Average Error: 40.1 → 1.0

Time: 34.6s

Precision: 64

Internal Precision: 1344

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

↓

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

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

Initial program 40.1
\[\frac{e^{x}}{e^{x} - 1}\]
Taylor expanded around 0 11.4
\[\leadsto \frac{e^{x}}{\color{blue}{x + \left(\frac{1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot {x}^{2}\right)}}\]
Simplified1.0
\[\leadsto \frac{e^{x}}{\color{blue}{x + \left(x \cdot x\right) \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right)}}\]
Final simplification1.0
\[\leadsto \frac{e^{x}}{x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)}\]

herbie shell --seed 2018219 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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