Error

Target

Derivation

1. Split input into 2 regimes

2. if x < -0.002056753941261893

   1. Initial program 0.0

      \[\frac{e^{x}}{e^{x} - 1}\]
   2. Using strategy rm

   3. Applied flip3--0.0

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}}}\]

   4. Taylor expanded around inf 0.0

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{{\left(e^{x}\right)}^{3} - 1}{{\left(e^{x}\right)}^{2} + \left(e^{x} + 1\right)}}}\]

   5. Simplified0.0

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{e^{x}}{\frac{e^{x} \cdot e^{x} + \left(e^{x} + 1\right)}{e^{x} \cdot e^{x}}} - \frac{1}{e^{x} \cdot e^{x} + \left(e^{x} + 1\right)}}}\]

   if -0.002056753941261893 < x

   1. Initial program 60.0

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

   2. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.002056753941261893:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x}}{\frac{\left(e^{x} + 1\right) + e^{x} \cdot e^{x}}{e^{x} \cdot e^{x}}} - \frac{1}{\left(e^{x} + 1\right) + e^{x} \cdot e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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