Average Error: 29.5 → 0.9
Time: 57.3s
Precision: 64
Internal Precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\frac{\left(e^{\left(-1 + \varepsilon\right) \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) + \left(\frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}{2}\]

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

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

  3. Taylor expanded around inf 29.5

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

  4. Simplified0.9

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

  5. Final simplification0.9

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

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(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))