Average Error: 29.6 → 0.9
Time: 23.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 100.06482820348397:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{\left(\varepsilon - 1\right) \cdot x} + \left(\frac{1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}} + \frac{e^{\left(\varepsilon - 1\right) \cdot x}}{\varepsilon}\right)\right) - \frac{1}{\varepsilon \cdot e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2}\\ \end{array}\]
\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 100.06482820348397:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r829229 = 1.0;
        double r829230 = eps;
        double r829231 = r829229 / r829230;
        double r829232 = r829229 + r829231;
        double r829233 = r829229 - r829230;
        double r829234 = x;
        double r829235 = r829233 * r829234;
        double r829236 = -r829235;
        double r829237 = exp(r829236);
        double r829238 = r829232 * r829237;
        double r829239 = r829231 - r829229;
        double r829240 = r829229 + r829230;
        double r829241 = r829240 * r829234;
        double r829242 = -r829241;
        double r829243 = exp(r829242);
        double r829244 = r829239 * r829243;
        double r829245 = r829238 - r829244;
        double r829246 = 2.0;
        double r829247 = r829245 / r829246;
        return r829247;
}
↓
double f(double x, double eps) {
        double r829248 = x;
        double r829249 = 100.06482820348397;
        bool r829250 = r829248 <= r829249;
        double r829251 = 0.6666666666666666;
        double r829252 = r829248 * r829248;
        double r829253 = r829252 * r829248;
        double r829254 = 2.0;
        double r829255 = r829254 - r829252;
        double r829256 = fma(r829251, r829253, r829255);
        double r829257 = r829256 / r829254;
        double r829258 = eps;
        double r829259 = 1.0;
        double r829260 = r829258 - r829259;
        double r829261 = r829260 * r829248;
        double r829262 = exp(r829261);
        double r829263 = fma(r829258, r829248, r829248);
        double r829264 = exp(r829263);
        double r829265 = r829259 / r829264;
        double r829266 = r829262 / r829258;
        double r829267 = r829265 + r829266;
        double r829268 = r829262 + r829267;
        double r829269 = r829258 * r829264;
        double r829270 = r829259 / r829269;
        double r829271 = r829268 - r829270;
        double r829272 = r829271 / r829254;
        double r829273 = r829250 ? r829257 : r829272;
        return r829273;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 100.06482820348397

    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. Simplified39.1

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

    3. Taylor expanded around 0 1.1

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

    4. Simplified1.1

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

    if 100.06482820348397 < x

    1. Initial program 0.3

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

    2. Simplified0.3

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

    3. Taylor expanded around inf 0.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 100.06482820348397:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{\left(\varepsilon - 1\right) \cdot x} + \left(\frac{1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}} + \frac{e^{\left(\varepsilon - 1\right) \cdot x}}{\varepsilon}\right)\right) - \frac{1}{\varepsilon \cdot e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))