Average Error: 29.4 → 1.1
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 13.138693286853812:\\ \;\;\;\;\frac{e^{\log 2 + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{e^{x \cdot \varepsilon - x}}{\varepsilon} + \left(e^{x \cdot \varepsilon - x} + e^{-\left(x \cdot \varepsilon + x\right)}\right)\right) - \frac{e^{-\left(x \cdot \varepsilon + x\right)}}{\varepsilon}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

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Your Program's Arguments

Results

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Derivation

  1. Split input into 2 regimes

  2. if x < 13.138693286853812

    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 1.3

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

    4. Applied add-exp-log1.4

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

    5. Taylor expanded around 0 1.4

      \[\leadsto \frac{e^{\color{blue}{\left(\log 2 + \frac{1}{3} \cdot {x}^{3}\right) - \frac{1}{2} \cdot {x}^{2}}}}{2}\]

    6. Simplified1.4

      \[\leadsto \frac{e^{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + \log 2}}}{2}\]

    if 13.138693286853812 < 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. Taylor expanded around -inf 0.3

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 13.138693286853812:\\ \;\;\;\;\frac{e^{\log 2 + \left(x \cdot x\right) \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{e^{x \cdot \varepsilon - x}}{\varepsilon} + \left(e^{x \cdot \varepsilon - x} + e^{-\left(x \cdot \varepsilon + x\right)}\right)\right) - \frac{e^{-\left(x \cdot \varepsilon + x\right)}}{\varepsilon}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 
(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))