\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]

↓

\[\left(\sin re \cdot 0.5\right) \cdot e^{im} + \frac{1}{e^{im}} \cdot \left(\sin re \cdot 0.5\right)\]

Derivation

Initial program 0.0
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\]
Simplified0.0
\[\leadsto \color{blue}{e^{im} \cdot \left(0.5 \cdot \sin re\right) + \frac{0.5 \cdot \sin re}{e^{im}}}\]
Using strategy rm
Applied div-inv0.0
\[\leadsto e^{im} \cdot \left(0.5 \cdot \sin re\right) + \color{blue}{\left(0.5 \cdot \sin re\right) \cdot \frac{1}{e^{im}}}\]
Final simplification0.0
\[\leadsto \left(\sin re \cdot 0.5\right) \cdot e^{im} + \frac{1}{e^{im}} \cdot \left(\sin re \cdot 0.5\right)\]

Reproduce

herbie shell --seed 2019091 
(FPCore (re im)
  :name "math.sin on complex, real part"
  (* (* 0.5 (sin re)) (+ (exp (- 0 im)) (exp im))))