Average Error: 29.4 → 1.1
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 245.57615583620168:\\ \;\;\;\;\frac{(\left(x \cdot x\right) \cdot \left(\frac{2}{3} \cdot x\right) + \left(2 - x \cdot x\right))_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - {e}^{\left(\left(-x\right) \cdot \left(\varepsilon + 1\right)\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 245.57615583620168

    1. Initial program 39.0

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

    2. Taylor expanded around 0 1.4

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

    3. Simplified1.4

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

    4. Taylor expanded around -inf 1.4

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

    5. Simplified1.4

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

    if 245.57615583620168 < x

    1. Initial program 0.0

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

    3. Applied *-un-lft-identity0.0

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

    4. Applied exp-prod0.0

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

    5. Simplified0.0

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 245.57615583620168:\\ \;\;\;\;\frac{(\left(x \cdot x\right) \cdot \left(\frac{2}{3} \cdot x\right) + \left(2 - x \cdot x\right))_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - {e}^{\left(\left(-x\right) \cdot \left(\varepsilon + 1\right)\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +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))