Hyperbolic sine

Average Error: 58.1 → 0.6

Time: 10.1min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

Error

Bits error versus x

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 simplification 0.6

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

Reproduce

herbie shell --seed 2020181 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (neg x))) 2.0))