Average Error: 28.9 → 0.6
Time: 5.1s
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 r93839 = x;
        double r93840 = exp(r93839);
        double r93841 = 2.0;
        double r93842 = r93840 - r93841;
        double r93843 = -r93839;
        double r93844 = exp(r93843);
        double r93845 = r93842 + r93844;
        return r93845;
}
↓
double f(double x) {
        double r93846 = x;
        double r93847 = 2.0;
        double r93848 = pow(r93846, r93847);
        double r93849 = 0.002777777777777778;
        double r93850 = 6.0;
        double r93851 = pow(r93846, r93850);
        double r93852 = r93849 * r93851;
        double r93853 = 0.08333333333333333;
        double r93854 = 4.0;
        double r93855 = pow(r93846, r93854);
        double r93856 = r93853 * r93855;
        double r93857 = r93852 + r93856;
        double r93858 = r93848 + r93857;
        return r93858;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.9
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 28.9

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

  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 2020064 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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