math.exp on complex, imaginary part

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]

2. Final simplification100.0%

   \[\leadsto e^{re} \cdot \sin im \]

1. Split input into 3 regimes

2. if (exp.f64 re) < 0.0

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. expm1-log1p-u100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

      2. expm1-udef100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{re} \cdot \sin im\right)} - 1} \]

      3. log1p-udef100.0%

         \[\leadsto e^{\color{blue}{\log \left(1 + e^{re} \cdot \sin im\right)}} - 1 \]

      4. add-exp-log100.0%

         \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right)} - 1 \]

   3. Applied egg-rr100.0%

      \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right) - 1} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if 0.0 < (exp.f64 re) < 1.0009999999999999

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 98.0%

      \[\leadsto \color{blue}{\sin im} \]

   if 1.0009999999999999 < (exp.f64 re)

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 82.9%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]
3. Recombined 3 regimes into one program.

4. Final simplification94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{re} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;e^{re} \leq 1.001:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

1. Split input into 4 regimes

2. if re < -85

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. expm1-log1p-u100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

      2. expm1-udef100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{re} \cdot \sin im\right)} - 1} \]

      3. log1p-udef100.0%

         \[\leadsto e^{\color{blue}{\log \left(1 + e^{re} \cdot \sin im\right)}} - 1 \]

      4. add-exp-log100.0%

         \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right)} - 1 \]

   3. Applied egg-rr100.0%

      \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right) - 1} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -85 < re < 0.00145

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 99.5%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. *-rgt-identity99.5%

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

      2. *-commutative99.5%

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

      3. associate-*l*99.5%

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

      4. distribute-lft-out99.5%

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

      5. distribute-lft-out99.5%

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

      6. associate-+l+99.5%

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

      7. +-commutative99.5%

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

      8. *-commutative99.5%

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

      9. unpow299.5%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   4. Simplified99.5%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)} \]

   if 0.00145 < re < 1.84999999999999997e154

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 78.9%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   if 1.84999999999999997e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. *-rgt-identity100.0%

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

      2. *-commutative100.0%

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

      3. associate-*l*100.0%

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

      4. distribute-lft-out100.0%

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

      5. distribute-lft-out100.0%

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

      6. associate-+l+100.0%

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

      7. +-commutative100.0%

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

      8. *-commutative100.0%

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

      9. unpow2100.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   4. Simplified100.0%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)} \]

   5. Taylor expanded in re around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin im \cdot {re}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative100.0%

         \[\leadsto \color{blue}{\left(\sin im \cdot {re}^{2}\right) \cdot 0.5} \]

      2. unpow2100.0%

         \[\leadsto \left(\sin im \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot 0.5 \]

      3. associate-*r*100.0%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)} \]

      4. associate-*r*100.0%

         \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   7. Simplified100.0%

      \[\leadsto \color{blue}{\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -85:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.00145:\\ \;\;\;\;\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

1. Split input into 4 regimes

2. if re < -1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. expm1-log1p-u100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

      2. expm1-udef100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{re} \cdot \sin im\right)} - 1} \]

      3. log1p-udef100.0%

         \[\leadsto e^{\color{blue}{\log \left(1 + e^{re} \cdot \sin im\right)}} - 1 \]

      4. add-exp-log100.0%

         \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right)} - 1 \]

   3. Applied egg-rr100.0%

      \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right) - 1} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -1 < re < 1.9000000000000001e-4

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 99.2%

      \[\leadsto \color{blue}{\sin im + \sin im \cdot re} \]
   3. Step-by-step derivation

      1. +-commutative99.2%

         \[\leadsto \color{blue}{\sin im \cdot re + \sin im} \]

      2. *-rgt-identity99.2%

         \[\leadsto \sin im \cdot re + \color{blue}{\sin im \cdot 1} \]

      3. distribute-lft-out99.2%

         \[\leadsto \color{blue}{\sin im \cdot \left(re + 1\right)} \]

   4. Simplified99.2%

      \[\leadsto \color{blue}{\sin im \cdot \left(re + 1\right)} \]

   if 1.9000000000000001e-4 < re < 1.84999999999999997e154

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 78.9%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   if 1.84999999999999997e154 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 100.0%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. *-rgt-identity100.0%

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

      2. *-commutative100.0%

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

      3. associate-*l*100.0%

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

      4. distribute-lft-out100.0%

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

      5. distribute-lft-out100.0%

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

      6. associate-+l+100.0%

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

      7. +-commutative100.0%

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

      8. *-commutative100.0%

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

      9. unpow2100.0%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   4. Simplified100.0%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)} \]

   5. Taylor expanded in re around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin im \cdot {re}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative100.0%

         \[\leadsto \color{blue}{\left(\sin im \cdot {re}^{2}\right) \cdot 0.5} \]

      2. unpow2100.0%

         \[\leadsto \left(\sin im \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot 0.5 \]

      3. associate-*r*100.0%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)} \]

      4. associate-*r*100.0%

         \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   7. Simplified100.0%

      \[\leadsto \color{blue}{\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 0.00019:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{elif}\;re \leq 1.85 \cdot 10^{+154}:\\ \;\;\;\;e^{re} \cdot im\\ \mathbf{else}:\\ \;\;\;\;\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if re < -1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. expm1-log1p-u100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

      2. expm1-udef100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{re} \cdot \sin im\right)} - 1} \]

      3. log1p-udef100.0%

         \[\leadsto e^{\color{blue}{\log \left(1 + e^{re} \cdot \sin im\right)}} - 1 \]

      4. add-exp-log100.0%

         \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right)} - 1 \]

   3. Applied egg-rr100.0%

      \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right) - 1} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -1 < re < 4.00000000000000033e-5

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 99.2%

      \[\leadsto \color{blue}{\sin im + \sin im \cdot re} \]
   3. Step-by-step derivation

      1. +-commutative99.2%

         \[\leadsto \color{blue}{\sin im \cdot re + \sin im} \]

      2. *-rgt-identity99.2%

         \[\leadsto \sin im \cdot re + \color{blue}{\sin im \cdot 1} \]

      3. distribute-lft-out99.2%

         \[\leadsto \color{blue}{\sin im \cdot \left(re + 1\right)} \]

   4. Simplified99.2%

      \[\leadsto \color{blue}{\sin im \cdot \left(re + 1\right)} \]

   if 4.00000000000000033e-5 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 82.9%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]
3. Recombined 3 regimes into one program.

4. Final simplification94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\sin im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{re} \cdot im\\ \end{array} \]

1. Split input into 3 regimes

2. if re < -54

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. expm1-log1p-u100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

      2. expm1-udef100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{re} \cdot \sin im\right)} - 1} \]

      3. log1p-udef100.0%

         \[\leadsto e^{\color{blue}{\log \left(1 + e^{re} \cdot \sin im\right)}} - 1 \]

      4. add-exp-log100.0%

         \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right)} - 1 \]

   3. Applied egg-rr100.0%

      \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right) - 1} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -54 < re < 14.199999999999999

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 96.7%

      \[\leadsto \color{blue}{\sin im} \]

   if 14.199999999999999 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 53.7%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. *-rgt-identity53.7%

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

      2. *-commutative53.7%

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

      3. associate-*l*53.7%

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

      4. distribute-lft-out53.7%

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

      5. distribute-lft-out53.7%

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

      6. associate-+l+53.7%

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

      7. +-commutative53.7%

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

      8. *-commutative53.7%

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

      9. unpow253.7%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   4. Simplified53.7%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)} \]

   5. Taylor expanded in re around inf 53.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin im \cdot {re}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative53.7%

         \[\leadsto \color{blue}{\left(\sin im \cdot {re}^{2}\right) \cdot 0.5} \]

      2. unpow253.7%

         \[\leadsto \left(\sin im \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot 0.5 \]

      3. associate-*r*53.7%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)} \]

      4. associate-*r*53.7%

         \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   7. Simplified53.7%

      \[\leadsto \color{blue}{\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   8. Taylor expanded in im around 0 55.4%

      \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   9. Step-by-step derivation

      1. associate-*r*55.4%

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

      2. *-commutative55.4%

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

      3. unpow255.4%

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

      4. associate-*r*55.4%

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

   10. Simplified55.4%

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

4. Final simplification85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -54:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 14.2:\\ \;\;\;\;\sin im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if re < -1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. expm1-log1p-u100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

      2. expm1-udef100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{re} \cdot \sin im\right)} - 1} \]

      3. log1p-udef100.0%

         \[\leadsto e^{\color{blue}{\log \left(1 + e^{re} \cdot \sin im\right)}} - 1 \]

      4. add-exp-log100.0%

         \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right)} - 1 \]

   3. Applied egg-rr100.0%

      \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right) - 1} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -1 < re < 2.7000000000000002

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 98.1%

      \[\leadsto \color{blue}{\sin im + \sin im \cdot re} \]
   3. Step-by-step derivation

      1. +-commutative98.1%

         \[\leadsto \color{blue}{\sin im \cdot re + \sin im} \]

      2. *-rgt-identity98.1%

         \[\leadsto \sin im \cdot re + \color{blue}{\sin im \cdot 1} \]

      3. distribute-lft-out98.1%

         \[\leadsto \color{blue}{\sin im \cdot \left(re + 1\right)} \]

   4. Simplified98.1%

      \[\leadsto \color{blue}{\sin im \cdot \left(re + 1\right)} \]

   5. Taylor expanded in im around 0 57.3%

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

   if 2.7000000000000002 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 53.7%

      \[\leadsto \color{blue}{\sin im + \left(\sin im \cdot re + 0.5 \cdot \left(\sin im \cdot {re}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. *-rgt-identity53.7%

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

      2. *-commutative53.7%

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

      3. associate-*l*53.7%

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

      4. distribute-lft-out53.7%

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

      5. distribute-lft-out53.7%

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

      6. associate-+l+53.7%

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

      7. +-commutative53.7%

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

      8. *-commutative53.7%

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

      9. unpow253.7%

         \[\leadsto \sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \color{blue}{\left(re \cdot re\right)}\right) \]

   4. Simplified53.7%

      \[\leadsto \color{blue}{\sin im \cdot \left(\left(re + 1\right) + 0.5 \cdot \left(re \cdot re\right)\right)} \]

   5. Taylor expanded in re around inf 53.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\sin im \cdot {re}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative53.7%

         \[\leadsto \color{blue}{\left(\sin im \cdot {re}^{2}\right) \cdot 0.5} \]

      2. unpow253.7%

         \[\leadsto \left(\sin im \cdot \color{blue}{\left(re \cdot re\right)}\right) \cdot 0.5 \]

      3. associate-*r*53.7%

         \[\leadsto \color{blue}{\sin im \cdot \left(\left(re \cdot re\right) \cdot 0.5\right)} \]

      4. associate-*r*53.7%

         \[\leadsto \sin im \cdot \color{blue}{\left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   7. Simplified53.7%

      \[\leadsto \color{blue}{\sin im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)} \]

   8. Taylor expanded in im around 0 55.4%

      \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   9. Step-by-step derivation

      1. associate-*r*55.4%

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

      2. *-commutative55.4%

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

      3. unpow255.4%

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

      4. associate-*r*55.4%

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

   10. Simplified55.4%

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

4. Final simplification68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 2.7:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re \cdot \left(re \cdot 0.5\right)\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if re < -66

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. expm1-log1p-u100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

      2. expm1-udef100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{re} \cdot \sin im\right)} - 1} \]

      3. log1p-udef100.0%

         \[\leadsto e^{\color{blue}{\log \left(1 + e^{re} \cdot \sin im\right)}} - 1 \]

      4. add-exp-log100.0%

         \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right)} - 1 \]

   3. Applied egg-rr100.0%

      \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right) - 1} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -66 < re < 1

   1. Initial program 99.9%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 58.4%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   3. Taylor expanded in re around 0 56.9%

      \[\leadsto \color{blue}{im} \]

   if 1 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 4.4%

      \[\leadsto \color{blue}{\sin im + \sin im \cdot re} \]
   3. Step-by-step derivation

      1. +-commutative4.4%

         \[\leadsto \color{blue}{\sin im \cdot re + \sin im} \]

      2. *-rgt-identity4.4%

         \[\leadsto \sin im \cdot re + \color{blue}{\sin im \cdot 1} \]

      3. distribute-lft-out4.4%

         \[\leadsto \color{blue}{\sin im \cdot \left(re + 1\right)} \]

   4. Simplified4.4%

      \[\leadsto \color{blue}{\sin im \cdot \left(re + 1\right)} \]

   5. Taylor expanded in im around 0 18.9%

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

   6. Taylor expanded in re around inf 18.9%

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

4. Final simplification58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -66:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

1. Split input into 2 regimes

2. if re < -1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]
   2. Step-by-step derivation

      1. expm1-log1p-u100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{re} \cdot \sin im\right)\right)} \]

      2. expm1-udef100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(e^{re} \cdot \sin im\right)} - 1} \]

      3. log1p-udef100.0%

         \[\leadsto e^{\color{blue}{\log \left(1 + e^{re} \cdot \sin im\right)}} - 1 \]

      4. add-exp-log100.0%

         \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right)} - 1 \]

   3. Applied egg-rr100.0%

      \[\leadsto \color{blue}{\left(1 + e^{re} \cdot \sin im\right) - 1} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{1} - 1 \]

   if -1 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 60.2%

      \[\leadsto \color{blue}{\sin im + \sin im \cdot re} \]
   3. Step-by-step derivation

      1. +-commutative60.2%

         \[\leadsto \color{blue}{\sin im \cdot re + \sin im} \]

      2. *-rgt-identity60.2%

         \[\leadsto \sin im \cdot re + \color{blue}{\sin im \cdot 1} \]

      3. distribute-lft-out60.2%

         \[\leadsto \color{blue}{\sin im \cdot \left(re + 1\right)} \]

   4. Simplified60.2%

      \[\leadsto \color{blue}{\sin im \cdot \left(re + 1\right)} \]

   5. Taylor expanded in im around 0 41.8%

      \[\leadsto \color{blue}{\left(1 + re\right) \cdot im} \]
3. Recombined 2 regimes into one program.

4. Final simplification58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(re + 1\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if re < 1

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in im around 0 75.1%

      \[\leadsto \color{blue}{e^{re} \cdot im} \]

   3. Taylor expanded in re around 0 35.7%

      \[\leadsto \color{blue}{im} \]

   if 1 < re

   1. Initial program 100.0%

      \[e^{re} \cdot \sin im \]

   2. Taylor expanded in re around 0 4.4%

      \[\leadsto \color{blue}{\sin im + \sin im \cdot re} \]
   3. Step-by-step derivation

      1. +-commutative4.4%

         \[\leadsto \color{blue}{\sin im \cdot re + \sin im} \]

      2. *-rgt-identity4.4%

         \[\leadsto \sin im \cdot re + \color{blue}{\sin im \cdot 1} \]

      3. distribute-lft-out4.4%

         \[\leadsto \color{blue}{\sin im \cdot \left(re + 1\right)} \]

   4. Simplified4.4%

      \[\leadsto \color{blue}{\sin im \cdot \left(re + 1\right)} \]

   5. Taylor expanded in im around 0 18.9%

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

   6. Taylor expanded in re around inf 18.9%

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

4. Final simplification30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1:\\ \;\;\;\;im\\ \mathbf{else}:\\ \;\;\;\;re \cdot im\\ \end{array} \]

1. Initial program 100.0%

   \[e^{re} \cdot \sin im \]

2. Taylor expanded in im around 0 77.2%

   \[\leadsto \color{blue}{e^{re} \cdot im} \]

3. Taylor expanded in re around 0 26.3%

   \[\leadsto \color{blue}{im} \]

4. Final simplification26.3%

   \[\leadsto im \]

herbie shell --seed 2023167 
(FPCore (re im)
  :name "math.exp on complex, imaginary part"
  :precision binary64
  (* (exp re) (sin im)))