Error

Target

Derivation

1. Split input into 2 regimes

2. if x < -0.00014688974838714762

   1. Initial program 0.1

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

   3. Applied flip--0.1

      \[\leadsto \frac{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}{x}\]

   4. Applied simplify0.0

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

   6. Applied flip--0.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{e^{x + x} \cdot e^{x + x} - 1 \cdot 1}{e^{x + x} + 1}}}{e^{x} + 1}}{x}\]

   7. Applied simplify0.0

      \[\leadsto \frac{\frac{\frac{\color{blue}{{\left(e^{2}\right)}^{\left(x + x\right)} - 1}}{e^{x + x} + 1}}{e^{x} + 1}}{x}\]
   8. Using strategy rm

   9. Applied flip-+0.0

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

   10. Applied associate-/r/0.0

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

   11. Applied simplify0.0

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

   if -0.00014688974838714762 < x

   1. Initial program 60.1

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

   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.

Runtime

Time bar (total: 27.7s)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' 
(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))