Average Error: 29.5 → 1.1
Time: 4.4m
Precision: 64
Internal Precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\frac{\log \left(e^{\left(e^{\left(-1 + \varepsilon\right) \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - \frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon}\right)}\right)}{2}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 29.5

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

  2. Simplified29.5

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

  4. Applied add-log-exp31.3

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

  5. Applied add-log-exp30.9

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

  6. Applied add-log-exp31.0

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

  7. Applied sum-log31.0

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

  8. Applied diff-log31.0

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

  9. Simplified1.1

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

  10. Final simplification1.1

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

Reproduce

herbie shell --seed 2019053 
(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))