Average Error: 29.5 → 1.1
Time: 11.1s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 201.192235912335804:\\ \;\;\;\;\frac{\sqrt[3]{{\left(2 + \left(x \cdot x\right) \cdot \left(0.66666666666666674 \cdot x - 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{e^{-x \cdot \left(1 - \varepsilon\right)}}{\varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}\right) - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 201.192235912335804:\\
\;\;\;\;\frac{\sqrt[3]{{\left(2 + \left(x \cdot x\right) \cdot \left(0.66666666666666674 \cdot x - 1\right)\right)}^{3}}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r41170 = 1.0;
        double r41171 = eps;
        double r41172 = r41170 / r41171;
        double r41173 = r41170 + r41172;
        double r41174 = r41170 - r41171;
        double r41175 = x;
        double r41176 = r41174 * r41175;
        double r41177 = -r41176;
        double r41178 = exp(r41177);
        double r41179 = r41173 * r41178;
        double r41180 = r41172 - r41170;
        double r41181 = r41170 + r41171;
        double r41182 = r41181 * r41175;
        double r41183 = -r41182;
        double r41184 = exp(r41183);
        double r41185 = r41180 * r41184;
        double r41186 = r41179 - r41185;
        double r41187 = 2.0;
        double r41188 = r41186 / r41187;
        return r41188;
}
↓
double f(double x, double eps) {
        double r41189 = x;
        double r41190 = 201.1922359123358;
        bool r41191 = r41189 <= r41190;
        double r41192 = 2.0;
        double r41193 = r41189 * r41189;
        double r41194 = 0.6666666666666667;
        double r41195 = r41194 * r41189;
        double r41196 = 1.0;
        double r41197 = r41195 - r41196;
        double r41198 = r41193 * r41197;
        double r41199 = r41192 + r41198;
        double r41200 = 3.0;
        double r41201 = pow(r41199, r41200);
        double r41202 = cbrt(r41201);
        double r41203 = r41202 / r41192;
        double r41204 = eps;
        double r41205 = r41196 - r41204;
        double r41206 = r41189 * r41205;
        double r41207 = -r41206;
        double r41208 = exp(r41207);
        double r41209 = r41208 / r41204;
        double r41210 = r41209 + r41208;
        double r41211 = r41196 * r41210;
        double r41212 = r41196 / r41204;
        double r41213 = r41212 - r41196;
        double r41214 = r41196 + r41204;
        double r41215 = r41214 * r41189;
        double r41216 = exp(r41215);
        double r41217 = r41213 / r41216;
        double r41218 = r41211 - r41217;
        double r41219 = r41218 / r41192;
        double r41220 = r41191 ? r41203 : r41219;
        return r41220;
}

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. Split input into 2 regimes

  2. if x < 201.1922359123358

    1. Initial program 39.2

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

    2. Simplified39.2

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

    3. Taylor expanded around 0 1.4

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube1.4

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right) \cdot \left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right)\right) \cdot \left(\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right)}}}{2}\]

    6. Simplified1.4

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(2 + \left(x \cdot x\right) \cdot \left(0.66666666666666674 \cdot x - 1\right)\right)}^{3}}}}{2}\]

    if 201.1922359123358 < x

    1. Initial program 0.0

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

    2. Simplified0.0

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.1

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 201.192235912335804:\\ \;\;\;\;\frac{\sqrt[3]{{\left(2 + \left(x \cdot x\right) \cdot \left(0.66666666666666674 \cdot x - 1\right)\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{e^{-x \cdot \left(1 - \varepsilon\right)}}{\varepsilon} + e^{-x \cdot \left(1 - \varepsilon\right)}\right) - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))