Average Error: 29.1 → 0.6

Time: 26.9s

Precision: 64

Internal Precision: 128

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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	29.1
Target	0.0
Herbie	0.6

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

Derivation

Initial program 29.1
\[\left(e^{x} - 2\right) + e^{-x}\]
Simplified29.1
\[\leadsto \color{blue}{\left(e^{x} - 2\right) - \frac{-1}{e^{x}}}\]
Taylor expanded around 0 0.6
\[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)}\]
Simplified0.6
\[\leadsto \color{blue}{{x}^{6} \cdot \frac{1}{360} + \left(x \cdot x + \frac{1}{12} \cdot {x}^{4}\right)}\]
Final simplification0.6
\[\leadsto \left(x \cdot x + \frac{1}{12} \cdot {x}^{4}\right) + {x}^{6} \cdot \frac{1}{360}\]

Reproduce

herbie shell --seed 2019093 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))