Average Error: 29.4 → 0.6

Time: 16.4s

Precision: 64

Internal Precision: 1344

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

↓

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

Error

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

Target

Original	29.4
Target	0.0
Herbie	0.6

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

Derivation

Initial program 29.4
\[\left(e^{x} - 2\right) + e^{-x}\]
Initial simplification29.4
\[\leadsto \left(e^{x} - 2\right) - \frac{-1}{e^{x}}\]
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
\[\leadsto \color{blue}{(\frac{1}{12} \cdot \left({x}^{4}\right) + \left((\frac{1}{360} \cdot \left({x}^{6}\right) + \left(x \cdot x\right))_*\right))_*}\]
Using strategy rm
Applied add-sqr-sqrt0.6
\[\leadsto (\frac{1}{12} \cdot \left({x}^{4}\right) + \color{blue}{\left(\sqrt{(\frac{1}{360} \cdot \left({x}^{6}\right) + \left(x \cdot x\right))_*} \cdot \sqrt{(\frac{1}{360} \cdot \left({x}^{6}\right) + \left(x \cdot x\right))_*}\right)})_*\]
Final simplification0.6
\[\leadsto (\frac{1}{12} \cdot \left({x}^{4}\right) + \left(\sqrt{(\frac{1}{360} \cdot \left({x}^{6}\right) + \left(x \cdot x\right))_*} \cdot \sqrt{(\frac{1}{360} \cdot \left({x}^{6}\right) + \left(x \cdot x\right))_*}\right))_*\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.6	0.6	0.0	0.6	0%

herbie shell --seed 2018290 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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