Average Error: 29.6 → 1.1
Time: 30.2s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.22706723216422863842467450012918561697:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, e^{x \cdot \left(-\left(\varepsilon + 1\right)\right)}, \left(\left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon} + e^{x \cdot \left(\varepsilon - 1\right)}\right) - \frac{e^{x \cdot \left(-\left(\varepsilon + 1\right)\right)}}{\varepsilon}\right) \cdot 1\right)}{2}\\ \end{array}\]
\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 2.22706723216422863842467450012918561697:\\
\;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right) + 2}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r2476267 = 1.0;
        double r2476268 = eps;
        double r2476269 = r2476267 / r2476268;
        double r2476270 = r2476267 + r2476269;
        double r2476271 = r2476267 - r2476268;
        double r2476272 = x;
        double r2476273 = r2476271 * r2476272;
        double r2476274 = -r2476273;
        double r2476275 = exp(r2476274);
        double r2476276 = r2476270 * r2476275;
        double r2476277 = r2476269 - r2476267;
        double r2476278 = r2476267 + r2476268;
        double r2476279 = r2476278 * r2476272;
        double r2476280 = -r2476279;
        double r2476281 = exp(r2476280);
        double r2476282 = r2476277 * r2476281;
        double r2476283 = r2476276 - r2476282;
        double r2476284 = 2.0;
        double r2476285 = r2476283 / r2476284;
        return r2476285;
}
↓
double f(double x, double eps) {
        double r2476286 = x;
        double r2476287 = 2.2270672321642286;
        bool r2476288 = r2476286 <= r2476287;
        double r2476289 = r2476286 * r2476286;
        double r2476290 = 0.6666666666666667;
        double r2476291 = r2476290 * r2476286;
        double r2476292 = 1.0;
        double r2476293 = r2476291 - r2476292;
        double r2476294 = r2476289 * r2476293;
        double r2476295 = 2.0;
        double r2476296 = r2476294 + r2476295;
        double r2476297 = r2476296 / r2476295;
        double r2476298 = eps;
        double r2476299 = r2476298 + r2476292;
        double r2476300 = -r2476299;
        double r2476301 = r2476286 * r2476300;
        double r2476302 = exp(r2476301);
        double r2476303 = r2476298 - r2476292;
        double r2476304 = r2476286 * r2476303;
        double r2476305 = exp(r2476304);
        double r2476306 = r2476305 / r2476298;
        double r2476307 = r2476306 + r2476305;
        double r2476308 = r2476302 / r2476298;
        double r2476309 = r2476307 - r2476308;
        double r2476310 = r2476309 * r2476292;
        double r2476311 = fma(r2476292, r2476302, r2476310);
        double r2476312 = r2476311 / r2476295;
        double r2476313 = r2476288 ? r2476297 : r2476312;
        return r2476313;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 2.2270672321642286

    1. Initial program 39.5

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

    2. Taylor expanded around 0 1.3

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

    3. Simplified1.3

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

    4. Taylor expanded around 0 1.3

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

    5. Simplified1.3

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

    if 2.2270672321642286 < x

    1. Initial program 0.6

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.6

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.22706723216422863842467450012918561697:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, e^{x \cdot \left(-\left(\varepsilon + 1\right)\right)}, \left(\left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon} + e^{x \cdot \left(\varepsilon - 1\right)}\right) - \frac{e^{x \cdot \left(-\left(\varepsilon + 1\right)\right)}}{\varepsilon}\right) \cdot 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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))