Average Error: 29.1 → 0.6
Time: 4.9s
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 r92790 = x;
        double r92791 = exp(r92790);
        double r92792 = 2.0;
        double r92793 = r92791 - r92792;
        double r92794 = -r92790;
        double r92795 = exp(r92794);
        double r92796 = r92793 + r92795;
        return r92796;
}
↓
double f(double x) {
        double r92797 = x;
        double r92798 = 2.0;
        double r92799 = pow(r92797, r92798);
        double r92800 = 0.002777777777777778;
        double r92801 = 6.0;
        double r92802 = pow(r92797, r92801);
        double r92803 = r92800 * r92802;
        double r92804 = 0.08333333333333333;
        double r92805 = 4.0;
        double r92806 = pow(r92797, r92805);
        double r92807 = r92804 * r92806;
        double r92808 = r92803 + r92807;
        double r92809 = r92799 + r92808;
        return r92809;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.1
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.1

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

  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 2020083 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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