Error

Derivation

1. Initial program 43.5

   \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

2. Initial simplification43.5

   \[\leadsto \frac{e^{x} \cdot \sin y - \frac{\sin y}{e^{x}}}{2}\]

3. Taylor expanded around 0 31.5

   \[\leadsto \color{blue}{\left(x \cdot y + \frac{1}{6} \cdot \left({x}^{3} \cdot y\right)\right) - \frac{1}{6} \cdot \left(x \cdot {y}^{3}\right)}\]

4. Simplified31.4

   \[\leadsto \color{blue}{x \cdot y + \left(\left(x \cdot y\right) \cdot \frac{1}{6}\right) \cdot \left(\left(y + x\right) \cdot \left(x - y\right)\right)}\]
5. Using strategy rm

6. Applied add-cube-cbrt31.4

   \[\leadsto x \cdot y + \color{blue}{\left(\sqrt[3]{\left(\left(x \cdot y\right) \cdot \frac{1}{6}\right) \cdot \left(\left(y + x\right) \cdot \left(x - y\right)\right)} \cdot \sqrt[3]{\left(\left(x \cdot y\right) \cdot \frac{1}{6}\right) \cdot \left(\left(y + x\right) \cdot \left(x - y\right)\right)}\right) \cdot \sqrt[3]{\left(\left(x \cdot y\right) \cdot \frac{1}{6}\right) \cdot \left(\left(y + x\right) \cdot \left(x - y\right)\right)}}\]

7. Final simplification31.4

   \[\leadsto x \cdot y + \sqrt[3]{\left(\left(y + x\right) \cdot \left(x - y\right)\right) \cdot \left(\left(x \cdot y\right) \cdot \frac{1}{6}\right)} \cdot \left(\sqrt[3]{\left(\left(y + x\right) \cdot \left(x - y\right)\right) \cdot \left(\left(x \cdot y\right) \cdot \frac{1}{6}\right)} \cdot \sqrt[3]{\left(\left(y + x\right) \cdot \left(x - y\right)\right) \cdot \left(\left(x \cdot y\right) \cdot \frac{1}{6}\right)}\right)\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018230 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))