Error

Target

Derivation

1. Split input into 2 regimes

2. if x < -0.0001573056087989334

   1. Initial program 0.0

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

   3. Applied flip--0.0

      \[\leadsto \frac{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}{x}\]
   4. Using strategy rm

   5. Applied flip3-+0.0

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

   6. Taylor expanded around -inf 0.0

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

   7. Simplified0.0

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

   if -0.0001573056087989334 < x

   1. Initial program 60.1

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

   2. Taylor expanded around 0 0.5

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

   3. Simplified0.5

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

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0001573056087989334:\\ \;\;\;\;\frac{\frac{e^{x} \cdot e^{x} - 1}{\frac{1 + e^{x} \cdot \left(e^{x} \cdot e^{x}\right)}{e^{x} \cdot e^{x} + \left(1 - e^{x}\right)}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right) + x}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 
(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))