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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -5.77763001913738 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-\left(re + im\right)\right) \cdot 2.0}\\ \mathbf{elif}\;im \le -5.064661185225939 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}} \cdot 0.5\\ \mathbf{elif}\;im \le 3.878550394816133 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re \cdot -2\right)} \cdot 0.5\\ \mathbf{elif}\;im \le 2.6002124496209947 \cdot 10^{+64}:\\ \;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]

double f(double re, double im) {
        double r1287058 = 0.5;
        double r1287059 = 2.0;
        double r1287060 = re;
        double r1287061 = r1287060 * r1287060;
        double r1287062 = im;
        double r1287063 = r1287062 * r1287062;
        double r1287064 = r1287061 + r1287063;
        double r1287065 = sqrt(r1287064);
        double r1287066 = r1287065 - r1287060;
        double r1287067 = r1287059 * r1287066;
        double r1287068 = sqrt(r1287067);
        double r1287069 = r1287058 * r1287068;
        return r1287069;
}

↓

double f(double re, double im) {
        double r1287070 = im;
        double r1287071 = -5.77763001913738e+77;
        bool r1287072 = r1287070 <= r1287071;
        double r1287073 = 0.5;
        double r1287074 = re;
        double r1287075 = r1287074 + r1287070;
        double r1287076 = -r1287075;
        double r1287077 = 2.0;
        double r1287078 = r1287076 * r1287077;
        double r1287079 = sqrt(r1287078);
        double r1287080 = r1287073 * r1287079;
        double r1287081 = -5.064661185225939e-112;
        bool r1287082 = r1287070 <= r1287081;
        double r1287083 = r1287070 * r1287070;
        double r1287084 = r1287077 * r1287083;
        double r1287085 = sqrt(r1287084);
        double r1287086 = r1287074 * r1287074;
        double r1287087 = r1287083 + r1287086;
        double r1287088 = sqrt(r1287087);
        double r1287089 = r1287088 + r1287074;
        double r1287090 = sqrt(r1287089);
        double r1287091 = r1287085 / r1287090;
        double r1287092 = r1287091 * r1287073;
        double r1287093 = 3.878550394816133e-144;
        bool r1287094 = r1287070 <= r1287093;
        double r1287095 = -2.0;
        double r1287096 = r1287074 * r1287095;
        double r1287097 = r1287077 * r1287096;
        double r1287098 = sqrt(r1287097);
        double r1287099 = r1287098 * r1287073;
        double r1287100 = 2.6002124496209947e+64;
        bool r1287101 = r1287070 <= r1287100;
        double r1287102 = r1287070 - r1287074;
        double r1287103 = r1287102 * r1287077;
        double r1287104 = sqrt(r1287103);
        double r1287105 = r1287104 * r1287073;
        double r1287106 = r1287101 ? r1287092 : r1287105;
        double r1287107 = r1287094 ? r1287099 : r1287106;
        double r1287108 = r1287082 ? r1287092 : r1287107;
        double r1287109 = r1287072 ? r1287080 : r1287108;
        return r1287109;
}

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

↓

\begin{array}{l}
\mathbf{if}\;im \le -5.77763001913738 \cdot 10^{+77}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(-\left(re + im\right)\right) \cdot 2.0}\\

\mathbf{elif}\;im \le -5.064661185225939 \cdot 10^{-112}:\\
\;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}} \cdot 0.5\\

\mathbf{elif}\;im \le 3.878550394816133 \cdot 10^{-144}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(re \cdot -2\right)} \cdot 0.5\\

\mathbf{elif}\;im \le 2.6002124496209947 \cdot 10^{+64}:\\
\;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}} \cdot 0.5\\

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

\end{array}

Derivation

Split input into 4 regimes
if im < -5.77763001913738e+77
1. Initial program 46.8
  \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
2. Using strategy rm
3. Applied +-commutative46.8
  \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{im \cdot im + re \cdot re}} - re\right)}\]
4. Taylor expanded around -inf 10.3
  \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(-\left(re + im\right)\right)}}\]
if -5.77763001913738e+77 < im < -5.064661185225939e-112 or 3.878550394816133e-144 < im < 2.6002124496209947e+64
1. Initial program 24.1
  \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
2. Using strategy rm
3. Applied flip--33.6
  \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]
4. Applied associate-*r/33.6
  \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} + re}}}\]
5. Applied sqrt-div33.7
  \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]
6. Simplified25.5
  \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
if -5.064661185225939e-112 < im < 3.878550394816133e-144
1. Initial program 39.9
  \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
2. Taylor expanded around -inf 35.7
  \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(-2 \cdot re\right)}}\]
if 2.6002124496209947e+64 < im
1. Initial program 46.0
  \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
2. Using strategy rm
3. Applied +-commutative46.0
  \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{im \cdot im + re \cdot re}} - re\right)}\]
4. Taylor expanded around inf 11.0
  \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(im - re\right)}}\]
Recombined 4 regimes into one program.
Final simplification22.9
\[\leadsto \begin{array}{l} \mathbf{if}\;im \le -5.77763001913738 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-\left(re + im\right)\right) \cdot 2.0}\\ \mathbf{elif}\;im \le -5.064661185225939 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}} \cdot 0.5\\ \mathbf{elif}\;im \le 3.878550394816133 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re \cdot -2\right)} \cdot 0.5\\ \mathbf{elif}\;im \le 2.6002124496209947 \cdot 10^{+64}:\\ \;\;\;\;\frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(im - re\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]

Error

Derivation

`if im < -5.77763001913738e+77`

`if -5.77763001913738e+77 < im < -5.064661185225939e-112 or 3.878550394816133e-144 < im < 2.6002124496209947e+64`

`if -5.064661185225939e-112 < im < 3.878550394816133e-144`

`if 2.6002124496209947e+64 < im`

Reproduce