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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{1 - e^{-x}} \le 1.0:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Original	40.5
Target	40.1
Herbie	0.5

Derivation

Split input into 2 regimes
if (/ 1 (- 1 (exp (- x)))) < 1.0
1. Initial program 2.4
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Using strategy rm
3. Applied clear-num2.4
  \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}}\]
4. Applied simplify1.1
  \[\leadsto \frac{1}{\color{blue}{1 - e^{-x}}}\]
if 1.0 < (/ 1 (- 1 (exp (- x))))
1. Initial program 61.2
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Taylor expanded around 0 0.2
  \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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

Error

Target

Derivation

`if (/ 1 (- 1 (exp (- x)))) < 1.0`

`if 1.0 < (/ 1 (- 1 (exp (- x))))`

Runtime