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Target

Derivation

1. Split input into 2 regimes

2. if x < -0.00017454211774255434

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt0.1

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

   4. Applied difference-of-sqr-10.1

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

   if -0.00017454211774255434 < x

   1. Initial program 60.0

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

   2. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]
   3. Using strategy rm

   4. Applied associate-+r+0.5

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

4. Final simplification0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00017454211774255434:\\ \;\;\;\;\frac{\left(\sqrt{e^{x}} - 1\right) \cdot \left(1 + \sqrt{e^{x}}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 
(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))

Details

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