Average Error: 29.7 → 1.1
Time: 7.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 254.14528551610528:\\ \;\;\;\;\frac{e^{e^{\log \left(\log \left(0.66666666666666674 \cdot {x}^{3} + \left(2 - 1 \cdot {x}^{2}\right)\right)\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{\sqrt[3]{-\left(1 + \varepsilon\right) \cdot x} \cdot \sqrt[3]{-\left(1 + \varepsilon\right) \cdot x}}\right)}^{\left(\sqrt[3]{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2}\\ \end{array}\]
\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 254.14528551610528:\\
\;\;\;\;\frac{e^{e^{\log \left(\log \left(0.66666666666666674 \cdot {x}^{3} + \left(2 - 1 \cdot {x}^{2}\right)\right)\right)}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{\sqrt[3]{-\left(1 + \varepsilon\right) \cdot x} \cdot \sqrt[3]{-\left(1 + \varepsilon\right) \cdot x}}\right)}^{\left(\sqrt[3]{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2}\\

\end{array}
double f(double x, double eps) {
        double r41158 = 1.0;
        double r41159 = eps;
        double r41160 = r41158 / r41159;
        double r41161 = r41158 + r41160;
        double r41162 = r41158 - r41159;
        double r41163 = x;
        double r41164 = r41162 * r41163;
        double r41165 = -r41164;
        double r41166 = exp(r41165);
        double r41167 = r41161 * r41166;
        double r41168 = r41160 - r41158;
        double r41169 = r41158 + r41159;
        double r41170 = r41169 * r41163;
        double r41171 = -r41170;
        double r41172 = exp(r41171);
        double r41173 = r41168 * r41172;
        double r41174 = r41167 - r41173;
        double r41175 = 2.0;
        double r41176 = r41174 / r41175;
        return r41176;
}
↓
double f(double x, double eps) {
        double r41177 = x;
        double r41178 = 254.14528551610528;
        bool r41179 = r41177 <= r41178;
        double r41180 = 0.6666666666666667;
        double r41181 = 3.0;
        double r41182 = pow(r41177, r41181);
        double r41183 = r41180 * r41182;
        double r41184 = 2.0;
        double r41185 = 1.0;
        double r41186 = 2.0;
        double r41187 = pow(r41177, r41186);
        double r41188 = r41185 * r41187;
        double r41189 = r41184 - r41188;
        double r41190 = r41183 + r41189;
        double r41191 = log(r41190);
        double r41192 = log(r41191);
        double r41193 = exp(r41192);
        double r41194 = exp(r41193);
        double r41195 = r41194 / r41184;
        double r41196 = eps;
        double r41197 = r41185 / r41196;
        double r41198 = r41185 + r41197;
        double r41199 = r41185 - r41196;
        double r41200 = r41199 * r41177;
        double r41201 = -r41200;
        double r41202 = exp(r41201);
        double r41203 = r41198 * r41202;
        double r41204 = r41197 - r41185;
        double r41205 = r41185 + r41196;
        double r41206 = r41205 * r41177;
        double r41207 = -r41206;
        double r41208 = cbrt(r41207);
        double r41209 = r41208 * r41208;
        double r41210 = exp(r41209);
        double r41211 = pow(r41210, r41208);
        double r41212 = r41204 * r41211;
        double r41213 = r41203 - r41212;
        double r41214 = r41213 / r41184;
        double r41215 = r41179 ? r41195 : r41214;
        return r41215;
}

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 < 254.14528551610528

    1. Initial program 38.8

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

    2. Taylor expanded around 0 1.3

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

    4. Applied associate--l+1.3

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

    6. Applied add-exp-log1.4

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

    8. Applied add-exp-log1.4

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

    if 254.14528551610528 < x

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied exp-prod0.1

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 254.14528551610528:\\ \;\;\;\;\frac{e^{e^{\log \left(\log \left(0.66666666666666674 \cdot {x}^{3} + \left(2 - 1 \cdot {x}^{2}\right)\right)\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{\sqrt[3]{-\left(1 + \varepsilon\right) \cdot x} \cdot \sqrt[3]{-\left(1 + \varepsilon\right) \cdot x}}\right)}^{\left(\sqrt[3]{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(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))