Average Error: 30.0 → 1.4
Time: 23.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.148667606658413279291391368818874701985 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(2 - \left(1 \cdot x\right) \cdot x\right) - \frac{\log \left(e^{\left(x \cdot \left(x \cdot x\right)\right) \cdot 2.77555756156289135105907917022705078125 \cdot 10^{-17}}\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{e^{x \cdot \left(1 + \varepsilon\right)}} - \left(\frac{\frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}} - \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon - 1\right)}\right)}{2}\\ \end{array}\]
\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 1.148667606658413279291391368818874701985 \cdot 10^{-9}:\\
\;\;\;\;\frac{\left(2 - \left(1 \cdot x\right) \cdot x\right) - \frac{\log \left(e^{\left(x \cdot \left(x \cdot x\right)\right) \cdot 2.77555756156289135105907917022705078125 \cdot 10^{-17}}\right)}{\varepsilon}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r2317090 = 1.0;
        double r2317091 = eps;
        double r2317092 = r2317090 / r2317091;
        double r2317093 = r2317090 + r2317092;
        double r2317094 = r2317090 - r2317091;
        double r2317095 = x;
        double r2317096 = r2317094 * r2317095;
        double r2317097 = -r2317096;
        double r2317098 = exp(r2317097);
        double r2317099 = r2317093 * r2317098;
        double r2317100 = r2317092 - r2317090;
        double r2317101 = r2317090 + r2317091;
        double r2317102 = r2317101 * r2317095;
        double r2317103 = -r2317102;
        double r2317104 = exp(r2317103);
        double r2317105 = r2317100 * r2317104;
        double r2317106 = r2317099 - r2317105;
        double r2317107 = 2.0;
        double r2317108 = r2317106 / r2317107;
        return r2317108;
}
↓
double f(double x, double eps) {
        double r2317109 = x;
        double r2317110 = 1.1486676066584133e-09;
        bool r2317111 = r2317109 <= r2317110;
        double r2317112 = 2.0;
        double r2317113 = 1.0;
        double r2317114 = r2317113 * r2317109;
        double r2317115 = r2317114 * r2317109;
        double r2317116 = r2317112 - r2317115;
        double r2317117 = r2317109 * r2317109;
        double r2317118 = r2317109 * r2317117;
        double r2317119 = 2.7755575615628914e-17;
        double r2317120 = r2317118 * r2317119;
        double r2317121 = exp(r2317120);
        double r2317122 = log(r2317121);
        double r2317123 = eps;
        double r2317124 = r2317122 / r2317123;
        double r2317125 = r2317116 - r2317124;
        double r2317126 = r2317125 / r2317112;
        double r2317127 = r2317113 + r2317123;
        double r2317128 = r2317109 * r2317127;
        double r2317129 = exp(r2317128);
        double r2317130 = r2317113 / r2317129;
        double r2317131 = r2317113 / r2317123;
        double r2317132 = r2317131 / r2317129;
        double r2317133 = r2317113 + r2317131;
        double r2317134 = r2317123 - r2317113;
        double r2317135 = r2317109 * r2317134;
        double r2317136 = exp(r2317135);
        double r2317137 = r2317133 * r2317136;
        double r2317138 = r2317132 - r2317137;
        double r2317139 = r2317130 - r2317138;
        double r2317140 = r2317139 / r2317112;
        double r2317141 = r2317111 ? r2317126 : r2317140;
        return r2317141;
}

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 < 1.1486676066584133e-09

    1. Initial program 39.6

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

    2. Simplified39.6

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

    3. Taylor expanded around 0 7.2

      \[\leadsto \frac{\color{blue}{2 - \left(1 \cdot {x}^{2} + 2.77555756156289135105907917022705078125 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon}\right)}}{2}\]

    4. Simplified7.2

      \[\leadsto \frac{\color{blue}{\left(2 - x \cdot \left(x \cdot 1\right)\right) - \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot 2.77555756156289135105907917022705078125 \cdot 10^{-17}}{\varepsilon}}}{2}\]
    5. Using strategy rm

    6. Applied add-log-exp1.3

      \[\leadsto \frac{\left(2 - x \cdot \left(x \cdot 1\right)\right) - \frac{\color{blue}{\log \left(e^{\left(x \cdot \left(x \cdot x\right)\right) \cdot 2.77555756156289135105907917022705078125 \cdot 10^{-17}}\right)}}{\varepsilon}}{2}\]

    if 1.1486676066584133e-09 < x

    1. Initial program 2.2

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

    2. Simplified2.2

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

    4. Applied div-sub2.2

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

    5. Applied associate-+l-2.0

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.148667606658413279291391368818874701985 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(2 - \left(1 \cdot x\right) \cdot x\right) - \frac{\log \left(e^{\left(x \cdot \left(x \cdot x\right)\right) \cdot 2.77555756156289135105907917022705078125 \cdot 10^{-17}}\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{e^{x \cdot \left(1 + \varepsilon\right)}} - \left(\frac{\frac{1}{\varepsilon}}{e^{x \cdot \left(1 + \varepsilon\right)}} - \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon - 1\right)}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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))