Average Error: 29.9 → 0.6
Time: 5.6s
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 r101705 = x;
        double r101706 = exp(r101705);
        double r101707 = 2.0;
        double r101708 = r101706 - r101707;
        double r101709 = -r101705;
        double r101710 = exp(r101709);
        double r101711 = r101708 + r101710;
        return r101711;
}
↓
double f(double x) {
        double r101712 = x;
        double r101713 = 2.0;
        double r101714 = pow(r101712, r101713);
        double r101715 = 0.002777777777777778;
        double r101716 = 6.0;
        double r101717 = pow(r101712, r101716);
        double r101718 = r101715 * r101717;
        double r101719 = 0.08333333333333333;
        double r101720 = 4.0;
        double r101721 = pow(r101712, r101720);
        double r101722 = r101719 * r101721;
        double r101723 = r101718 + r101722;
        double r101724 = r101714 + r101723;
        return r101724;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

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

  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 2020033 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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