Average Error: 29.6 → 1.1
Time: 27.9s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 80.15112264149943:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2 - x \cdot x\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} - \frac{e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}}{\varepsilon}\right) + \left(e^{\varepsilon \cdot x - x} + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right)}{2}\\ \end{array}\]
\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 80.15112264149943:\\
\;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2 - x \cdot x\right)\right)\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1305246 = 1.0;
        double r1305247 = eps;
        double r1305248 = r1305246 / r1305247;
        double r1305249 = r1305246 + r1305248;
        double r1305250 = r1305246 - r1305247;
        double r1305251 = x;
        double r1305252 = r1305250 * r1305251;
        double r1305253 = -r1305252;
        double r1305254 = exp(r1305253);
        double r1305255 = r1305249 * r1305254;
        double r1305256 = r1305248 - r1305246;
        double r1305257 = r1305246 + r1305247;
        double r1305258 = r1305257 * r1305251;
        double r1305259 = -r1305258;
        double r1305260 = exp(r1305259);
        double r1305261 = r1305256 * r1305260;
        double r1305262 = r1305255 - r1305261;
        double r1305263 = 2.0;
        double r1305264 = r1305262 / r1305263;
        return r1305264;
}
↓
double f(double x, double eps) {
        double r1305265 = x;
        double r1305266 = 80.15112264149943;
        bool r1305267 = r1305265 <= r1305266;
        double r1305268 = 0.6666666666666666;
        double r1305269 = r1305268 * r1305265;
        double r1305270 = r1305265 * r1305265;
        double r1305271 = 2.0;
        double r1305272 = r1305271 - r1305270;
        double r1305273 = fma(r1305269, r1305270, r1305272);
        double r1305274 = /* ERROR: no posit support in C */;
        double r1305275 = /* ERROR: no posit support in C */;
        double r1305276 = r1305275 / r1305271;
        double r1305277 = eps;
        double r1305278 = fma(r1305265, r1305277, r1305265);
        double r1305279 = -r1305278;
        double r1305280 = exp(r1305279);
        double r1305281 = r1305280 / r1305277;
        double r1305282 = r1305280 - r1305281;
        double r1305283 = r1305277 * r1305265;
        double r1305284 = r1305283 - r1305265;
        double r1305285 = exp(r1305284);
        double r1305286 = r1305285 / r1305277;
        double r1305287 = r1305285 + r1305286;
        double r1305288 = r1305282 + r1305287;
        double r1305289 = r1305288 / r1305271;
        double r1305290 = r1305267 ? r1305276 : r1305289;
        return r1305290;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 80.15112264149943

    1. Initial program 39.3

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

    2. Simplified39.3

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

    3. Taylor expanded around 0 1.2

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

    4. Simplified1.2

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

    6. Applied insert-posit161.4

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

    if 80.15112264149943 < 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

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

    3. Taylor expanded around -inf 0.2

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

    4. Simplified0.2

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 80.15112264149943:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\frac{2}{3} \cdot x, x \cdot x, 2 - x \cdot x\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} - \frac{e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}}{\varepsilon}\right) + \left(e^{\varepsilon \cdot x - x} + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 +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))