Average Error: 30.1 → 0.6
Time: 4.8s
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 r137920 = x;
        double r137921 = exp(r137920);
        double r137922 = 2.0;
        double r137923 = r137921 - r137922;
        double r137924 = -r137920;
        double r137925 = exp(r137924);
        double r137926 = r137923 + r137925;
        return r137926;
}
↓
double f(double x) {
        double r137927 = x;
        double r137928 = 2.0;
        double r137929 = pow(r137927, r137928);
        double r137930 = 0.002777777777777778;
        double r137931 = 6.0;
        double r137932 = pow(r137927, r137931);
        double r137933 = r137930 * r137932;
        double r137934 = 0.08333333333333333;
        double r137935 = 4.0;
        double r137936 = pow(r137927, r137935);
        double r137937 = r137934 * r137936;
        double r137938 = r137933 + r137937;
        double r137939 = r137929 + r137938;
        return r137939;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  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 2020065 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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