Average Error: 58.1 → 0.6
Time: 6.6s
Precision: 64
Internal Precision: 128
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{2 \cdot x + \left({x}^{5} \cdot \frac{1}{60} + \frac{1}{3} \cdot {x}^{3}\right)}{2}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.1

    \[\frac{e^{x} - e^{-x}}{2}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2}\]

  3. Final simplification0.6

    \[\leadsto \frac{2 \cdot x + \left({x}^{5} \cdot \frac{1}{60} + \frac{1}{3} \cdot {x}^{3}\right)}{2}\]

Reproduce

herbie shell --seed 2019026 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))