Average Error: 29.6 → 1.8
Time: 28.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 8.501320620640498795819319236142604285806 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left({x}^{3}, 0.6666666666666667406815349750104360282421, 2\right), \mathsf{fma}\left({x}^{3}, 0.6666666666666667406815349750104360282421, 2\right), \left(-1 \cdot 1\right) \cdot {x}^{4}\right)}{\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(x, 0.6666666666666667406815349750104360282421, 1\right), 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{1 + \frac{1}{\varepsilon}}\right) \cdot \left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot e^{-\left(1 - \varepsilon\right) \cdot x}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]
\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 8.501320620640498795819319236142604285806 \cdot 10^{-15}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left({x}^{3}, 0.6666666666666667406815349750104360282421, 2\right), \mathsf{fma}\left({x}^{3}, 0.6666666666666667406815349750104360282421, 2\right), \left(-1 \cdot 1\right) \cdot {x}^{4}\right)}{\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(x, 0.6666666666666667406815349750104360282421, 1\right), 2\right)}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{1 + \frac{1}{\varepsilon}}\right) \cdot \left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot e^{-\left(1 - \varepsilon\right) \cdot x}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\

\end{array}
double f(double x, double eps) {
        double r42941 = 1.0;
        double r42942 = eps;
        double r42943 = r42941 / r42942;
        double r42944 = r42941 + r42943;
        double r42945 = r42941 - r42942;
        double r42946 = x;
        double r42947 = r42945 * r42946;
        double r42948 = -r42947;
        double r42949 = exp(r42948);
        double r42950 = r42944 * r42949;
        double r42951 = r42943 - r42941;
        double r42952 = r42941 + r42942;
        double r42953 = r42952 * r42946;
        double r42954 = -r42953;
        double r42955 = exp(r42954);
        double r42956 = r42951 * r42955;
        double r42957 = r42950 - r42956;
        double r42958 = 2.0;
        double r42959 = r42957 / r42958;
        return r42959;
}
↓
double f(double x, double eps) {
        double r42960 = x;
        double r42961 = 8.501320620640499e-15;
        bool r42962 = r42960 <= r42961;
        double r42963 = 3.0;
        double r42964 = pow(r42960, r42963);
        double r42965 = 0.6666666666666667;
        double r42966 = 2.0;
        double r42967 = fma(r42964, r42965, r42966);
        double r42968 = 1.0;
        double r42969 = r42968 * r42968;
        double r42970 = -r42969;
        double r42971 = 4.0;
        double r42972 = pow(r42960, r42971);
        double r42973 = r42970 * r42972;
        double r42974 = fma(r42967, r42967, r42973);
        double r42975 = 2.0;
        double r42976 = pow(r42960, r42975);
        double r42977 = fma(r42960, r42965, r42968);
        double r42978 = fma(r42976, r42977, r42966);
        double r42979 = r42974 / r42978;
        double r42980 = r42979 / r42966;
        double r42981 = eps;
        double r42982 = r42968 / r42981;
        double r42983 = r42968 + r42982;
        double r42984 = cbrt(r42983);
        double r42985 = r42984 * r42984;
        double r42986 = r42968 - r42981;
        double r42987 = r42986 * r42960;
        double r42988 = -r42987;
        double r42989 = exp(r42988);
        double r42990 = r42984 * r42989;
        double r42991 = r42985 * r42990;
        double r42992 = r42982 - r42968;
        double r42993 = r42968 + r42981;
        double r42994 = r42993 * r42960;
        double r42995 = -r42994;
        double r42996 = exp(r42995);
        double r42997 = r42992 * r42996;
        double r42998 = r42991 - r42997;
        double r42999 = r42998 / r42966;
        double r43000 = r42962 ? r42980 : r42999;
        return r43000;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 8.501320620640499e-15

    1. Initial program 38.9

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

    2. Taylor expanded around 0 1.1

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

    3. Simplified1.1

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

    5. Applied log1p-expm1-u1.1

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right)\right)\right)} - 1 \cdot {x}^{2}}{2}\]
    6. Using strategy rm

    7. Applied flip--1.1

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right)\right)\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right)\right)\right) - \left(1 \cdot {x}^{2}\right) \cdot \left(1 \cdot {x}^{2}\right)}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right)\right)\right) + 1 \cdot {x}^{2}}}}{2}\]

    8. Simplified1.1

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left({x}^{3}, 0.6666666666666667406815349750104360282421, 2\right), \mathsf{fma}\left({x}^{3}, 0.6666666666666667406815349750104360282421, 2\right), \left(-1 \cdot 1\right) \cdot {x}^{4}\right)}}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(0.6666666666666667406815349750104360282421, {x}^{3}, 2\right)\right)\right) + 1 \cdot {x}^{2}}}{2}\]

    9. Simplified1.1

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left({x}^{3}, 0.6666666666666667406815349750104360282421, 2\right), \mathsf{fma}\left({x}^{3}, 0.6666666666666667406815349750104360282421, 2\right), \left(-1 \cdot 1\right) \cdot {x}^{4}\right)}{\color{blue}{\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(x, 0.6666666666666667406815349750104360282421, 1\right), 2\right)}}}{2}\]

    if 8.501320620640499e-15 < x

    1. Initial program 3.7

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

    3. Applied add-cube-cbrt3.7

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

    4. Applied associate-*l*3.7

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 8.501320620640498795819319236142604285806 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left({x}^{3}, 0.6666666666666667406815349750104360282421, 2\right), \mathsf{fma}\left({x}^{3}, 0.6666666666666667406815349750104360282421, 2\right), \left(-1 \cdot 1\right) \cdot {x}^{4}\right)}{\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(x, 0.6666666666666667406815349750104360282421, 1\right), 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot \sqrt[3]{1 + \frac{1}{\varepsilon}}\right) \cdot \left(\sqrt[3]{1 + \frac{1}{\varepsilon}} \cdot e^{-\left(1 - \varepsilon\right) \cdot x}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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))