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Target

Derivation

1. Split input into 2 regimes

2. if x < -0.002285063158101973

   1. Initial program 0.0

      \[\frac{e^{x}}{e^{x} - 1}\]

   2. Taylor expanded around inf 0.0

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

   4. Applied add-sqr-sqrt0.0

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

   5. Applied difference-of-sqr-10.0

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

   if -0.002285063158101973 < x

   1. Initial program 60.1

      \[\frac{e^{x}}{e^{x} - 1}\]

   2. Taylor expanded around 0 0.8

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

4. Final simplification0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.002285063158101973:\\ \;\;\;\;\frac{e^{x}}{\left(\sqrt{e^{x}} - 1\right) \cdot \left(1 + \sqrt{e^{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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

Details

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