Average Error: 29.6 → 0.6
Time: 18.3s
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 r77745 = x;
        double r77746 = exp(r77745);
        double r77747 = 2.0;
        double r77748 = r77746 - r77747;
        double r77749 = -r77745;
        double r77750 = exp(r77749);
        double r77751 = r77748 + r77750;
        return r77751;
}
↓
double f(double x) {
        double r77752 = x;
        double r77753 = 2.0;
        double r77754 = pow(r77752, r77753);
        double r77755 = 0.002777777777777778;
        double r77756 = 6.0;
        double r77757 = pow(r77752, r77756);
        double r77758 = r77755 * r77757;
        double r77759 = 0.08333333333333333;
        double r77760 = 4.0;
        double r77761 = pow(r77752, r77760);
        double r77762 = r77759 * r77761;
        double r77763 = r77758 + r77762;
        double r77764 = r77754 + r77763;
        return r77764;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.6

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

  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 2019325 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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