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Target

Derivation

1. Split input into 2 regimes

2. if x < -0.0018581611268275176

   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 *-un-lft-identity0.0

      \[\leadsto \frac{\color{blue}{1 \cdot e^{x}}}{e^{x} - 1}\]

   5. Applied associate-/l*0.0

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}}\]
   6. Using strategy rm

   7. Applied div-inv0.0

      \[\leadsto \frac{1}{\color{blue}{\left(e^{x} - 1\right) \cdot \frac{1}{e^{x}}}}\]

   if -0.0018581611268275176 < 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

      \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0018581611268275176:\\ \;\;\;\;\frac{1}{\left(e^{x} - 1\right) \cdot \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Runtime

Time bar (total: 14.2s)Debug logProfile

herbie shell --seed 2018340 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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