Average Error: 29.5 → 1.1
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.8053804953360366:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}\right) \cdot \left(\left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}\right) \cdot \left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + \left(\left(e^{\left(-1 - \varepsilon\right) \cdot x} - \frac{e^{\left(-1 - \varepsilon\right) \cdot x}}{\varepsilon}\right) + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right)}{2}\\ \end{array}\]

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 < 1.8053804953360366

    1. Initial program 39.2

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

    2. Initial simplification39.2

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

    3. Taylor expanded around 0 1.3

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

    5. Applied add-cbrt-cube1.3

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

    if 1.8053804953360366 < 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. Initial simplification0.4

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

    3. Taylor expanded around inf 0.4

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

    4. Simplified0.4

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.8053804953360366:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}\right) \cdot \left(\left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}\right) \cdot \left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + \left(\left(e^{\left(-1 - \varepsilon\right) \cdot x} - \frac{e^{\left(-1 - \varepsilon\right) \cdot x}}{\varepsilon}\right) + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right)}{2}\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018234 
(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))