Hyperbolic tangent

Average Error: 58.0 → 1.9

Time: 3.2s

Precision: binary64

Math

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

↓

\[x + \left(\frac{2}{15} \cdot {x}^{5} - \frac{1}{3} \cdot {x}^{3}\right)\]

Error

Bits error versus x

Derivation

1. Initial program 58.0

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

2. Taylor expanded around 0 1.9

   \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]
3. Using strategy rm

4. Applied associate--l+ 1.9

   \[\leadsto \color{blue}{x + \left(\frac{2}{15} \cdot {x}^{5} - \frac{1}{3} \cdot {x}^{3}\right)}\]

5. Final simplification 1.9

   \[\leadsto x + \left(\frac{2}{15} \cdot {x}^{5} - \frac{1}{3} \cdot {x}^{3}\right)\]

Reproduce

herbie shell --seed 2020174 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (neg x))) (+ (exp x) (exp (neg x)))))