Average Error: 44.3 → 0
Time: 1.6m
Precision: 64
Internal precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le 921224087.6603982:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\frac{1 + \frac{1}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}}}\right)}^3 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes.

  2. if x < 921224087.6603982

    1. Initial program 51.0

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

    2. Applied taylor 0

      \[\leadsto \frac{2}{2}\]

    3. Taylor expanded around 0 0

      \[\leadsto \frac{\color{blue}{2}}{2}\]

    4. Applied simplify 0

      \[\leadsto \color{blue}{1}\]

    if 921224087.6603982 < x

    1. Initial program 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 add-cube-cbrt 0

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

    4. Applied simplify 0

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

Runtime

Time bar (total: 1.6m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(2802247271 501202877 2967845228 3304626069 2587726502 1941762487)'
(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))