Average Error: 29.3 → 0.6
Time: 24.0s
Precision: 64
Internal Precision: 1408
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[(\frac{1}{12} \cdot \left({x}^{4}\right) + \left(\sqrt{(\left({x}^{6}\right) \cdot \frac{1}{360} + \left(x \cdot x\right))_*} \cdot \sqrt{(\left({x}^{6}\right) \cdot \frac{1}{360} + \left(x \cdot x\right))_*}\right))_*\]

Error

Bits error versus x

Target

Original29.3
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.3

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

  2. Taylor expanded around 0 0.6

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

  3. Applied simplify0.6

    \[\leadsto \color{blue}{(\frac{1}{12} \cdot \left({x}^{4}\right) + \left((\left({x}^{6}\right) \cdot \frac{1}{360} + \left(x \cdot x\right))_*\right))_*}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.6

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

Runtime

Time bar (total: 24.0s)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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