Average Error: 34.3 → 0.1
Time: 8.4s
Precision: 64
Internal precision: 1408
\[\left(e^{x} - 2\right) + e^{-x}\]
⬇
\[{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)\]

Error

Bits error versus x

Target

Original34.3
Comparison9.1
Herbie0.1
\[ 4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^2 \]

Derivation

  1. Initial program 34.3

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

  2. Applied taylor 0.1

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

  3. Taylor expanded around 0 0.1

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

Runtime

Time bar (total: 8.4s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1067901057 3396600083 3715501224 3126139233 3908045574 1593683916)'
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :target
  (* 4 (sqr (sinh (/ x 2))))

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