Average Error: 29.2 → 0.7
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{360} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\]

Error

Bits error versus x

Target

Original29.2
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.2

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

  2. Simplified29.1

    \[\leadsto \color{blue}{\left(e^{x} - 2\right) - \frac{-1}{e^{x}}}\]

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019091 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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