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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\log \left(e^{1 - e^{-x}}\right)} \le 195.2204320355037:\\ \;\;\;\;\frac{1}{\log \left(e^{1 - e^{-x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)\\ \end{array}\]

Original	40.7
Target	40.4
Herbie	0.6

Derivation

Split input into 2 regimes
if (/ 1 (log (exp (- 1 (exp (- x)))))) < 195.2204320355037
1. Initial program 2.3
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Using strategy rm
3. Applied clear-num2.3
  \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}}\]
4. Applied simplify1.4
  \[\leadsto \frac{1}{\color{blue}{1 - e^{-x}}}\]
5. Using strategy rm
6. Applied add-log-exp1.8
  \[\leadsto \frac{1}{\color{blue}{\log \left(e^{1 - e^{-x}}\right)}}\]
if 195.2204320355037 < (/ 1 (log (exp (- 1 (exp (- x))))))
1. Initial program 61.4
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{x} + \frac{1}{12} \cdot x\right)}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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

Error

Target

Derivation

`if (/ 1 (log (exp (- 1 (exp (- x)))))) < 195.2204320355037`

`if 195.2204320355037 < (/ 1 (log (exp (- 1 (exp (- x))))))`

Runtime