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

↓

\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.7310163699296692:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x + x} - 1}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Original	40.1
Target	39.8
Herbie	0.6

Derivation

Split input into 2 regimes
if (exp x) < 0.7310163699296692
1. Initial program 0.0
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Using strategy rm
3. Applied flip--0.0
  \[\leadsto \frac{e^{x}}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}\]
4. Applied simplify0.0
  \[\leadsto \frac{e^{x}}{\frac{\color{blue}{e^{x + x} - 1}}{e^{x} + 1}}\]
if 0.7310163699296692 < (exp x)
1. Initial program 59.9
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Taylor expanded around 0 1.0
  \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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

Error

Target

Derivation

`if (exp x) < 0.7310163699296692`

`if 0.7310163699296692 < (exp x)`

Runtime