Average Error: 40.3 → 0.7

Time: 23.4s

Precision: 64

Internal Precision: 1344

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

↓

\[\frac{\sqrt{e^{x}}}{x + \left({x}^{5} \cdot \frac{1}{1920} + \frac{1}{24} \cdot {x}^{3}\right)}\]

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

Initial program 40.3
\[\frac{e^{x}}{e^{x} - 1}\]
Initial simplification40.3
\[\leadsto \frac{e^{x}}{e^{x} - 1}\]
Using strategy rm
Applied add-sqr-sqrt40.3
\[\leadsto \frac{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}}}{e^{x} - 1}\]
Applied associate-/l*40.3
\[\leadsto \color{blue}{\frac{\sqrt{e^{x}}}{\frac{e^{x} - 1}{\sqrt{e^{x}}}}}\]
Taylor expanded around 0 0.7
\[\leadsto \frac{\sqrt{e^{x}}}{\color{blue}{x + \left(\frac{1}{24} \cdot {x}^{3} + \frac{1}{1920} \cdot {x}^{5}\right)}}\]
Final simplification0.7
\[\leadsto \frac{\sqrt{e^{x}}}{x + \left({x}^{5} \cdot \frac{1}{1920} + \frac{1}{24} \cdot {x}^{3}\right)}\]

	Baseline	Herbie	Oracle	Span	%
Regimes	0.7	0.7	0.3	0.4	0%

herbie shell --seed 2018339 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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