Average Error: 29.1 → 0.8
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.584917519819828:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) - \frac{-2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} + \left(\frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon} - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 1.584917519819828

    1. Initial program 38.6

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

    2. Simplified38.6

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

    3. Taylor expanded around 0 1.0

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

    4. Simplified1.0

      \[\leadsto \frac{\color{blue}{\left(2 - x \cdot x\right) - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-2}{3}}}{2}\]

    5. Taylor expanded around inf 1.0

      \[\leadsto \frac{\left(2 - x \cdot x\right) - \color{blue}{{x}^{3}} \cdot \frac{-2}{3}}{2}\]

    6. Simplified1.0

      \[\leadsto \frac{\left(2 - x \cdot x\right) - \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot \frac{-2}{3}}{2}\]

    if 1.584917519819828 < x

    1. Initial program 0.3

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

    2. Simplified0.3

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

    4. Applied associate--l+0.3

      \[\leadsto \frac{\color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} + \left(\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon} - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.584917519819828:\\ \;\;\;\;\frac{\left(2 - x \cdot x\right) - \frac{-2}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} + \left(\frac{e^{\left(-1 + \varepsilon\right) \cdot x}}{\varepsilon} - \left(\frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon} - e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019088 
(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))