Average Error: 40.0 → 1.1

Time: 20.7s

Precision: 64

Internal Precision: 128

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

↓

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

Error

Target

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

Initial program 40.0
\[\frac{e^{x}}{e^{x} - 1}\]
Taylor expanded around 0 11.7
\[\leadsto \frac{e^{x}}{\color{blue}{x + \left(\frac{1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot {x}^{2}\right)}}\]
Simplified1.1
\[\leadsto \frac{e^{x}}{\color{blue}{x + \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)}}\]
Final simplification1.1
\[\leadsto \frac{e^{x}}{x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)}\]

herbie shell --seed 2019051 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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