Complex division, real part

1. Split input into 3 regimes

2. if c < -1.1499999999999999e81 or 1.00000000000000007e70 < c

   1. Initial program 35.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 72.3%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. unpow272.3%

         \[\leadsto \frac{a}{c} + \frac{d \cdot b}{\color{blue}{c \cdot c}} \]

      2. times-frac82.7%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{d}{c} \cdot \frac{b}{c}} \]

   4. Simplified82.7%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}} \]
   5. Step-by-step derivation

      1. clear-num83.6%

         \[\leadsto \frac{a}{c} + \frac{d}{c} \cdot \color{blue}{\frac{1}{\frac{c}{b}}} \]

      2. un-div-inv83.6%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{\frac{d}{c}}{\frac{c}{b}}} \]

   6. Applied egg-rr83.6%

      \[\leadsto \frac{a}{c} + \color{blue}{\frac{\frac{d}{c}}{\frac{c}{b}}} \]

   if -1.1499999999999999e81 < c < -2.1999999999999999e-97 or 2.2000000000000001e-80 < c < 1.00000000000000007e70

   1. Initial program 87.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   if -2.1999999999999999e-97 < c < 2.2000000000000001e-80

   1. Initial program 67.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. *-un-lft-identity67.7%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt67.7%

         \[\leadsto \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac67.7%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-def67.7%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]

      5. fma-def67.7%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-def83.5%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr83.5%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   4. Taylor expanded in c around 0 58.5%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + \frac{c \cdot a}{d}\right)} \]

   5. Taylor expanded in c around 0 93.5%

      \[\leadsto \color{blue}{\frac{1}{d}} \cdot \left(b + \frac{c \cdot a}{d}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+81}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a \cdot c}{d}\right)\\ \mathbf{elif}\;c \leq 10^{+70}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \end{array} \]

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < +inf.0

   1. Initial program 77.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. *-un-lft-identity77.7%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt77.7%

         \[\leadsto \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac77.8%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-def77.8%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]

      5. fma-def77.8%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-def93.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr93.9%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 0.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 45.6%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. unpow245.6%

         \[\leadsto \frac{a}{c} + \frac{d \cdot b}{\color{blue}{c \cdot c}} \]

      2. times-frac56.9%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{d}{c} \cdot \frac{b}{c}} \]

   4. Simplified56.9%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}} \]
   5. Step-by-step derivation

      1. clear-num56.9%

         \[\leadsto \frac{a}{c} + \frac{d}{c} \cdot \color{blue}{\frac{1}{\frac{c}{b}}} \]

      2. un-div-inv57.0%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{\frac{d}{c}}{\frac{c}{b}}} \]

   6. Applied egg-rr57.0%

      \[\leadsto \frac{a}{c} + \color{blue}{\frac{\frac{d}{c}}{\frac{c}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \end{array} \]

1. Split input into 2 regimes

2. if c < -2.2e22 or -1.8499999999999998e-71 < c < -1.5499999999999999e-94 or 5.2000000000000001e67 < c

   1. Initial program 45.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 71.0%

      \[\leadsto \color{blue}{\frac{a}{c}} \]

   if -2.2e22 < c < -1.8499999999999998e-71 or -1.5499999999999999e-94 < c < 5.2000000000000001e67

   1. Initial program 74.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. *-un-lft-identity74.0%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt74.0%

         \[\leadsto \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac74.0%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-def74.0%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]

      5. fma-def74.0%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-def85.4%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr85.4%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   4. Taylor expanded in c around 0 50.6%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + \frac{c \cdot a}{d}\right)} \]

   5. Taylor expanded in c around 0 81.8%

      \[\leadsto \color{blue}{\frac{1}{d}} \cdot \left(b + \frac{c \cdot a}{d}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{+22} \lor \neg \left(c \leq -1.85 \cdot 10^{-71}\right) \land \left(c \leq -1.55 \cdot 10^{-94} \lor \neg \left(c \leq 5.2 \cdot 10^{+67}\right)\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a \cdot c}{d}\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if c < -1.49999999999999996e-21 or -6.8000000000000002e-67 < c < -1.5000000000000001e-94 or 5.00000000000000027e31 < c

   1. Initial program 51.6%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 71.0%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. unpow271.0%

         \[\leadsto \frac{a}{c} + \frac{d \cdot b}{\color{blue}{c \cdot c}} \]

      2. times-frac77.7%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{d}{c} \cdot \frac{b}{c}} \]

   4. Simplified77.7%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}} \]

   if -1.49999999999999996e-21 < c < -6.8000000000000002e-67 or -1.5000000000000001e-94 < c < 5.00000000000000027e31

   1. Initial program 71.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. *-un-lft-identity71.2%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt71.2%

         \[\leadsto \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac71.2%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-def71.2%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]

      5. fma-def71.2%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-def84.2%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr84.2%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   4. Taylor expanded in c around 0 54.8%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + \frac{c \cdot a}{d}\right)} \]

   5. Taylor expanded in c around 0 86.4%

      \[\leadsto \color{blue}{\frac{1}{d}} \cdot \left(b + \frac{c \cdot a}{d}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.5 \cdot 10^{-21} \lor \neg \left(c \leq -6.8 \cdot 10^{-67} \lor \neg \left(c \leq -1.5 \cdot 10^{-94}\right) \land c \leq 5 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a \cdot c}{d}\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if c < -1.45e-21 or -3.3999999999999998e-61 < c < -1.5499999999999999e-94 or 6.4000000000000001e31 < c

   1. Initial program 51.6%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 71.0%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. unpow271.0%

         \[\leadsto \frac{a}{c} + \frac{d \cdot b}{\color{blue}{c \cdot c}} \]

      2. times-frac77.7%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{d}{c} \cdot \frac{b}{c}} \]

   4. Simplified77.7%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}} \]
   5. Step-by-step derivation

      1. clear-num78.3%

         \[\leadsto \frac{a}{c} + \frac{d}{c} \cdot \color{blue}{\frac{1}{\frac{c}{b}}} \]

      2. un-div-inv78.4%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{\frac{d}{c}}{\frac{c}{b}}} \]

   6. Applied egg-rr78.4%

      \[\leadsto \frac{a}{c} + \color{blue}{\frac{\frac{d}{c}}{\frac{c}{b}}} \]

   if -1.45e-21 < c < -3.3999999999999998e-61 or -1.5499999999999999e-94 < c < 6.4000000000000001e31

   1. Initial program 71.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. *-un-lft-identity71.2%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt71.2%

         \[\leadsto \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac71.2%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-def71.2%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]

      5. fma-def71.2%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-def84.2%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr84.2%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   4. Taylor expanded in c around 0 54.8%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + \frac{c \cdot a}{d}\right)} \]

   5. Taylor expanded in c around 0 86.4%

      \[\leadsto \color{blue}{\frac{1}{d}} \cdot \left(b + \frac{c \cdot a}{d}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{-21} \lor \neg \left(c \leq -3.4 \cdot 10^{-61}\right) \land \left(c \leq -1.55 \cdot 10^{-94} \lor \neg \left(c \leq 6.4 \cdot 10^{+31}\right)\right):\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a \cdot c}{d}\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if c < -6.7999999999999997e-22 or -1.89999999999999997e-60 < c < -1.84999999999999993e-96 or 5.39999999999999971e31 < c

   1. Initial program 51.6%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 71.0%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. unpow271.0%

         \[\leadsto \frac{a}{c} + \frac{d \cdot b}{\color{blue}{c \cdot c}} \]

      2. times-frac77.7%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{d}{c} \cdot \frac{b}{c}} \]

   4. Simplified77.7%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}} \]
   5. Step-by-step derivation

      1. clear-num78.3%

         \[\leadsto \frac{a}{c} + \frac{d}{c} \cdot \color{blue}{\frac{1}{\frac{c}{b}}} \]

      2. un-div-inv78.4%

         \[\leadsto \frac{a}{c} + \color{blue}{\frac{\frac{d}{c}}{\frac{c}{b}}} \]

   6. Applied egg-rr78.4%

      \[\leadsto \frac{a}{c} + \color{blue}{\frac{\frac{d}{c}}{\frac{c}{b}}} \]

   if -6.7999999999999997e-22 < c < -1.89999999999999997e-60

   1. Initial program 69.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 86.8%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. unpow286.8%

         \[\leadsto \frac{b}{d} + \frac{c \cdot a}{\color{blue}{d \cdot d}} \]

      2. times-frac86.8%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{c}{d} \cdot \frac{a}{d}} \]

   4. Simplified86.8%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}} \]

   if -1.84999999999999993e-96 < c < 5.39999999999999971e31

   1. Initial program 71.3%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. *-un-lft-identity71.3%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt71.3%

         \[\leadsto \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac71.3%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-def71.3%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]

      5. fma-def71.3%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-def84.1%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr84.1%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   4. Taylor expanded in c around 0 53.7%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + \frac{c \cdot a}{d}\right)} \]

   5. Taylor expanded in c around 0 86.4%

      \[\leadsto \color{blue}{\frac{1}{d}} \cdot \left(b + \frac{c \cdot a}{d}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq -1.9 \cdot 10^{-60}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq -1.85 \cdot 10^{-96} \lor \neg \left(c \leq 5.4 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a \cdot c}{d}\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if c < -1.62000000000000009e-15 or -4.69999999999999988e-68 < c < -1.02000000000000004e-97 or 1.02e68 < c

   1. Initial program 49.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 68.4%

      \[\leadsto \color{blue}{\frac{a}{c}} \]

   if -1.62000000000000009e-15 < c < -4.69999999999999988e-68 or -1.02000000000000004e-97 < c < 2.0999999999999999e-164 or 2.69999999999999988e-89 < c < 1.02e68

   1. Initial program 71.8%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 69.9%

      \[\leadsto \color{blue}{\frac{b}{d}} \]

   if 2.0999999999999999e-164 < c < 2.69999999999999988e-89

   1. Initial program 80.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. *-un-lft-identity80.2%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt80.2%

         \[\leadsto \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac80.6%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-def80.6%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]

      5. fma-def80.6%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-def90.1%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr90.1%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   4. Taylor expanded in c around 0 51.2%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + \frac{c \cdot a}{d}\right)} \]

   5. Taylor expanded in c around 0 90.7%

      \[\leadsto \color{blue}{\frac{1}{d}} \cdot \left(b + \frac{c \cdot a}{d}\right) \]

   6. Taylor expanded in d around 0 62.7%

      \[\leadsto \color{blue}{\frac{c \cdot a}{{d}^{2}}} \]
   7. Step-by-step derivation

      1. unpow262.7%

         \[\leadsto \frac{c \cdot a}{\color{blue}{d \cdot d}} \]

      2. associate-/l*62.7%

         \[\leadsto \color{blue}{\frac{c}{\frac{d \cdot d}{a}}} \]

      3. associate-*r/72.3%

         \[\leadsto \frac{c}{\color{blue}{d \cdot \frac{d}{a}}} \]

   8. Simplified72.3%

      \[\leadsto \color{blue}{\frac{c}{d \cdot \frac{d}{a}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.62 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -4.7 \cdot 10^{-68}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq -1.02 \cdot 10^{-97}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{-164}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{-89}:\\ \;\;\;\;\frac{c}{d \cdot \frac{d}{a}}\\ \mathbf{elif}\;c \leq 1.02 \cdot 10^{+68}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

1. Split input into 3 regimes

2. if c < -1.95e124 or 5.49999999999999968e67 < c

   1. Initial program 35.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 75.4%

      \[\leadsto \color{blue}{\frac{a}{c}} \]

   if -1.95e124 < c < -8.1999999999999996e-98

   1. Initial program 84.1%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in a around inf 56.5%

      \[\leadsto \color{blue}{\frac{c \cdot a}{{d}^{2} + {c}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l*62.8%

         \[\leadsto \color{blue}{\frac{c}{\frac{{d}^{2} + {c}^{2}}{a}}} \]

      2. associate-/r/62.9%

         \[\leadsto \color{blue}{\frac{c}{{d}^{2} + {c}^{2}} \cdot a} \]

      3. unpow262.9%

         \[\leadsto \frac{c}{\color{blue}{d \cdot d} + {c}^{2}} \cdot a \]

      4. unpow262.9%

         \[\leadsto \frac{c}{d \cdot d + \color{blue}{c \cdot c}} \cdot a \]

      5. fma-udef62.9%

         \[\leadsto \frac{c}{\color{blue}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \cdot a \]

   4. Simplified62.9%

      \[\leadsto \color{blue}{\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a} \]
   5. Step-by-step derivation

      1. fma-udef62.9%

         \[\leadsto \frac{c}{\color{blue}{d \cdot d + c \cdot c}} \cdot a \]

   6. Applied egg-rr62.9%

      \[\leadsto \frac{c}{\color{blue}{d \cdot d + c \cdot c}} \cdot a \]

   if -8.1999999999999996e-98 < c < 5.49999999999999968e67

   1. Initial program 72.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Step-by-step derivation

      1. *-un-lft-identity72.0%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{c \cdot c + d \cdot d} \]

      2. add-sqr-sqrt72.0%

         \[\leadsto \frac{1 \cdot \left(a \cdot c + b \cdot d\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      3. times-frac71.9%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-def71.9%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}} \]

      5. fma-def71.9%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d}} \]

      6. hypot-def84.1%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   3. Applied egg-rr84.1%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   4. Taylor expanded in c around 0 51.2%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + \frac{c \cdot a}{d}\right)} \]

   5. Taylor expanded in c around 0 83.8%

      \[\leadsto \color{blue}{\frac{1}{d}} \cdot \left(b + \frac{c \cdot a}{d}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.95 \cdot 10^{+124}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -8.2 \cdot 10^{-98}:\\ \;\;\;\;a \cdot \frac{c}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 5.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a \cdot c}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

1. Split input into 2 regimes

2. if c < -1.45000000000000009e-15 or -4.20000000000000016e-68 < c < -1.34999999999999993e-97 or 5.49999999999999968e67 < c

   1. Initial program 49.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 68.4%

      \[\leadsto \color{blue}{\frac{a}{c}} \]

   if -1.45000000000000009e-15 < c < -4.20000000000000016e-68 or -1.34999999999999993e-97 < c < 5.49999999999999968e67

   1. Initial program 72.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 66.8%

      \[\leadsto \color{blue}{\frac{b}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -4.2 \cdot 10^{-68} \lor \neg \left(c \leq -1.35 \cdot 10^{-97}\right) \land c \leq 5.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

1. Initial program 60.7%

   \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

2. Taylor expanded in c around inf 44.0%

   \[\leadsto \color{blue}{\frac{a}{c}} \]

3. Final simplification44.0%

   \[\leadsto \frac{a}{c} \]

herbie shell --seed 2023167 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))