Average Error: 30.0 → 0.8
Time: 1.2m
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\frac{\left(\left(\frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon} - \frac{e^{\left(-1 - \varepsilon\right) \cdot x}}{\varepsilon}\right) + e^{\left(\sqrt[3]{\left(-1 - \varepsilon\right) \cdot x} \cdot \sqrt[3]{\left(-1 - \varepsilon\right) \cdot x}\right) \cdot \sqrt[3]{\left(-1 - \varepsilon\right) \cdot x}}\right) + e^{\left(\varepsilon + -1\right) \cdot x}}{2}\]
\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}
↓
\frac{\left(\left(\frac{e^{\left(\varepsilon + -1\right) \cdot x}}{\varepsilon} - \frac{e^{\left(-1 - \varepsilon\right) \cdot x}}{\varepsilon}\right) + e^{\left(\sqrt[3]{\left(-1 - \varepsilon\right) \cdot x} \cdot \sqrt[3]{\left(-1 - \varepsilon\right) \cdot x}\right) \cdot \sqrt[3]{\left(-1 - \varepsilon\right) \cdot x}}\right) + e^{\left(\varepsilon + -1\right) \cdot x}}{2}
double f(double x, double eps) {
        double r5836221 = 1.0;
        double r5836222 = eps;
        double r5836223 = r5836221 / r5836222;
        double r5836224 = r5836221 + r5836223;
        double r5836225 = r5836221 - r5836222;
        double r5836226 = x;
        double r5836227 = r5836225 * r5836226;
        double r5836228 = -r5836227;
        double r5836229 = exp(r5836228);
        double r5836230 = r5836224 * r5836229;
        double r5836231 = r5836223 - r5836221;
        double r5836232 = r5836221 + r5836222;
        double r5836233 = r5836232 * r5836226;
        double r5836234 = -r5836233;
        double r5836235 = exp(r5836234);
        double r5836236 = r5836231 * r5836235;
        double r5836237 = r5836230 - r5836236;
        double r5836238 = 2.0;
        double r5836239 = r5836237 / r5836238;
        return r5836239;
}
↓
double f(double x, double eps) {
        double r5836240 = eps;
        double r5836241 = -1.0;
        double r5836242 = r5836240 + r5836241;
        double r5836243 = x;
        double r5836244 = r5836242 * r5836243;
        double r5836245 = exp(r5836244);
        double r5836246 = r5836245 / r5836240;
        double r5836247 = r5836241 - r5836240;
        double r5836248 = r5836247 * r5836243;
        double r5836249 = exp(r5836248);
        double r5836250 = r5836249 / r5836240;
        double r5836251 = r5836246 - r5836250;
        double r5836252 = cbrt(r5836248);
        double r5836253 = r5836252 * r5836252;
        double r5836254 = r5836253 * r5836252;
        double r5836255 = exp(r5836254);
        double r5836256 = r5836251 + r5836255;
        double r5836257 = r5836256 + r5836245;
        double r5836258 = 2.0;
        double r5836259 = r5836257 / r5836258;
        return r5836259;
}

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 30.0

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

  2. Simplified30.0

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

  4. Applied associate--l+25.2

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

  6. Applied associate--r-0.8

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

  8. Applied add-cube-cbrt0.8

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

  9. Final simplification0.8

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

Reproduce

herbie shell --seed 2019128 
(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))