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Target

Derivation

1. Split input into 2 regimes

2. if x < -0.00020187990748026416

   1. Initial program 0.1

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

   3. Applied add-log-exp0.1

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

   if -0.00020187990748026416 < x

   1. Initial program 59.9

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

   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 add-cube-cbrt0.5

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

   5. Taylor expanded around 0 0.5

      \[\leadsto \left(\sqrt[3]{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)} \cdot \color{blue}{\left(\frac{1}{6} \cdot x + \left(\frac{1}{36} \cdot {x}^{2} + 1\right)\right)}\right) \cdot \sqrt[3]{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]

   6. Taylor expanded around 0 0.5

      \[\leadsto \left(\color{blue}{\left(\frac{1}{6} \cdot x + \left(\frac{1}{36} \cdot {x}^{2} + 1\right)\right)} \cdot \left(\frac{1}{6} \cdot x + \left(\frac{1}{36} \cdot {x}^{2} + 1\right)\right)\right) \cdot \sqrt[3]{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00020187990748026416:\\ \;\;\;\;\frac{\log \left(e^{e^{x} - 1}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 + \frac{1}{36} \cdot {x}^{2}\right) + \frac{1}{6} \cdot x\right) \cdot \left(\left(1 + \frac{1}{36} \cdot {x}^{2}\right) + \frac{1}{6} \cdot x\right)\right) \cdot \sqrt[3]{\left(\frac{1}{6} \cdot {x}^{2} + 1\right) + x \cdot \frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(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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