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\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
double f(double x, double eps) {
        double r1169662 = 1.0;
        double r1169663 = eps;
        double r1169664 = r1169662 / r1169663;
        double r1169665 = r1169662 + r1169664;
        double r1169666 = r1169662 - r1169663;
        double r1169667 = x;
        double r1169668 = r1169666 * r1169667;
        double r1169669 = -r1169668;
        double r1169670 = exp(r1169669);
        double r1169671 = r1169665 * r1169670;
        double r1169672 = r1169664 - r1169662;
        double r1169673 = r1169662 + r1169663;
        double r1169674 = r1169673 * r1169667;
        double r1169675 = -r1169674;
        double r1169676 = exp(r1169675);
        double r1169677 = r1169672 * r1169676;
        double r1169678 = r1169671 - r1169677;
        double r1169679 = 2.0;
        double r1169680 = r1169678 / r1169679;
        return r1169680;
}

Reproduce

herbie shell --seed 2019135 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))