Average Error: 29.5 → 1.1
Time: 29.6s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.9334006562055708:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \frac{2}{3} \cdot x, 2\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(-1 - \varepsilon\right) \cdot x} + \frac{e^{x \cdot \varepsilon - x}}{\varepsilon}\right) - \frac{e^{\left(-1 - \varepsilon\right) \cdot x}}{\varepsilon}\right) + e^{x \cdot \varepsilon - x}}{2}\\ \end{array}\]
\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}
↓
\begin{array}{l}
\mathbf{if}\;x \le 1.9334006562055708:\\
\;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \frac{2}{3} \cdot x, 2\right) - x \cdot x}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\left(e^{\left(-1 - \varepsilon\right) \cdot x} + \frac{e^{x \cdot \varepsilon - x}}{\varepsilon}\right) - \frac{e^{\left(-1 - \varepsilon\right) \cdot x}}{\varepsilon}\right) + e^{x \cdot \varepsilon - x}}{2}\\

\end{array}
double f(double x, double eps) {
        double r1759140 = 1.0;
        double r1759141 = eps;
        double r1759142 = r1759140 / r1759141;
        double r1759143 = r1759140 + r1759142;
        double r1759144 = r1759140 - r1759141;
        double r1759145 = x;
        double r1759146 = r1759144 * r1759145;
        double r1759147 = -r1759146;
        double r1759148 = exp(r1759147);
        double r1759149 = r1759143 * r1759148;
        double r1759150 = r1759142 - r1759140;
        double r1759151 = r1759140 + r1759141;
        double r1759152 = r1759151 * r1759145;
        double r1759153 = -r1759152;
        double r1759154 = exp(r1759153);
        double r1759155 = r1759150 * r1759154;
        double r1759156 = r1759149 - r1759155;
        double r1759157 = 2.0;
        double r1759158 = r1759156 / r1759157;
        return r1759158;
}
↓
double f(double x, double eps) {
        double r1759159 = x;
        double r1759160 = 1.9334006562055708;
        bool r1759161 = r1759159 <= r1759160;
        double r1759162 = r1759159 * r1759159;
        double r1759163 = 0.6666666666666666;
        double r1759164 = r1759163 * r1759159;
        double r1759165 = 2.0;
        double r1759166 = fma(r1759162, r1759164, r1759165);
        double r1759167 = r1759166 - r1759162;
        double r1759168 = r1759167 / r1759165;
        double r1759169 = -1.0;
        double r1759170 = eps;
        double r1759171 = r1759169 - r1759170;
        double r1759172 = r1759171 * r1759159;
        double r1759173 = exp(r1759172);
        double r1759174 = r1759159 * r1759170;
        double r1759175 = r1759174 - r1759159;
        double r1759176 = exp(r1759175);
        double r1759177 = r1759176 / r1759170;
        double r1759178 = r1759173 + r1759177;
        double r1759179 = r1759173 / r1759170;
        double r1759180 = r1759178 - r1759179;
        double r1759181 = r1759180 + r1759176;
        double r1759182 = r1759181 / r1759165;
        double r1759183 = r1759161 ? r1759168 : r1759182;
        return r1759183;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 1.9334006562055708

    1. Initial program 39.0

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

    2. Simplified39.0

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

    3. Taylor expanded around 0 1.2

      \[\leadsto \frac{\color{blue}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}}{2}\]

    4. Simplified1.2

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot \frac{2}{3}, 2\right) - x \cdot x}}{2}\]

    if 1.9334006562055708 < x

    1. Initial program 0.6

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

    2. Simplified0.6

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

    3. Taylor expanded around inf 0.6

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

    4. Simplified0.5

      \[\leadsto \frac{\color{blue}{e^{x \cdot \varepsilon - x} + \left(\left(e^{x \cdot \left(-1 - \varepsilon\right)} + \frac{e^{x \cdot \varepsilon - x}}{\varepsilon}\right) - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.9334006562055708:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \frac{2}{3} \cdot x, 2\right) - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\left(-1 - \varepsilon\right) \cdot x} + \frac{e^{x \cdot \varepsilon - x}}{\varepsilon}\right) - \frac{e^{\left(-1 - \varepsilon\right) \cdot x}}{\varepsilon}\right) + e^{x \cdot \varepsilon - x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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))