Average Error: 29.6 → 1.0
Time: 38.4s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\frac{\log \left(e^{\left(\left(\frac{e^{\left(\varepsilon - 1\right) \cdot x}}{\varepsilon} - \frac{e^{\left(-1 - \varepsilon\right) \cdot x}}{\varepsilon}\right) + e^{\left(-1 - \varepsilon\right) \cdot x}\right) + e^{\left(\varepsilon - 1\right) \cdot x}}\right)}{2}\]
\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{\log \left(e^{\left(\left(\frac{e^{\left(\varepsilon - 1\right) \cdot x}}{\varepsilon} - \frac{e^{\left(-1 - \varepsilon\right) \cdot x}}{\varepsilon}\right) + e^{\left(-1 - \varepsilon\right) \cdot x}\right) + e^{\left(\varepsilon - 1\right) \cdot x}}\right)}{2}
double f(double x, double eps) {
        double r957238 = 1.0;
        double r957239 = eps;
        double r957240 = r957238 / r957239;
        double r957241 = r957238 + r957240;
        double r957242 = r957238 - r957239;
        double r957243 = x;
        double r957244 = r957242 * r957243;
        double r957245 = -r957244;
        double r957246 = exp(r957245);
        double r957247 = r957241 * r957246;
        double r957248 = r957240 - r957238;
        double r957249 = r957238 + r957239;
        double r957250 = r957249 * r957243;
        double r957251 = -r957250;
        double r957252 = exp(r957251);
        double r957253 = r957248 * r957252;
        double r957254 = r957247 - r957253;
        double r957255 = 2.0;
        double r957256 = r957254 / r957255;
        return r957256;
}
↓
double f(double x, double eps) {
        double r957257 = eps;
        double r957258 = 1.0;
        double r957259 = r957257 - r957258;
        double r957260 = x;
        double r957261 = r957259 * r957260;
        double r957262 = exp(r957261);
        double r957263 = r957262 / r957257;
        double r957264 = -1.0;
        double r957265 = r957264 - r957257;
        double r957266 = r957265 * r957260;
        double r957267 = exp(r957266);
        double r957268 = r957267 / r957257;
        double r957269 = r957263 - r957268;
        double r957270 = r957269 + r957267;
        double r957271 = r957270 + r957262;
        double r957272 = exp(r957271);
        double r957273 = log(r957272);
        double r957274 = 2.0;
        double r957275 = r957273 / r957274;
        return r957275;
}

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 29.6

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

  2. Simplified29.6

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

  4. Applied add-log-exp29.7

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

  5. Applied add-log-exp31.4

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

  6. Applied diff-log31.4

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

  7. Applied add-log-exp31.0

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

  8. Applied add-log-exp31.1

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

  9. Applied sum-log31.1

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

  10. Applied diff-log31.1

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

  11. Simplified1.0

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

  12. Final simplification1.0

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

Reproduce

herbie shell --seed 2019153 
(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))