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Target

Derivation

1. Split input into 2 regimes

2. if x < -0.00017053935770967545

   1. Initial program 0.0

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

   3. Applied add-log-exp0.0

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

   5. Applied flip3--0.0

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

   if -0.00017053935770967545 < x

   1. Initial program 60.1

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

   2. Taylor expanded around 0 0.4

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

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00017053935770967545:\\ \;\;\;\;\frac{\log \left(e^{\frac{{\left(e^{x}\right)}^{3} - 1}{\left(e^{x} + 1\right) + e^{x} \cdot e^{x}}}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{2} + \left({x}^{2} \cdot \frac{1}{6} + 1\right)\\ \end{array}\]

Runtime

Time bar (total: 6.2s)Debug logProfile

herbie shell --seed 2018290 
(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))