Average Error: 29.5 → 1.0
Time: 6.1s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.591957276208114:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2}\\ \end{array}\]
\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 3.591957276208114:\\
\;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r47361 = 1.0;
        double r47362 = eps;
        double r47363 = r47361 / r47362;
        double r47364 = r47361 + r47363;
        double r47365 = r47361 - r47362;
        double r47366 = x;
        double r47367 = r47365 * r47366;
        double r47368 = -r47367;
        double r47369 = exp(r47368);
        double r47370 = r47364 * r47369;
        double r47371 = r47363 - r47361;
        double r47372 = r47361 + r47362;
        double r47373 = r47372 * r47366;
        double r47374 = -r47373;
        double r47375 = exp(r47374);
        double r47376 = r47371 * r47375;
        double r47377 = r47370 - r47376;
        double r47378 = 2.0;
        double r47379 = r47377 / r47378;
        return r47379;
}
↓
double f(double x, double eps) {
        double r47380 = x;
        double r47381 = 3.591957276208114;
        bool r47382 = r47380 <= r47381;
        double r47383 = 3.0;
        double r47384 = pow(r47380, r47383);
        double r47385 = 0.6666666666666667;
        double r47386 = 2.0;
        double r47387 = 1.0;
        double r47388 = 2.0;
        double r47389 = pow(r47380, r47388);
        double r47390 = r47387 * r47389;
        double r47391 = r47386 - r47390;
        double r47392 = fma(r47384, r47385, r47391);
        double r47393 = r47392 / r47386;
        double r47394 = eps;
        double r47395 = r47387 / r47394;
        double r47396 = r47387 + r47395;
        double r47397 = r47387 - r47394;
        double r47398 = r47397 * r47380;
        double r47399 = -r47398;
        double r47400 = exp(r47399);
        double r47401 = r47395 - r47387;
        double r47402 = r47387 + r47394;
        double r47403 = r47402 * r47380;
        double r47404 = -r47403;
        double r47405 = exp(r47404);
        double r47406 = r47401 * r47405;
        double r47407 = -r47406;
        double r47408 = fma(r47396, r47400, r47407);
        double r47409 = r47408 / r47386;
        double r47410 = r47382 ? r47393 : r47409;
        return r47410;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 3.591957276208114

    1. Initial program 39.0

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

    2. Taylor expanded around 0 1.2

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.2

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}}{2}\]

    if 3.591957276208114 < x

    1. Initial program 0.4

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

    3. Applied fma-neg0.4

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.591957276208114:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.66666666666666674, 2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))