Average Error: 43.8 → 0.5
Time: 10.8s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.01415223685758477:\\ \;\;\;\;\frac{e^{x} + \frac{-1}{e^{x}}}{2} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < -0.014152236857584766

    1. Initial program 4.1

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

    2. Simplified4.1

      \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{2} \cdot \sin y}\]
    3. Using strategy rm

    4. Applied sub-neg4.1

      \[\leadsto \frac{\color{blue}{e^{x} + \left(-e^{-x}\right)}}{2} \cdot \sin y\]

    5. Simplified4.2

      \[\leadsto \frac{e^{x} + \color{blue}{\frac{-1}{e^{x}}}}{2} \cdot \sin y\]

    if -0.014152236857584766 < x

    1. Initial program 44.0

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

    2. Simplified44.0

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

    3. Taylor expanded around 0 0.5

      \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2} \cdot \sin y\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.01415223685758477:\\ \;\;\;\;\frac{e^{x} + \frac{-1}{e^{x}}}{2} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2} \cdot \sin y\\ \end{array}\]

Reproduce

herbie shell --seed 2020129 
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2.0) (cos y)) (* (/ (- (exp x) (exp (- x))) 2.0) (sin y)))))