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Target

Derivation

1. Split input into 2 regimes

2. if x < -0.00011434802268200287

   1. Initial program 0.1

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

   3. Applied div-inv0.1

      \[\leadsto \color{blue}{\left(e^{x} - 1\right) \cdot \frac{1}{x}}\]
   4. Using strategy rm

   5. Applied flip--0.1

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

   6. Applied associate-*l/0.1

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

   7. Simplified0.0

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

   9. Applied flip-+0.0

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

   if -0.00011434802268200287 < x

   1. Initial program 60.3

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

   2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00011434802268200287:\\ \;\;\;\;\frac{\frac{-1 + e^{x + x}}{x}}{\frac{e^{x} \cdot e^{x} - 1}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{2} + \left({x}^{2} \cdot \frac{1}{6} + 1\right)\\ \end{array}\]

Runtime

Time bar (total: 10.0s)Debug logProfile

herbie shell --seed 2018295 
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

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