Average Error: 29.2 → 1.0
Time: 4.5m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 45.9097282073001:\\ \;\;\;\;\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 45.9097282073001

    1. Initial program 38.7

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

    2. Taylor expanded around 0 1.3

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

    if 45.9097282073001 < x

    1. Initial program 0.2

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

    3. Applied pow10.2

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

    4. Applied pow10.2

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

    5. Applied pow-prod-down0.2

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

    6. Applied simplify0.2

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

  4. Applied simplify1.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le 45.9097282073001:\\ \;\;\;\;\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array}}\]

Runtime

Time bar (total: 4.5m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' 
(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))