Average Error: 57.8 → 0.8
Time: 6.9s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{{x}^{3} \cdot \frac{1}{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}\]

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.8

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

  2. Taylor expanded around 0 0.8

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

  3. Final simplification0.8

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

Runtime

Time bar (total: 6.9s)Debug logProfile

herbie shell --seed 2018215 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))