Average Error: 29.1 → 1.1
Time: 5.4s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 251.67631196577193:\\ \;\;\;\;\frac{{\left(0.33333333333333337 \cdot {x}^{3} + 1\right)}^{3} - {\left(0.5 \cdot {x}^{2}\right)}^{3}}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) \cdot \left(\left(0.33333333333333337 \cdot {x}^{3} + 1\right) + 0.5 \cdot {x}^{2}\right) + 0.5 \cdot \left(0.5 \cdot {x}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}} \cdot \sqrt{\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}}\\ \end{array}\]
\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 251.67631196577193:\\
\;\;\;\;\frac{{\left(0.33333333333333337 \cdot {x}^{3} + 1\right)}^{3} - {\left(0.5 \cdot {x}^{2}\right)}^{3}}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) \cdot \left(\left(0.33333333333333337 \cdot {x}^{3} + 1\right) + 0.5 \cdot {x}^{2}\right) + 0.5 \cdot \left(0.5 \cdot {x}^{4}\right)}\\

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

\end{array}
double f(double x, double eps) {
        double r38997 = 1.0;
        double r38998 = eps;
        double r38999 = r38997 / r38998;
        double r39000 = r38997 + r38999;
        double r39001 = r38997 - r38998;
        double r39002 = x;
        double r39003 = r39001 * r39002;
        double r39004 = -r39003;
        double r39005 = exp(r39004);
        double r39006 = r39000 * r39005;
        double r39007 = r38999 - r38997;
        double r39008 = r38997 + r38998;
        double r39009 = r39008 * r39002;
        double r39010 = -r39009;
        double r39011 = exp(r39010);
        double r39012 = r39007 * r39011;
        double r39013 = r39006 - r39012;
        double r39014 = 2.0;
        double r39015 = r39013 / r39014;
        return r39015;
}
↓
double f(double x, double eps) {
        double r39016 = x;
        double r39017 = 251.67631196577193;
        bool r39018 = r39016 <= r39017;
        double r39019 = 0.33333333333333337;
        double r39020 = 3.0;
        double r39021 = pow(r39016, r39020);
        double r39022 = r39019 * r39021;
        double r39023 = 1.0;
        double r39024 = r39022 + r39023;
        double r39025 = pow(r39024, r39020);
        double r39026 = 0.5;
        double r39027 = 2.0;
        double r39028 = pow(r39016, r39027);
        double r39029 = r39026 * r39028;
        double r39030 = pow(r39029, r39020);
        double r39031 = r39025 - r39030;
        double r39032 = r39024 + r39029;
        double r39033 = r39024 * r39032;
        double r39034 = 4.0;
        double r39035 = pow(r39016, r39034);
        double r39036 = r39026 * r39035;
        double r39037 = r39026 * r39036;
        double r39038 = r39033 + r39037;
        double r39039 = r39031 / r39038;
        double r39040 = eps;
        double r39041 = r39023 / r39040;
        double r39042 = r39023 + r39041;
        double r39043 = r39023 - r39040;
        double r39044 = r39043 * r39016;
        double r39045 = exp(r39044);
        double r39046 = r39042 / r39045;
        double r39047 = 2.0;
        double r39048 = r39046 / r39047;
        double r39049 = r39041 - r39023;
        double r39050 = r39023 + r39040;
        double r39051 = r39050 * r39016;
        double r39052 = exp(r39051);
        double r39053 = r39049 / r39052;
        double r39054 = r39053 / r39047;
        double r39055 = r39048 - r39054;
        double r39056 = sqrt(r39055);
        double r39057 = r39056 * r39056;
        double r39058 = r39018 ? r39039 : r39057;
        return r39058;
}

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 < 251.67631196577193

    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. Simplified38.8

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

    3. Taylor expanded around 0 1.4

      \[\leadsto \color{blue}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\]
    4. Using strategy rm

    5. Applied flip3--1.4

      \[\leadsto \color{blue}{\frac{{\left(0.33333333333333337 \cdot {x}^{3} + 1\right)}^{3} - {\left(0.5 \cdot {x}^{2}\right)}^{3}}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) \cdot \left(0.33333333333333337 \cdot {x}^{3} + 1\right) + \left(\left(0.5 \cdot {x}^{2}\right) \cdot \left(0.5 \cdot {x}^{2}\right) + \left(0.33333333333333337 \cdot {x}^{3} + 1\right) \cdot \left(0.5 \cdot {x}^{2}\right)\right)}}\]

    6. Simplified1.4

      \[\leadsto \frac{{\left(0.33333333333333337 \cdot {x}^{3} + 1\right)}^{3} - {\left(0.5 \cdot {x}^{2}\right)}^{3}}{\color{blue}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) \cdot \left(\left(0.33333333333333337 \cdot {x}^{3} + 1\right) + 0.5 \cdot {x}^{2}\right) + 0.5 \cdot \left(0.5 \cdot {x}^{4}\right)}}\]

    if 251.67631196577193 < x

    1. Initial program 0.2

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

    2. Simplified0.2

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

    4. Applied add-sqr-sqrt0.2

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 251.67631196577193:\\ \;\;\;\;\frac{{\left(0.33333333333333337 \cdot {x}^{3} + 1\right)}^{3} - {\left(0.5 \cdot {x}^{2}\right)}^{3}}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) \cdot \left(\left(0.33333333333333337 \cdot {x}^{3} + 1\right) + 0.5 \cdot {x}^{2}\right) + 0.5 \cdot \left(0.5 \cdot {x}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}} \cdot \sqrt{\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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))