Average Error: 30.0 → 0.9
Time: 2.6m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 12.409794484669835:\\ \;\;\;\;\frac{\log \left(e^{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right)}\right) + \left(2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} + e^{\varepsilon \cdot x - x}\right) - e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 12.409794484669835

    1. Initial program 39.3

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

    2. Taylor expanded around 0 1.1

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

    3. Taylor expanded around -inf 62.0

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

    4. Simplified1.1

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

    6. Applied add-log-exp1.1

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

    if 12.409794484669835 < x

    1. Initial program 0.4

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

    2. Taylor expanded around inf 0.4

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 12.409794484669835:\\ \;\;\;\;\frac{\log \left(e^{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right)}\right) + \left(2 - x \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} + e^{\varepsilon \cdot x - x}\right) - e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018242 +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))