Average Error: 29.2 → 0.6
Time: 17.2s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)
double f(double x) {
        double r79775 = x;
        double r79776 = exp(r79775);
        double r79777 = 2.0;
        double r79778 = r79776 - r79777;
        double r79779 = -r79775;
        double r79780 = exp(r79779);
        double r79781 = r79778 + r79780;
        return r79781;
}
↓
double f(double x) {
        double r79782 = x;
        double r79783 = 2.0;
        double r79784 = pow(r79782, r79783);
        double r79785 = 0.002777777777777778;
        double r79786 = 6.0;
        double r79787 = pow(r79782, r79786);
        double r79788 = r79785 * r79787;
        double r79789 = 0.08333333333333333;
        double r79790 = 4.0;
        double r79791 = pow(r79782, r79790);
        double r79792 = r79789 * r79791;
        double r79793 = r79788 + r79792;
        double r79794 = r79784 + r79793;
        return r79794;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.2

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

  2. Taylor expanded around 0 0.6

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

  3. Final simplification0.6

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

Reproduce

herbie shell --seed 2019208 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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