Average Error: 29.3 → 0.9
Time: 27.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 307.0243008404694:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \varepsilon - x}, 1 + \frac{1}{\varepsilon}, \frac{\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{1 - \frac{1}{\varepsilon}}}{\sqrt[3]{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}} \cdot \frac{\sqrt[3]{1 - \frac{1}{\varepsilon}}}{\sqrt[3]{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}\right)}{2}\\ \end{array}\]
\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 307.0243008404694:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r1498076 = 1.0;
        double r1498077 = eps;
        double r1498078 = r1498076 / r1498077;
        double r1498079 = r1498076 + r1498078;
        double r1498080 = r1498076 - r1498077;
        double r1498081 = x;
        double r1498082 = r1498080 * r1498081;
        double r1498083 = -r1498082;
        double r1498084 = exp(r1498083);
        double r1498085 = r1498079 * r1498084;
        double r1498086 = r1498078 - r1498076;
        double r1498087 = r1498076 + r1498077;
        double r1498088 = r1498087 * r1498081;
        double r1498089 = -r1498088;
        double r1498090 = exp(r1498089);
        double r1498091 = r1498086 * r1498090;
        double r1498092 = r1498085 - r1498091;
        double r1498093 = 2.0;
        double r1498094 = r1498092 / r1498093;
        return r1498094;
}
↓
double f(double x, double eps) {
        double r1498095 = x;
        double r1498096 = 307.0243008404694;
        bool r1498097 = r1498095 <= r1498096;
        double r1498098 = 0.6666666666666666;
        double r1498099 = r1498095 * r1498095;
        double r1498100 = r1498099 * r1498095;
        double r1498101 = 2.0;
        double r1498102 = r1498101 - r1498099;
        double r1498103 = fma(r1498098, r1498100, r1498102);
        double r1498104 = r1498103 / r1498101;
        double r1498105 = eps;
        double r1498106 = r1498095 * r1498105;
        double r1498107 = r1498106 - r1498095;
        double r1498108 = exp(r1498107);
        double r1498109 = 1.0;
        double r1498110 = r1498109 / r1498105;
        double r1498111 = r1498109 + r1498110;
        double r1498112 = r1498109 - r1498110;
        double r1498113 = cbrt(r1498112);
        double r1498114 = r1498113 * r1498113;
        double r1498115 = fma(r1498095, r1498105, r1498095);
        double r1498116 = exp(r1498115);
        double r1498117 = cbrt(r1498116);
        double r1498118 = r1498117 * r1498117;
        double r1498119 = r1498114 / r1498118;
        double r1498120 = r1498113 / r1498117;
        double r1498121 = r1498119 * r1498120;
        double r1498122 = fma(r1498108, r1498111, r1498121);
        double r1498123 = r1498122 / r1498101;
        double r1498124 = r1498097 ? r1498104 : r1498123;
        return r1498124;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 307.0243008404694

    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. Simplified38.9

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

    3. Taylor expanded around 0 1.2

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

    4. Simplified1.2

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

    if 307.0243008404694 < x

    1. Initial program 0.0

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

    2. Simplified0.0

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

    4. Applied add-cube-cbrt0.0

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

    5. Applied add-cube-cbrt0.0

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

    6. Applied times-frac0.0

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 307.0243008404694:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{3}, \left(x \cdot x\right) \cdot x, 2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x \cdot \varepsilon - x}, 1 + \frac{1}{\varepsilon}, \frac{\sqrt[3]{1 - \frac{1}{\varepsilon}} \cdot \sqrt[3]{1 - \frac{1}{\varepsilon}}}{\sqrt[3]{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}} \cdot \frac{\sqrt[3]{1 - \frac{1}{\varepsilon}}}{\sqrt[3]{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 +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))