Average Error: 57.9 → 0.6
Time: 23.2s
Precision: 64
Internal Precision: 1344
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) + x \cdot 2\right) + {x}^{5} \cdot \frac{1}{60}}{2}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.9

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

  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. Applied simplify0.6

    \[\leadsto \color{blue}{\frac{(x \cdot \left((\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_*\right) + \left({x}^{5} \cdot \frac{1}{60}\right))_*}{2}}\]
  4. Using strategy rm

  5. Applied fma-udef0.6

    \[\leadsto \frac{\color{blue}{x \cdot (\frac{1}{3} \cdot \left(x \cdot x\right) + 2)_* + {x}^{5} \cdot \frac{1}{60}}}{2}\]
  6. Using strategy rm

  7. Applied fma-udef0.6

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

  8. Applied distribute-lft-in0.6

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

Runtime

Time bar (total: 23.2s)Debug logProfile

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))