Average Error: 29.7 → 1.0
Time: 27.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 100.79533204800275:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \mathsf{log1p}\left(\left(\left(\mathsf{expm1}\left(\frac{2}{3} \cdot x\right)\right)\right)\right), 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\mathsf{fma}\left(x, \varepsilon, -x\right)} + e^{\mathsf{fma}\left(\varepsilon, -x, -x\right)}\right) + \frac{e^{\mathsf{fma}\left(x, \varepsilon, -x\right)}}{\varepsilon}\right) - \frac{e^{\mathsf{fma}\left(\varepsilon, -x, -x\right)}}{\varepsilon}}{2}\\ \end{array}\]
\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 100.79533204800275:\\
\;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \mathsf{log1p}\left(\left(\left(\mathsf{expm1}\left(\frac{2}{3} \cdot x\right)\right)\right)\right), 2 - x \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1390927 = 1.0;
        double r1390928 = eps;
        double r1390929 = r1390927 / r1390928;
        double r1390930 = r1390927 + r1390929;
        double r1390931 = r1390927 - r1390928;
        double r1390932 = x;
        double r1390933 = r1390931 * r1390932;
        double r1390934 = -r1390933;
        double r1390935 = exp(r1390934);
        double r1390936 = r1390930 * r1390935;
        double r1390937 = r1390929 - r1390927;
        double r1390938 = r1390927 + r1390928;
        double r1390939 = r1390938 * r1390932;
        double r1390940 = -r1390939;
        double r1390941 = exp(r1390940);
        double r1390942 = r1390937 * r1390941;
        double r1390943 = r1390936 - r1390942;
        double r1390944 = 2.0;
        double r1390945 = r1390943 / r1390944;
        return r1390945;
}
↓
double f(double x, double eps) {
        double r1390946 = x;
        double r1390947 = 100.79533204800275;
        bool r1390948 = r1390946 <= r1390947;
        double r1390949 = r1390946 * r1390946;
        double r1390950 = 0.6666666666666666;
        double r1390951 = r1390950 * r1390946;
        double r1390952 = expm1(r1390951);
        double r1390953 = /* ERROR: no posit support in C */;
        double r1390954 = /* ERROR: no posit support in C */;
        double r1390955 = log1p(r1390954);
        double r1390956 = 2.0;
        double r1390957 = r1390956 - r1390949;
        double r1390958 = fma(r1390949, r1390955, r1390957);
        double r1390959 = r1390958 / r1390956;
        double r1390960 = eps;
        double r1390961 = -r1390946;
        double r1390962 = fma(r1390946, r1390960, r1390961);
        double r1390963 = exp(r1390962);
        double r1390964 = fma(r1390960, r1390961, r1390961);
        double r1390965 = exp(r1390964);
        double r1390966 = r1390963 + r1390965;
        double r1390967 = r1390963 / r1390960;
        double r1390968 = r1390966 + r1390967;
        double r1390969 = r1390965 / r1390960;
        double r1390970 = r1390968 - r1390969;
        double r1390971 = r1390970 / r1390956;
        double r1390972 = r1390948 ? r1390959 : r1390971;
        return r1390972;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 100.79533204800275

    1. Initial program 39.6

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

    2. Simplified39.7

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

    3. Taylor expanded around 0 1.3

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

    4. Simplified1.3

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

    6. Applied log1p-expm1-u1.3

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2}{3} \cdot x\right)\right)}, 2 - x \cdot x\right)}{2}\]
    7. Using strategy rm

    8. Applied insert-posit161.4

      \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{log1p}\left(\color{blue}{\left(\left(\mathsf{expm1}\left(\frac{2}{3} \cdot x\right)\right)\right)}\right), 2 - x \cdot x\right)}{2}\]

    if 100.79533204800275 < 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. Simplified0.1

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

    3. Taylor expanded around inf 0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 100.79533204800275:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \mathsf{log1p}\left(\left(\left(\mathsf{expm1}\left(\frac{2}{3} \cdot x\right)\right)\right)\right), 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\mathsf{fma}\left(x, \varepsilon, -x\right)} + e^{\mathsf{fma}\left(\varepsilon, -x, -x\right)}\right) + \frac{e^{\mathsf{fma}\left(x, \varepsilon, -x\right)}}{\varepsilon}\right) - \frac{e^{\mathsf{fma}\left(\varepsilon, -x, -x\right)}}{\varepsilon}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +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))