Average Error: 29.7 → 1.0
Time: 29.0s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\frac{e^{x \cdot \varepsilon - x} + \left(\left(\frac{e^{x \cdot \varepsilon - x}}{\varepsilon} - \frac{e^{\left(-x\right) - x \cdot \varepsilon}}{\varepsilon}\right) + e^{\left(-x\right) - x \cdot \varepsilon}\right)}{2}\]
\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}
↓
\frac{e^{x \cdot \varepsilon - x} + \left(\left(\frac{e^{x \cdot \varepsilon - x}}{\varepsilon} - \frac{e^{\left(-x\right) - x \cdot \varepsilon}}{\varepsilon}\right) + e^{\left(-x\right) - x \cdot \varepsilon}\right)}{2}
double f(double x, double eps) {
        double r1735532 = 1.0;
        double r1735533 = eps;
        double r1735534 = r1735532 / r1735533;
        double r1735535 = r1735532 + r1735534;
        double r1735536 = r1735532 - r1735533;
        double r1735537 = x;
        double r1735538 = r1735536 * r1735537;
        double r1735539 = -r1735538;
        double r1735540 = exp(r1735539);
        double r1735541 = r1735535 * r1735540;
        double r1735542 = r1735534 - r1735532;
        double r1735543 = r1735532 + r1735533;
        double r1735544 = r1735543 * r1735537;
        double r1735545 = -r1735544;
        double r1735546 = exp(r1735545);
        double r1735547 = r1735542 * r1735546;
        double r1735548 = r1735541 - r1735547;
        double r1735549 = 2.0;
        double r1735550 = r1735548 / r1735549;
        return r1735550;
}
↓
double f(double x, double eps) {
        double r1735551 = x;
        double r1735552 = eps;
        double r1735553 = r1735551 * r1735552;
        double r1735554 = r1735553 - r1735551;
        double r1735555 = exp(r1735554);
        double r1735556 = r1735555 / r1735552;
        double r1735557 = -r1735551;
        double r1735558 = r1735557 - r1735553;
        double r1735559 = exp(r1735558);
        double r1735560 = r1735559 / r1735552;
        double r1735561 = r1735556 - r1735560;
        double r1735562 = r1735561 + r1735559;
        double r1735563 = r1735555 + r1735562;
        double r1735564 = 2.0;
        double r1735565 = r1735563 / r1735564;
        return r1735565;
}

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. Initial program 29.7

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

  2. Simplified24.9

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

  4. Applied associate--r-1.0

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

  5. Final simplification1.0

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

Reproduce

herbie shell --seed 2019134 
(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))