Error

Target

Derivation

1. Split input into 2 regimes

2. if (+ (- (exp x) 2) (exp (- x))) < 0.00020891152756186138

   1. Initial program 29.8

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

   2. Taylor expanded around 0 0.0

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

   3. Simplified0.0

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

   5. Applied add-sqr-sqrt0.0

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

   if 0.00020891152756186138 < (+ (- (exp x) 2) (exp (- x)))

   1. Initial program 1.8

      \[\left(e^{x} - 2\right) + e^{-x}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \le 0.00020891152756186138:\\ \;\;\;\;\sqrt{(\frac{1}{12} \cdot \left({x}^{4}\right) + \left((\frac{1}{360} \cdot \left({x}^{6}\right) + \left(x \cdot x\right))_*\right))_*} \cdot \sqrt{(\frac{1}{12} \cdot \left({x}^{4}\right) + \left((\frac{1}{360} \cdot \left({x}^{6}\right) + \left(x \cdot x\right))_*\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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

Details

Time bar (total: 18.8s)Debug log