Error

Target

Derivation

1. Split input into 2 regimes

2. if x < -0.00016966990026214703

   1. Initial program 0.1

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

   3. Applied flip3--0.1

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

   4. Applied associate-/l/0.1

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

   5. Applied simplify0.1

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

   if -0.00016966990026214703 < x

   1. Initial program 60.0

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

   2. Taylor expanded around 0 0.4

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

4. Applied simplify0.3

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -0.00016966990026214703:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{3} - 1}{x \cdot \left(e^{x} \cdot e^{x} + \left(1 + e^{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{1}{2} + 1\right) + \frac{1}{6} \cdot {x}^{2}\\ \end{array}}\]

Runtime

Time bar (total: 33.3s)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(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))