Average Error: 29.6 → 1.0
Time: 25.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 336.4224144429294369729177560657262802124:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{-\left(1 - \varepsilon\right) \cdot x}\right)\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]
\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 336.4224144429294369729177560657262802124:\\
\;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{-\left(1 - \varepsilon\right) \cdot x}\right)\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\

\end{array}
double f(double x, double eps) {
        double r43046 = 1.0;
        double r43047 = eps;
        double r43048 = r43046 / r43047;
        double r43049 = r43046 + r43048;
        double r43050 = r43046 - r43047;
        double r43051 = x;
        double r43052 = r43050 * r43051;
        double r43053 = -r43052;
        double r43054 = exp(r43053);
        double r43055 = r43049 * r43054;
        double r43056 = r43048 - r43046;
        double r43057 = r43046 + r43047;
        double r43058 = r43057 * r43051;
        double r43059 = -r43058;
        double r43060 = exp(r43059);
        double r43061 = r43056 * r43060;
        double r43062 = r43055 - r43061;
        double r43063 = 2.0;
        double r43064 = r43062 / r43063;
        return r43064;
}
↓
double f(double x, double eps) {
        double r43065 = x;
        double r43066 = 336.42241444292944;
        bool r43067 = r43065 <= r43066;
        double r43068 = 0.6666666666666667;
        double r43069 = 3.0;
        double r43070 = pow(r43065, r43069);
        double r43071 = 2.0;
        double r43072 = fma(r43068, r43070, r43071);
        double r43073 = 1.0;
        double r43074 = 2.0;
        double r43075 = pow(r43065, r43074);
        double r43076 = r43073 * r43075;
        double r43077 = r43072 - r43076;
        double r43078 = r43077 / r43071;
        double r43079 = eps;
        double r43080 = r43073 / r43079;
        double r43081 = r43073 + r43080;
        double r43082 = r43073 - r43079;
        double r43083 = r43082 * r43065;
        double r43084 = -r43083;
        double r43085 = exp(r43084);
        double r43086 = log1p(r43085);
        double r43087 = expm1(r43086);
        double r43088 = r43081 * r43087;
        double r43089 = r43080 - r43073;
        double r43090 = r43073 + r43079;
        double r43091 = r43090 * r43065;
        double r43092 = -r43091;
        double r43093 = exp(r43092);
        double r43094 = r43089 * r43093;
        double r43095 = r43088 - r43094;
        double r43096 = r43095 / r43071;
        double r43097 = r43067 ? r43078 : r43096;
        return r43097;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 336.42241444292944

    1. Initial program 39.1

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

    2. Taylor expanded around 0 1.2

      \[\leadsto \frac{\color{blue}{\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.2

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}}{2}\]

    if 336.42241444292944 < 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. Using strategy rm

    3. Applied expm1-log1p-u0.2

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{-\left(1 - \varepsilon\right) \cdot x}\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 336.4224144429294369729177560657262802124:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(e^{-\left(1 - \varepsilon\right) \cdot x}\right)\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(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))