Average Error: 29.4 → 1.2
Time: 20.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\frac{\left(\left(\frac{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}}{\varepsilon} - \frac{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}}{\varepsilon}\right) + e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}\right) \cdot 1 + 1 \cdot e^{\left(\varepsilon - 1\right) \cdot x}}{2}\]
\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}
↓
\frac{\left(\left(\frac{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}}{\varepsilon} - \frac{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}}{\varepsilon}\right) + e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}\right) \cdot 1 + 1 \cdot e^{\left(\varepsilon - 1\right) \cdot x}}{2}
double f(double x, double eps) {
        double r32954 = 1.0;
        double r32955 = eps;
        double r32956 = r32954 / r32955;
        double r32957 = r32954 + r32956;
        double r32958 = r32954 - r32955;
        double r32959 = x;
        double r32960 = r32958 * r32959;
        double r32961 = -r32960;
        double r32962 = exp(r32961);
        double r32963 = r32957 * r32962;
        double r32964 = r32956 - r32954;
        double r32965 = r32954 + r32955;
        double r32966 = r32965 * r32959;
        double r32967 = -r32966;
        double r32968 = exp(r32967);
        double r32969 = r32964 * r32968;
        double r32970 = r32963 - r32969;
        double r32971 = 2.0;
        double r32972 = r32970 / r32971;
        return r32972;
}
↓
double f(double x, double eps) {
        double r32973 = x;
        double r32974 = exp(r32973);
        double r32975 = eps;
        double r32976 = 1.0;
        double r32977 = r32975 - r32976;
        double r32978 = pow(r32974, r32977);
        double r32979 = r32978 / r32975;
        double r32980 = -r32973;
        double r32981 = exp(r32980);
        double r32982 = r32975 + r32976;
        double r32983 = pow(r32981, r32982);
        double r32984 = r32983 / r32975;
        double r32985 = r32979 - r32984;
        double r32986 = r32980 * r32982;
        double r32987 = exp(r32986);
        double r32988 = r32985 + r32987;
        double r32989 = r32988 * r32976;
        double r32990 = r32977 * r32973;
        double r32991 = exp(r32990);
        double r32992 = r32976 * r32991;
        double r32993 = r32989 + r32992;
        double r32994 = 2.0;
        double r32995 = r32993 / r32994;
        return r32995;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.4

    \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

  2. Taylor expanded around inf 29.4

    \[\leadsto \frac{\color{blue}{\left(1 \cdot e^{x \cdot \varepsilon - 1 \cdot x} + \left(1 \cdot e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)} + 1 \cdot \frac{e^{x \cdot \varepsilon - 1 \cdot x}}{\varepsilon}\right)\right) - 1 \cdot \frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon}}}{2}\]

  3. Simplified24.7

    \[\leadsto \frac{\color{blue}{e^{x \cdot \left(\varepsilon - 1\right)} \cdot 1 + 1 \cdot \left(\left(e^{-x \cdot \left(\varepsilon + 1\right)} + \frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon}\right) - \frac{e^{-x \cdot \left(\varepsilon + 1\right)}}{\varepsilon}\right)}}{2}\]
  4. Using strategy rm

  5. Applied associate--l+0.9

    \[\leadsto \frac{e^{x \cdot \left(\varepsilon - 1\right)} \cdot 1 + 1 \cdot \color{blue}{\left(e^{-x \cdot \left(\varepsilon + 1\right)} + \left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon} - \frac{e^{-x \cdot \left(\varepsilon + 1\right)}}{\varepsilon}\right)\right)}}{2}\]

  6. Simplified1.2

    \[\leadsto \frac{e^{x \cdot \left(\varepsilon - 1\right)} \cdot 1 + 1 \cdot \left(e^{-x \cdot \left(\varepsilon + 1\right)} + \color{blue}{\left(\frac{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}}{\varepsilon} - \frac{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}}{\varepsilon}\right)}\right)}{2}\]

  7. Final simplification1.2

    \[\leadsto \frac{\left(\left(\frac{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}}{\varepsilon} - \frac{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}}{\varepsilon}\right) + e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}\right) \cdot 1 + 1 \cdot e^{\left(\varepsilon - 1\right) \cdot x}}{2}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))