Average Error: 29.2 → 0.9
Time: 20.8s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\left(\frac{0.5}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(\frac{\frac{0.5}{\varepsilon}}{e^{x \cdot \left(\varepsilon + 1\right)}} - \frac{\frac{0.5}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}}\right)\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
\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}
↓
\left(\frac{0.5}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(\frac{\frac{0.5}{\varepsilon}}{e^{x \cdot \left(\varepsilon + 1\right)}} - \frac{\frac{0.5}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}}\right)\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}
double f(double x, double eps) {
        double r59524 = 1.0;
        double r59525 = eps;
        double r59526 = r59524 / r59525;
        double r59527 = r59524 + r59526;
        double r59528 = r59524 - r59525;
        double r59529 = x;
        double r59530 = r59528 * r59529;
        double r59531 = -r59530;
        double r59532 = exp(r59531);
        double r59533 = r59527 * r59532;
        double r59534 = r59526 - r59524;
        double r59535 = r59524 + r59525;
        double r59536 = r59535 * r59529;
        double r59537 = -r59536;
        double r59538 = exp(r59537);
        double r59539 = r59534 * r59538;
        double r59540 = r59533 - r59539;
        double r59541 = 2.0;
        double r59542 = r59540 / r59541;
        return r59542;
}
↓
double f(double x, double eps) {
        double r59543 = 0.5;
        double r59544 = x;
        double r59545 = 1.0;
        double r59546 = eps;
        double r59547 = r59545 - r59546;
        double r59548 = r59544 * r59547;
        double r59549 = exp(r59548);
        double r59550 = r59543 / r59549;
        double r59551 = r59543 / r59546;
        double r59552 = r59546 + r59545;
        double r59553 = r59544 * r59552;
        double r59554 = exp(r59553);
        double r59555 = r59551 / r59554;
        double r59556 = r59551 / r59549;
        double r59557 = r59555 - r59556;
        double r59558 = r59550 - r59557;
        double r59559 = r59545 + r59546;
        double r59560 = r59559 * r59544;
        double r59561 = exp(r59560);
        double r59562 = r59545 / r59561;
        double r59563 = 2.0;
        double r59564 = r59562 / r59563;
        double r59565 = r59558 + r59564;
        return r59565;
}

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 29.2

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

  2. Simplified29.2

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

  4. Applied div-sub29.2

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

  5. Applied div-sub29.2

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

  6. Applied associate--r-24.6

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

  7. Taylor expanded around inf 24.6

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

  8. Simplified0.9

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

  9. Final simplification0.9

    \[\leadsto \left(\frac{0.5}{e^{x \cdot \left(1 - \varepsilon\right)}} - \left(\frac{\frac{0.5}{\varepsilon}}{e^{x \cdot \left(\varepsilon + 1\right)}} - \frac{\frac{0.5}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}}\right)\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]

Reproduce

herbie shell --seed 2019291 
(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))