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Target

Derivation

1. Initial program 39.7

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

2. Taylor expanded around 0 11.8

   \[\leadsto \frac{e^{x}}{\color{blue}{x + \left(\frac{1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot {x}^{2}\right)}}\]

3. Simplified1.1

   \[\leadsto \frac{e^{x}}{\color{blue}{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}}\]
4. Using strategy rm

5. Applied *-un-lft-identity1.1

   \[\leadsto \frac{\color{blue}{1 \cdot e^{x}}}{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}\]

6. Applied associate-/l*1.1

   \[\leadsto \color{blue}{\frac{1}{\frac{x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)}{e^{x}}}}\]

7. Final simplification1.1

   \[\leadsto \frac{1}{\frac{\left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + x}{e^{x}}}\]

Reproduce

herbie shell --seed 2019008 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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

Details

Time bar (total: 16.4s)Debug log