Average Error: 29.7 → 0.6
Time: 6.1s
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 r81834 = x;
        double r81835 = exp(r81834);
        double r81836 = 2.0;
        double r81837 = r81835 - r81836;
        double r81838 = -r81834;
        double r81839 = exp(r81838);
        double r81840 = r81837 + r81839;
        return r81840;
}
↓
double f(double x) {
        double r81841 = x;
        double r81842 = 2.0;
        double r81843 = pow(r81841, r81842);
        double r81844 = 0.002777777777777778;
        double r81845 = 6.0;
        double r81846 = pow(r81841, r81845);
        double r81847 = r81844 * r81846;
        double r81848 = 0.08333333333333333;
        double r81849 = 4.0;
        double r81850 = pow(r81841, r81849);
        double r81851 = r81848 * r81850;
        double r81852 = r81847 + r81851;
        double r81853 = r81843 + r81852;
        return r81853;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.7

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

  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 2020081 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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