exp2 (problem 3.3.7)

Average Error: 29.9 → 0.7

Time: 4.1s

Precision: binary64

Math

\[\left(e^{x} - 2\right) + e^{-x}\]

↓

\[x \cdot x + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\]

Error

Bits error versus x

Target

Original	29.9
Target	0.1
Herbie	0.7

\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

1. Initial program 29.9

   \[\left(e^{x} - 2\right) + e^{-x}\]

2. Simplified 29.9

   \[\leadsto \color{blue}{e^{x} + \left(e^{-x} - 2\right)}\]

3. Taylor expanded around 0 0.7

   \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}\]

4. Simplified 0.7

   \[\leadsto \color{blue}{x \cdot x + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)}\]

5. Final simplification 0.7

   \[\leadsto x \cdot x + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\]

Reproduce

herbie shell --seed 2020185 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :herbie-expected 1.5

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (neg x))))