Average Error: 30.1 → 0.6
Time: 8.1s
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 r101921 = x;
        double r101922 = exp(r101921);
        double r101923 = 2.0;
        double r101924 = r101922 - r101923;
        double r101925 = -r101921;
        double r101926 = exp(r101925);
        double r101927 = r101924 + r101926;
        return r101927;
}
↓
double f(double x) {
        double r101928 = x;
        double r101929 = 2.0;
        double r101930 = pow(r101928, r101929);
        double r101931 = 0.002777777777777778;
        double r101932 = 6.0;
        double r101933 = pow(r101928, r101932);
        double r101934 = r101931 * r101933;
        double r101935 = 0.08333333333333333;
        double r101936 = 4.0;
        double r101937 = pow(r101928, r101936);
        double r101938 = r101935 * r101937;
        double r101939 = r101934 + r101938;
        double r101940 = r101930 + r101939;
        return r101940;
}

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 2020024 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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