Derivation

Split input into 2 regimes
if (exp x) < 0.9984662786877636
1. Initial program 0.0
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Using strategy rm
3. Applied flip3--0.0
  \[\leadsto \frac{e^{x}}{\color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}}}\]
4. Applied associate-/r/0.0
  \[\leadsto \color{blue}{\frac{e^{x}}{{\left(e^{x}\right)}^{3} - {1}^{3}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)}\]
5. Simplified0.0
  \[\leadsto \frac{e^{x}}{{\left(e^{x}\right)}^{3} - {1}^{3}} \cdot \color{blue}{\left(e^{x + x} + \left(1 + e^{x}\right)\right)}\]
if 0.9984662786877636 < (exp x)
1. Initial program 60.1
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Taylor expanded around 0 0.7
  \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.5
\[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9984662786877636:\\ \;\;\;\;\frac{e^{x}}{{\left(e^{x}\right)}^{3} - {1}^{3}} \cdot \left(\left(e^{x} + 1\right) + e^{x + x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Runtime

herbie shell --seed 2018217 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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

In
Out