_divideComplex, real part

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < +inf.0

   1. Initial program 78.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Step-by-step derivation

      1. *-un-lft-identity78.8%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]

      2. add-sqr-sqrt78.8%

         \[\leadsto \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      3. times-frac78.7%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      4. hypot-def78.7%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      5. fma-def78.7%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      6. hypot-def94.4%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr94.4%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 0.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 42.8%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}} \]
   3. Step-by-step derivation

      1. unpow242.8%

         \[\leadsto \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{\color{blue}{y.re \cdot y.re}} \]

      2. times-frac54.9%

         \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]

   4. Simplified54.9%

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

4. Final simplification86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]

1. Split input into 4 regimes

2. if y.re < -48000

   1. Initial program 48.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Step-by-step derivation

      1. *-un-lft-identity48.9%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]

      2. add-sqr-sqrt48.9%

         \[\leadsto \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      3. times-frac48.9%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      4. hypot-def48.9%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      5. fma-def48.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      6. hypot-def61.3%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr61.3%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   4. Taylor expanded in y.re around -inf 76.3%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-1 \cdot \frac{y.im \cdot x.im}{y.re} + -1 \cdot x.re\right)} \]
   5. Step-by-step derivation

      1. neg-mul-176.3%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-1 \cdot \frac{y.im \cdot x.im}{y.re} + \color{blue}{\left(-x.re\right)}\right) \]

      2. unsub-neg76.3%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-1 \cdot \frac{y.im \cdot x.im}{y.re} - x.re\right)} \]

      3. mul-1-neg76.3%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\left(-\frac{y.im \cdot x.im}{y.re}\right)} - x.re\right) \]

      4. associate-/l*81.5%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(-\color{blue}{\frac{y.im}{\frac{y.re}{x.im}}}\right) - x.re\right) \]

      5. distribute-neg-frac81.5%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{-y.im}{\frac{y.re}{x.im}}} - x.re\right) \]

   6. Simplified81.5%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{-y.im}{\frac{y.re}{x.im}} - x.re\right)} \]

   if -48000 < y.re < 5.00000000000000033e-153

   1. Initial program 72.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 78.7%

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im}} \]
   3. Step-by-step derivation

      1. +-commutative78.7%

         \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]

      2. unpow278.7%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re \cdot y.re}{\color{blue}{y.im \cdot y.im}} \]

      3. associate-/l*78.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{y.im \cdot y.im}{y.re}}} \]

   4. Simplified78.9%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}} \]
   5. Step-by-step derivation

      1. clear-num78.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{1}{\frac{\frac{y.im \cdot y.im}{y.re}}{x.re}}} \]

      2. inv-pow78.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{{\left(\frac{\frac{y.im \cdot y.im}{y.re}}{x.re}\right)}^{-1}} \]

      3. associate-/l*84.5%

         \[\leadsto \frac{x.im}{y.im} + {\left(\frac{\color{blue}{\frac{y.im}{\frac{y.re}{y.im}}}}{x.re}\right)}^{-1} \]

   6. Applied egg-rr84.5%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{{\left(\frac{\frac{y.im}{\frac{y.re}{y.im}}}{x.re}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-184.5%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{1}{\frac{\frac{y.im}{\frac{y.re}{y.im}}}{x.re}}} \]

      2. associate-/l/87.1%

         \[\leadsto \frac{x.im}{y.im} + \frac{1}{\color{blue}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}} \]

   8. Simplified87.1%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}} \]

   if 5.00000000000000033e-153 < y.re < 1.64999999999999995e80

   1. Initial program 78.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   if 1.64999999999999995e80 < y.re

   1. Initial program 41.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 74.8%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}} \]
   3. Step-by-step derivation

      1. unpow274.8%

         \[\leadsto \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{\color{blue}{y.re \cdot y.re}} \]

      2. times-frac85.2%

         \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]

   4. Simplified85.2%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]
3. Recombined 4 regimes into one program.

4. Final simplification84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -48000:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-153}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]

1. Split input into 3 regimes

2. if y.re < -2.6e5 or 1.1000000000000001e82 < y.re

   1. Initial program 45.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 74.5%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}} \]
   3. Step-by-step derivation

      1. unpow274.5%

         \[\leadsto \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{\color{blue}{y.re \cdot y.re}} \]

      2. times-frac83.1%

         \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]

   4. Simplified83.1%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]

   if -2.6e5 < y.re < 6.50000000000000032e-153

   1. Initial program 72.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 78.7%

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im}} \]
   3. Step-by-step derivation

      1. +-commutative78.7%

         \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]

      2. unpow278.7%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re \cdot y.re}{\color{blue}{y.im \cdot y.im}} \]

      3. associate-/l*78.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{y.im \cdot y.im}{y.re}}} \]

   4. Simplified78.9%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}} \]
   5. Step-by-step derivation

      1. clear-num78.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{1}{\frac{\frac{y.im \cdot y.im}{y.re}}{x.re}}} \]

      2. inv-pow78.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{{\left(\frac{\frac{y.im \cdot y.im}{y.re}}{x.re}\right)}^{-1}} \]

      3. associate-/l*84.5%

         \[\leadsto \frac{x.im}{y.im} + {\left(\frac{\color{blue}{\frac{y.im}{\frac{y.re}{y.im}}}}{x.re}\right)}^{-1} \]

   6. Applied egg-rr84.5%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{{\left(\frac{\frac{y.im}{\frac{y.re}{y.im}}}{x.re}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-184.5%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{1}{\frac{\frac{y.im}{\frac{y.re}{y.im}}}{x.re}}} \]

      2. associate-/l/87.1%

         \[\leadsto \frac{x.im}{y.im} + \frac{1}{\color{blue}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}} \]

   8. Simplified87.1%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}} \]

   if 6.50000000000000032e-153 < y.re < 1.1000000000000001e82

   1. Initial program 78.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
3. Recombined 3 regimes into one program.

4. Final simplification84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -260000:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{+82}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]

1. Split input into 3 regimes

2. if y.re < -9.5e19 or 1.09999999999999998e60 < y.re

   1. Initial program 43.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 70.9%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   if -9.5e19 < y.re < -4.5000000000000001e-40 or -1.65e-112 < y.re < 1.09999999999999998e60

   1. Initial program 74.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 67.1%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   if -4.5000000000000001e-40 < y.re < -1.65e-112

   1. Initial program 81.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in x.re around inf 71.1%

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.re}^{2} + {y.im}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l*66.4%

         \[\leadsto \color{blue}{\frac{x.re}{\frac{{y.re}^{2} + {y.im}^{2}}{y.re}}} \]

      2. associate-/r/57.3%

         \[\leadsto \color{blue}{\frac{x.re}{{y.re}^{2} + {y.im}^{2}} \cdot y.re} \]

      3. unpow257.3%

         \[\leadsto \frac{x.re}{\color{blue}{y.re \cdot y.re} + {y.im}^{2}} \cdot y.re \]

      4. unpow257.3%

         \[\leadsto \frac{x.re}{y.re \cdot y.re + \color{blue}{y.im \cdot y.im}} \cdot y.re \]

      5. +-commutative57.3%

         \[\leadsto \frac{x.re}{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}} \cdot y.re \]

      6. fma-udef57.3%

         \[\leadsto \frac{x.re}{\color{blue}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}} \cdot y.re \]

   4. Simplified57.3%

      \[\leadsto \color{blue}{\frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re} \]
   5. Step-by-step derivation

      1. fma-udef57.3%

         \[\leadsto \frac{x.re}{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}} \cdot y.re \]

      2. +-commutative57.3%

         \[\leadsto \frac{x.re}{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}} \cdot y.re \]

   6. Applied egg-rr57.3%

      \[\leadsto \frac{x.re}{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}} \cdot y.re \]
3. Recombined 3 regimes into one program.

4. Final simplification67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -4.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -1.65 \cdot 10^{-112}:\\ \;\;\;\;y.re \cdot \frac{x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{+60}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

1. Split input into 2 regimes

2. if y.re < -125 or 8.40000000000000005e80 < y.re

   1. Initial program 45.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 74.5%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}} \]
   3. Step-by-step derivation

      1. unpow274.5%

         \[\leadsto \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{\color{blue}{y.re \cdot y.re}} \]

      2. times-frac83.1%

         \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]

   4. Simplified83.1%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]

   if -125 < y.re < 8.40000000000000005e80

   1. Initial program 74.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 73.4%

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im}} \]
   3. Step-by-step derivation

      1. +-commutative73.4%

         \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]

      2. unpow273.4%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re \cdot y.re}{\color{blue}{y.im \cdot y.im}} \]

      3. associate-/l*73.5%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{y.im \cdot y.im}{y.re}}} \]

   4. Simplified73.5%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}} \]
   5. Step-by-step derivation

      1. clear-num73.5%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{1}{\frac{\frac{y.im \cdot y.im}{y.re}}{x.re}}} \]

      2. inv-pow73.5%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{{\left(\frac{\frac{y.im \cdot y.im}{y.re}}{x.re}\right)}^{-1}} \]

      3. associate-/l*78.8%

         \[\leadsto \frac{x.im}{y.im} + {\left(\frac{\color{blue}{\frac{y.im}{\frac{y.re}{y.im}}}}{x.re}\right)}^{-1} \]

   6. Applied egg-rr78.8%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{{\left(\frac{\frac{y.im}{\frac{y.re}{y.im}}}{x.re}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-178.8%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{1}{\frac{\frac{y.im}{\frac{y.re}{y.im}}}{x.re}}} \]

      2. associate-/l/81.4%

         \[\leadsto \frac{x.im}{y.im} + \frac{1}{\color{blue}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}} \]

   8. Simplified81.4%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -125 \lor \neg \left(y.re \leq 8.4 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}}\\ \end{array} \]

1. Split input into 2 regimes

2. if y.re < -8.80000000000000034e23 or 8.20000000000000003e80 < y.re

   1. Initial program 42.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 72.6%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   if -8.80000000000000034e23 < y.re < 8.20000000000000003e80

   1. Initial program 74.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 71.8%

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im}} \]
   3. Step-by-step derivation

      1. +-commutative71.8%

         \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]

      2. *-commutative71.8%

         \[\leadsto \frac{x.im}{y.im} + \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2}} \]

      3. unpow271.8%

         \[\leadsto \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{\color{blue}{y.im \cdot y.im}} \]

      4. times-frac76.5%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]

   4. Simplified76.5%

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

4. Final simplification75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.8 \cdot 10^{+23} \lor \neg \left(y.re \leq 8.2 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

1. Split input into 2 regimes

2. if y.re < -450 or 9.2000000000000008e78 < y.re

   1. Initial program 45.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 74.5%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}} \]
   3. Step-by-step derivation

      1. unpow274.5%

         \[\leadsto \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{\color{blue}{y.re \cdot y.re}} \]

      2. times-frac83.1%

         \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]

   4. Simplified83.1%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]

   if -450 < y.re < 9.2000000000000008e78

   1. Initial program 74.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 73.4%

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im}} \]
   3. Step-by-step derivation

      1. +-commutative73.4%

         \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]

      2. unpow273.4%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re \cdot y.re}{\color{blue}{y.im \cdot y.im}} \]

      3. associate-/l*73.5%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{y.im \cdot y.im}{y.re}}} \]

   4. Simplified73.5%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}} \]

   5. Taylor expanded in y.im around 0 73.5%

      \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{\frac{{y.im}^{2}}{y.re}}} \]
   6. Step-by-step derivation

      1. unpow273.5%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\frac{\color{blue}{y.im \cdot y.im}}{y.re}} \]

      2. associate-*r/78.8%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{y.im \cdot \frac{y.im}{y.re}}} \]

      3. *-commutative78.8%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{\frac{y.im}{y.re} \cdot y.im}} \]

      4. associate-/r/78.9%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{\frac{y.im}{\frac{y.re}{y.im}}}} \]

   7. Simplified78.9%

      \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{\frac{y.im}{\frac{y.re}{y.im}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -450 \lor \neg \left(y.re \leq 9.2 \cdot 10^{+78}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im}{\frac{y.re}{y.im}}}\\ \end{array} \]

1. Split input into 2 regimes

2. if y.re < -4.8000000000000001e24 or 1.16000000000000007e79 < y.re

   1. Initial program 42.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 72.6%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   if -4.8000000000000001e24 < y.re < 1.16000000000000007e79

   1. Initial program 74.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 71.8%

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im}} \]
   3. Step-by-step derivation

      1. +-commutative71.8%

         \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]

      2. unpow271.8%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re \cdot y.re}{\color{blue}{y.im \cdot y.im}} \]

      3. associate-/l*72.0%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{y.im \cdot y.im}{y.re}}} \]

   4. Simplified72.0%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}} \]

   5. Taylor expanded in y.im around 0 72.0%

      \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{\frac{{y.im}^{2}}{y.re}}} \]
   6. Step-by-step derivation

      1. unpow272.0%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\frac{\color{blue}{y.im \cdot y.im}}{y.re}} \]

      2. associate-*r/77.0%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{y.im \cdot \frac{y.im}{y.re}}} \]

      3. *-commutative77.0%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{\frac{y.im}{y.re} \cdot y.im}} \]

      4. associate-/r/77.1%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{\frac{y.im}{\frac{y.re}{y.im}}}} \]

   7. Simplified77.1%

      \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{\frac{y.im}{\frac{y.re}{y.im}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im}{\frac{y.re}{y.im}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

1. Split input into 3 regimes

2. if y.re < -3.8000000000000004e22 or 2.10000000000000017e56 < y.re

   1. Initial program 43.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 70.9%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   if -3.8000000000000004e22 < y.re < -4.9999999999999996e-41 or -4.9999999999999997e-113 < y.re < 2.10000000000000017e56

   1. Initial program 74.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 67.1%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

   if -4.9999999999999996e-41 < y.re < -4.9999999999999997e-113

   1. Initial program 81.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in x.re around inf 71.1%

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.re}^{2} + {y.im}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l*66.4%

         \[\leadsto \color{blue}{\frac{x.re}{\frac{{y.re}^{2} + {y.im}^{2}}{y.re}}} \]

      2. associate-/r/57.3%

         \[\leadsto \color{blue}{\frac{x.re}{{y.re}^{2} + {y.im}^{2}} \cdot y.re} \]

      3. unpow257.3%

         \[\leadsto \frac{x.re}{\color{blue}{y.re \cdot y.re} + {y.im}^{2}} \cdot y.re \]

      4. unpow257.3%

         \[\leadsto \frac{x.re}{y.re \cdot y.re + \color{blue}{y.im \cdot y.im}} \cdot y.re \]

      5. +-commutative57.3%

         \[\leadsto \frac{x.re}{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}} \cdot y.re \]

      6. fma-udef57.3%

         \[\leadsto \frac{x.re}{\color{blue}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}} \cdot y.re \]

   4. Simplified57.3%

      \[\leadsto \color{blue}{\frac{x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot y.re} \]
   5. Step-by-step derivation

      1. fma-udef57.3%

         \[\leadsto \frac{x.re}{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}} \cdot y.re \]

      2. +-commutative57.3%

         \[\leadsto \frac{x.re}{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}} \cdot y.re \]

   6. Applied egg-rr57.3%

      \[\leadsto \frac{x.re}{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}} \cdot y.re \]

   7. Taylor expanded in y.re around 0 52.4%

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative52.4%

         \[\leadsto \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2}} \]

      2. unpow252.4%

         \[\leadsto \frac{y.re \cdot x.re}{\color{blue}{y.im \cdot y.im}} \]

      3. times-frac57.0%

         \[\leadsto \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]

   9. Simplified57.0%

      \[\leadsto \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
3. Recombined 3 regimes into one program.

4. Final simplification67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-41}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-113}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

1. Split input into 2 regimes

2. if y.re < -4.49999999999999979e23 or 2.9000000000000002e57 < y.re

   1. Initial program 43.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 70.9%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   if -4.49999999999999979e23 < y.re < 2.9000000000000002e57

   1. Initial program 75.5%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 63.1%

      \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{+57}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

1. Initial program 62.2%

   \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

2. Taylor expanded in y.re around 0 44.1%

   \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

3. Final simplification44.1%

   \[\leadsto \frac{x.im}{y.im} \]

herbie shell --seed 2023166 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))