Average Error: 58.9 → 0.1
Time: 24.4s
Precision: 64
Internal precision: 128
\[\frac{e^{x} - e^{-x}}{2}\]
⬇
\[\frac{\frac{1}{3} \cdot {x}^3 + \left({x}^{5} \cdot \frac{1}{60} + \left(x + x\right)\right)}{2}\]

Error

Bits error versus x

Derivation

  1. Initial program 58.9

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

  2. Applied taylor 0.1

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

  3. Taylor expanded around 0 0.1

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

  4. Applied simplify 0.1

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

Runtime

Time bar (total: 24.4s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(2096387104 3042562760 2267621671 2362568811 1492117559 1700784281)'
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))