Average Error: 29.5 → 1.0
Time: 6.5s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 295.27480666079055:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - 1 \cdot \left(\frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon} - e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}\right)}{2}\\ \end{array}\]
\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 295.27480666079055:\\
\;\;\;\;\frac{\left(\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}} + 2\right) - 1 \cdot {x}^{2}}{2}\\

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

\end{array}
double f(double x, double eps) {
        double r33532 = 1.0;
        double r33533 = eps;
        double r33534 = r33532 / r33533;
        double r33535 = r33532 + r33534;
        double r33536 = r33532 - r33533;
        double r33537 = x;
        double r33538 = r33536 * r33537;
        double r33539 = -r33538;
        double r33540 = exp(r33539);
        double r33541 = r33535 * r33540;
        double r33542 = r33534 - r33532;
        double r33543 = r33532 + r33533;
        double r33544 = r33543 * r33537;
        double r33545 = -r33544;
        double r33546 = exp(r33545);
        double r33547 = r33542 * r33546;
        double r33548 = r33541 - r33547;
        double r33549 = 2.0;
        double r33550 = r33548 / r33549;
        return r33550;
}
↓
double f(double x, double eps) {
        double r33551 = x;
        double r33552 = 295.27480666079055;
        bool r33553 = r33551 <= r33552;
        double r33554 = 0.6666666666666667;
        double r33555 = 3.0;
        double r33556 = pow(r33551, r33555);
        double r33557 = r33554 * r33556;
        double r33558 = cbrt(r33557);
        double r33559 = r33558 * r33558;
        double r33560 = r33559 * r33558;
        double r33561 = 2.0;
        double r33562 = r33560 + r33561;
        double r33563 = 1.0;
        double r33564 = 2.0;
        double r33565 = pow(r33551, r33564);
        double r33566 = r33563 * r33565;
        double r33567 = r33562 - r33566;
        double r33568 = r33567 / r33561;
        double r33569 = eps;
        double r33570 = r33563 / r33569;
        double r33571 = r33563 + r33570;
        double r33572 = r33563 - r33569;
        double r33573 = r33572 * r33551;
        double r33574 = -r33573;
        double r33575 = exp(r33574);
        double r33576 = r33571 * r33575;
        double r33577 = r33551 * r33569;
        double r33578 = r33563 * r33551;
        double r33579 = r33577 + r33578;
        double r33580 = -r33579;
        double r33581 = exp(r33580);
        double r33582 = r33581 / r33569;
        double r33583 = r33582 - r33581;
        double r33584 = r33563 * r33583;
        double r33585 = r33576 - r33584;
        double r33586 = r33585 / r33561;
        double r33587 = r33553 ? r33568 : r33586;
        return r33587;
}

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 < 295.27480666079055

    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 1.3

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

    4. Applied add-cube-cbrt1.3

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

    if 295.27480666079055 < 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. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 295.27480666079055:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{0.66666666666666674 \cdot {x}^{3}} \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}}\right) \cdot \sqrt[3]{0.66666666666666674 \cdot {x}^{3}} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - 1 \cdot \left(\frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon} - e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))