Average Error: 29.6 → 1.8
Time: 7.1s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.0640289263225317 \cdot 10^{-25}:\\ \;\;\;\;\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]
\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 1.0640289263225317 \cdot 10^{-25}:\\
\;\;\;\;\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}\\

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

\end{array}
double f(double x, double eps) {
        double r51471 = 1.0;
        double r51472 = eps;
        double r51473 = r51471 / r51472;
        double r51474 = r51471 + r51473;
        double r51475 = r51471 - r51472;
        double r51476 = x;
        double r51477 = r51475 * r51476;
        double r51478 = -r51477;
        double r51479 = exp(r51478);
        double r51480 = r51474 * r51479;
        double r51481 = r51473 - r51471;
        double r51482 = r51471 + r51472;
        double r51483 = r51482 * r51476;
        double r51484 = -r51483;
        double r51485 = exp(r51484);
        double r51486 = r51481 * r51485;
        double r51487 = r51480 - r51486;
        double r51488 = 2.0;
        double r51489 = r51487 / r51488;
        return r51489;
}
↓
double f(double x, double eps) {
        double r51490 = x;
        double r51491 = 1.0640289263225317e-25;
        bool r51492 = r51490 <= r51491;
        double r51493 = 0.33333333333333337;
        double r51494 = 3.0;
        double r51495 = pow(r51490, r51494);
        double r51496 = r51493 * r51495;
        double r51497 = 1.0;
        double r51498 = r51496 + r51497;
        double r51499 = 0.5;
        double r51500 = 2.0;
        double r51501 = pow(r51490, r51500);
        double r51502 = r51499 * r51501;
        double r51503 = r51498 - r51502;
        double r51504 = eps;
        double r51505 = r51497 / r51504;
        double r51506 = r51497 + r51505;
        double r51507 = r51497 - r51504;
        double r51508 = r51507 * r51490;
        double r51509 = exp(r51508);
        double r51510 = r51506 / r51509;
        double r51511 = 2.0;
        double r51512 = r51510 / r51511;
        double r51513 = r51497 + r51504;
        double r51514 = r51513 * r51490;
        double r51515 = exp(r51514);
        double r51516 = r51505 / r51515;
        double r51517 = r51516 / r51511;
        double r51518 = r51512 - r51517;
        double r51519 = r51497 / r51515;
        double r51520 = r51519 / r51511;
        double r51521 = r51518 + r51520;
        double r51522 = r51492 ? r51503 : r51521;
        return r51522;
}

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 < 1.0640289263225317e-25

    1. Initial program 38.3

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

    2. Simplified38.3

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

    3. Taylor expanded around 0 1.1

      \[\leadsto \color{blue}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\]

    if 1.0640289263225317e-25 < x

    1. Initial program 4.6

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

    2. Simplified4.6

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

    4. Applied div-sub4.6

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

    5. Applied div-sub4.6

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

    6. Applied associate--r-3.9

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.0640289263225317 \cdot 10^{-25}:\\ \;\;\;\;\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))