-div45.3

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

    6. Simplified34.0

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

  4. Final simplification25.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.60085321524529635506480666887989434939 \cdot 10^{111}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le 1.569656924578216529544264149613932691613 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{\sqrt{im \cdot im + re \cdot re}}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}}\right) - re\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))