Average Error: 29.5 → 0.7

Time: 28.7s

Precision: 64

Internal Precision: 128

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

↓

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

Error

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

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Target

Original	29.5
Target	0.0
Herbie	0.7

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

Derivation

Initial program 29.5
\[\left(e^{x} - 2\right) + e^{-x}\]
Initial simplification29.5
\[\leadsto \left(e^{x} - 2\right) - \frac{-1}{e^{x}}\]
Taylor expanded around 0 0.7
\[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)}\]
Final simplification0.7
\[\leadsto {x}^{2} + \left({x}^{6} \cdot \frac{1}{360} + \frac{1}{12} \cdot {x}^{4}\right)\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.7	0.7	0.0	0.7	0%

herbie shell --seed 2018353 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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