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Target

Derivation

1. Split input into 2 regimes

2. if x < -0.00010271566703864994

   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 -0.00010271566703864994 < x

   1. Initial program 60.2

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

   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. Using strategy rm

   4. Applied add-log-exp0.4

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

   5. Applied add-log-exp0.4

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

   6. Applied sum-log0.4

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

   7. Simplified0.4

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

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00010271566703864994:\\ \;\;\;\;\frac{\log \left(e^{e^{x} - 1}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\frac{1}{2} \cdot x + 1\right) + \frac{1}{6} \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]

Runtime

Time bar (total: 12.0s)Debug logProfile

herbie shell --seed 2018351 
(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))