Average Error: 30.1 → 0.6
Time: 10.1s
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 r124826 = x;
        double r124827 = exp(r124826);
        double r124828 = 2.0;
        double r124829 = r124827 - r124828;
        double r124830 = -r124826;
        double r124831 = exp(r124830);
        double r124832 = r124829 + r124831;
        return r124832;
}
↓
double f(double x) {
        double r124833 = x;
        double r124834 = 2.0;
        double r124835 = pow(r124833, r124834);
        double r124836 = 0.002777777777777778;
        double r124837 = 6.0;
        double r124838 = pow(r124833, r124837);
        double r124839 = r124836 * r124838;
        double r124840 = 0.08333333333333333;
        double r124841 = 4.0;
        double r124842 = pow(r124833, r124841);
        double r124843 = r124840 * r124842;
        double r124844 = r124839 + r124843;
        double r124845 = r124835 + r124844;
        return r124845;
}

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 2020042 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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