Error

Target

Derivation

1. Split input into 2 regimes

2. if (exp x) < 0.499999998769616

   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

      \[\leadsto \frac{e^{x}}{\frac{\color{blue}{e^{x + x} - 1}}{e^{x} + 1}}\]
   5. Using strategy rm

   6. Applied flip--0.0

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

   7. Applied associate-/l/0.0

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

   if 0.499999998769616 < (exp x)

   1. Initial program 59.7

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

   2. Taylor expanded around 0 1.1

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

4. Applied simplify0.8

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;e^{x} \le 0.499999998769616:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x + x} \cdot e^{x + x} - 1}{\left(e^{x + x} + 1\right) \cdot \left(e^{x} + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}}\]

Runtime

Time bar (total: 59.8s)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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