Average Error: 29.7 → 4.3
Time: 7.4s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.79043542849065522529920941679858858976 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\mathsf{fma}\left(x, \varepsilon, 1 \cdot x\right)}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\\ \end{array}\]
\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 3.79043542849065522529920941679858858976 \cdot 10^{-16}:\\
\;\;\;\;\mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r58159 = 1.0;
        double r58160 = eps;
        double r58161 = r58159 / r58160;
        double r58162 = r58159 + r58161;
        double r58163 = r58159 - r58160;
        double r58164 = x;
        double r58165 = r58163 * r58164;
        double r58166 = -r58165;
        double r58167 = exp(r58166);
        double r58168 = r58162 * r58167;
        double r58169 = r58161 - r58159;
        double r58170 = r58159 + r58160;
        double r58171 = r58170 * r58164;
        double r58172 = -r58171;
        double r58173 = exp(r58172);
        double r58174 = r58169 * r58173;
        double r58175 = r58168 - r58174;
        double r58176 = 2.0;
        double r58177 = r58175 / r58176;
        return r58177;
}
↓
double f(double x, double eps) {
        double r58178 = x;
        double r58179 = 3.790435428490655e-16;
        bool r58180 = r58178 <= r58179;
        double r58181 = 1.3877787807814457e-17;
        double r58182 = cbrt(r58178);
        double r58183 = 2.0;
        double r58184 = exp(r58182);
        double r58185 = cbrt(r58184);
        double r58186 = log(r58185);
        double r58187 = r58183 * r58186;
        double r58188 = r58187 + r58186;
        double r58189 = r58182 * r58188;
        double r58190 = 3.0;
        double r58191 = pow(r58189, r58190);
        double r58192 = eps;
        double r58193 = r58192 / r58178;
        double r58194 = r58191 / r58193;
        double r58195 = 1.0;
        double r58196 = 0.5;
        double r58197 = pow(r58178, r58183);
        double r58198 = r58196 * r58197;
        double r58199 = r58195 - r58198;
        double r58200 = fma(r58181, r58194, r58199);
        double r58201 = r58195 * r58178;
        double r58202 = fma(r58178, r58192, r58201);
        double r58203 = -r58202;
        double r58204 = exp(r58203);
        double r58205 = 2.0;
        double r58206 = r58204 / r58205;
        double r58207 = r58195 / r58192;
        double r58208 = r58195 - r58207;
        double r58209 = r58195 + r58207;
        double r58210 = r58195 - r58192;
        double r58211 = r58210 * r58178;
        double r58212 = exp(r58211);
        double r58213 = r58205 * r58212;
        double r58214 = r58209 / r58213;
        double r58215 = fma(r58206, r58208, r58214);
        double r58216 = r58180 ? r58200 : r58215;
        return r58216;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 3.790435428490655e-16

    1. Initial program 38.4

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

    2. Simplified38.5

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

    3. Taylor expanded around 0 6.2

      \[\leadsto \color{blue}{\left(1.387778780781445675529539585113525390625 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon} + 1\right) - 0.5 \cdot {x}^{2}}\]

    4. Simplified6.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt6.2

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)\]

    7. Applied unpow-prod-down6.2

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{\color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot {\left(\sqrt[3]{x}\right)}^{3}}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)\]

    8. Applied associate-/l*6.2

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \color{blue}{\frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\frac{\varepsilon}{{\left(\sqrt[3]{x}\right)}^{3}}}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    9. Simplified6.2

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}}{\color{blue}{\frac{\varepsilon}{x}}}, 1 - 0.5 \cdot {x}^{2}\right)\]
    10. Using strategy rm

    11. Applied add-log-exp4.6

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \color{blue}{\log \left(e^{\sqrt[3]{x}}\right)}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]
    12. Using strategy rm

    13. Applied add-cube-cbrt4.4

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \log \color{blue}{\left(\left(\sqrt[3]{e^{\sqrt[3]{x}}} \cdot \sqrt[3]{e^{\sqrt[3]{x}}}\right) \cdot \sqrt[3]{e^{\sqrt[3]{x}}}\right)}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    14. Applied log-prod4.4

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \color{blue}{\left(\log \left(\sqrt[3]{e^{\sqrt[3]{x}}} \cdot \sqrt[3]{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right)}\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    15. Simplified4.4

      \[\leadsto \mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\]

    if 3.790435428490655e-16 < x

    1. Initial program 3.9

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

    2. Simplified3.9

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

    3. Taylor expanded around 0 3.9

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

    4. Simplified3.9

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

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.79043542849065522529920941679858858976 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(1.387778780781445675529539585113525390625 \cdot 10^{-17}, \frac{{\left(\sqrt[3]{x} \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{x}}}\right)\right)\right)}^{3}}{\frac{\varepsilon}{x}}, 1 - 0.5 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-\mathsf{fma}\left(x, \varepsilon, 1 \cdot x\right)}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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))