Average Error: 30.1 → 1.0
Time: 11.7s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 95.541640315829881:\\ \;\;\;\;\frac{\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]
\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 95.541640315829881:\\
\;\;\;\;\frac{\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{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 r28620 = 1.0;
        double r28621 = eps;
        double r28622 = r28620 / r28621;
        double r28623 = r28620 + r28622;
        double r28624 = r28620 - r28621;
        double r28625 = x;
        double r28626 = r28624 * r28625;
        double r28627 = -r28626;
        double r28628 = exp(r28627);
        double r28629 = r28623 * r28628;
        double r28630 = r28622 - r28620;
        double r28631 = r28620 + r28621;
        double r28632 = r28631 * r28625;
        double r28633 = -r28632;
        double r28634 = exp(r28633);
        double r28635 = r28630 * r28634;
        double r28636 = r28629 - r28635;
        double r28637 = 2.0;
        double r28638 = r28636 / r28637;
        return r28638;
}
↓
double f(double x, double eps) {
        double r28639 = x;
        double r28640 = 95.54164031582988;
        bool r28641 = r28639 <= r28640;
        double r28642 = 0.6666666666666667;
        double r28643 = 3.0;
        double r28644 = pow(r28639, r28643);
        double r28645 = r28642 * r28644;
        double r28646 = cbrt(r28645);
        double r28647 = r28646 * r28646;
        double r28648 = r28647 * r28646;
        double r28649 = 2.0;
        double r28650 = 1.0;
        double r28651 = 2.0;
        double r28652 = pow(r28639, r28651);
        double r28653 = r28650 * r28652;
        double r28654 = r28649 - r28653;
        double r28655 = r28648 + r28654;
        double r28656 = r28655 / r28649;
        double r28657 = eps;
        double r28658 = r28650 / r28657;
        double r28659 = r28650 + r28658;
        double r28660 = r28650 - r28657;
        double r28661 = r28660 * r28639;
        double r28662 = -r28661;
        double r28663 = exp(r28662);
        double r28664 = cbrt(r28663);
        double r28665 = r28664 * r28664;
        double r28666 = r28665 * r28664;
        double r28667 = r28659 * r28666;
        double r28668 = r28658 - r28650;
        double r28669 = r28650 + r28657;
        double r28670 = r28669 * r28639;
        double r28671 = -r28670;
        double r28672 = exp(r28671);
        double r28673 = r28668 * r28672;
        double r28674 = r28667 - r28673;
        double r28675 = r28674 / r28649;
        double r28676 = r28641 ? r28656 : r28675;
        return r28676;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 95.54164031582988

    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. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]

    3. Simplified1.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.66666666666666674, {x}^{3}, 2\right) - 1 \cdot {x}^{2}}}{2}\]
    4. Using strategy rm

    5. Applied fma-udef1.3

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right)} - 1 \cdot {x}^{2}}{2}\]

    6. Applied associate--l+1.3

      \[\leadsto \frac{\color{blue}{0.66666666666666674 \cdot {x}^{3} + \left(2 - 1 \cdot {x}^{2}\right)}}{2}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt1.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\]

    if 95.54164031582988 < x

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt0.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 95.541640315829881:\\ \;\;\;\;\frac{\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))