Average Error: 29.1 → 1.1
Time: 6.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 273.9633798024049156083492562174797058105:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.6666666666666667406815349750104360282421, 2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2}\\ \end{array}\]
\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 273.9633798024049156083492562174797058105:\\
\;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.6666666666666667406815349750104360282421, 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 r43014 = 1.0;
        double r43015 = eps;
        double r43016 = r43014 / r43015;
        double r43017 = r43014 + r43016;
        double r43018 = r43014 - r43015;
        double r43019 = x;
        double r43020 = r43018 * r43019;
        double r43021 = -r43020;
        double r43022 = exp(r43021);
        double r43023 = r43017 * r43022;
        double r43024 = r43016 - r43014;
        double r43025 = r43014 + r43015;
        double r43026 = r43025 * r43019;
        double r43027 = -r43026;
        double r43028 = exp(r43027);
        double r43029 = r43024 * r43028;
        double r43030 = r43023 - r43029;
        double r43031 = 2.0;
        double r43032 = r43030 / r43031;
        return r43032;
}
↓
double f(double x, double eps) {
        double r43033 = x;
        double r43034 = 273.9633798024049;
        bool r43035 = r43033 <= r43034;
        double r43036 = 3.0;
        double r43037 = pow(r43033, r43036);
        double r43038 = 0.6666666666666667;
        double r43039 = 2.0;
        double r43040 = 1.0;
        double r43041 = 2.0;
        double r43042 = pow(r43033, r43041);
        double r43043 = r43040 * r43042;
        double r43044 = r43039 - r43043;
        double r43045 = fma(r43037, r43038, r43044);
        double r43046 = r43045 / r43039;
        double r43047 = eps;
        double r43048 = r43040 / r43047;
        double r43049 = r43040 + r43048;
        double r43050 = r43040 - r43047;
        double r43051 = r43050 * r43033;
        double r43052 = -r43051;
        double r43053 = exp(r43052);
        double r43054 = r43048 - r43040;
        double r43055 = r43040 + r43047;
        double r43056 = r43055 * r43033;
        double r43057 = -r43056;
        double r43058 = exp(r43057);
        double r43059 = r43054 * r43058;
        double r43060 = -r43059;
        double r43061 = fma(r43049, r43053, r43060);
        double r43062 = r43061 / r43039;
        double r43063 = r43035 ? r43046 : r43062;
        return r43063;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 273.9633798024049

    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 1.4

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

    3. Simplified1.4

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

    if 273.9633798024049 < x

    1. Initial program 0.1

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

    3. Applied fma-neg0.1

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 273.9633798024049156083492562174797058105:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, 0.6666666666666667406815349750104360282421, 2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{-\left(1 - \varepsilon\right) \cdot x}, -\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))