Average Error: 29.8 → 0.6
Time: 5.8s
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 r105868 = x;
        double r105869 = exp(r105868);
        double r105870 = 2.0;
        double r105871 = r105869 - r105870;
        double r105872 = -r105868;
        double r105873 = exp(r105872);
        double r105874 = r105871 + r105873;
        return r105874;
}
↓
double f(double x) {
        double r105875 = x;
        double r105876 = 2.0;
        double r105877 = pow(r105875, r105876);
        double r105878 = 0.002777777777777778;
        double r105879 = 6.0;
        double r105880 = pow(r105875, r105879);
        double r105881 = r105878 * r105880;
        double r105882 = 0.08333333333333333;
        double r105883 = 4.0;
        double r105884 = pow(r105875, r105883);
        double r105885 = r105882 * r105884;
        double r105886 = r105881 + r105885;
        double r105887 = r105877 + r105886;
        return r105887;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

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

  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 2020002 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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