Average Error: 29.4 → 0.7

Time: 13.2s

Precision: 64

Internal Precision: 1344

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

↓

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

Error

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

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Out

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Target

Original	29.4
Target	0.0
Herbie	0.7

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

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
\[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)}\]
Using strategy rm
Applied add-log-exp0.7
\[\leadsto {x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + \color{blue}{\log \left(e^{\frac{1}{360} \cdot {x}^{6}}\right)}\right)\]
Final simplification0.7
\[\leadsto \left(\log \left(e^{{x}^{6} \cdot \frac{1}{360}}\right) + \frac{1}{12} \cdot {x}^{4}\right) + {x}^{2}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.7	0.7	0.0	0.6	0%

herbie shell --seed 2018296 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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