Derivation

Split input into 2 regimes
if x < -0.001888855552195268
1. Initial program 0.0
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Initial simplification0.0
  \[\leadsto \frac{e^{x}}{e^{x} - 1}\]
3. Using strategy rm
4. Applied flip--0.0
  \[\leadsto \frac{e^{x}}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}\]
5. Using strategy rm
6. Applied add-cube-cbrt0.0
  \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \sqrt[3]{e^{x}}}}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}\]
7. Applied associate-/l*0.0
  \[\leadsto \color{blue}{\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\frac{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}{\sqrt[3]{e^{x}}}}}\]
if -0.001888855552195268 < x
1. Initial program 60.2
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Initial simplification60.2
  \[\leadsto \frac{e^{x}}{e^{x} - 1}\]
3. Taylor expanded around 0 1.0
  \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.6
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.001888855552195268:\\ \;\;\;\;\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\frac{\frac{e^{x} \cdot e^{x} - 1}{e^{x} + 1}}{\sqrt[3]{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + x \cdot \frac{1}{12}\\ \end{array}\]

Runtime

herbie shell --seed 2018234 
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))

In
Out