Error

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Target

Derivation

1. Split input into 2 regimes

2. if x < -0.00017968162493656685

   1. Initial program 0.0

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

   3. Applied div-inv0.0

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

   if -0.00017968162493656685 < x

   1. Initial program 60.3

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00017968162493656685:\\ \;\;\;\;\left(e^{x} - 1\right) \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + {x}^{2} \cdot \frac{1}{6}\right) + x \cdot \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(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

Time bar (total: 5.5s)Debug log