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Target

Derivation

1. Split input into 2 regimes

2. if (+ (- (exp x) 2) (exp (- x))) < 0.014331332012292242

   1. Initial program 30.4

      \[\left(e^{x} - 2\right) + e^{-x}\]

   2. Initial simplification30.4

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

   3. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)}\]

   if 0.014331332012292242 < (+ (- (exp x) 2) (exp (- x)))

   1. Initial program 0.6

      \[\left(e^{x} - 2\right) + e^{-x}\]

   2. Initial simplification0.6

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

   4. Applied sub-neg0.6

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

   5. Applied associate--l+0.6

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-x} + \left(e^{x} - 2\right) \le 0.014331332012292242:\\ \;\;\;\;{x}^{2} + \left({x}^{4} \cdot \frac{1}{12} + \frac{1}{360} \cdot {x}^{6}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(\left(-2\right) - \frac{-1}{e^{x}}\right)\\ \end{array}\]

Runtime

Time bar (total: 19.6s)Debug logProfile

herbie shell --seed 2018215 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))