Average Error: 29.0 → 0.9
Time: 5.8m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 282.1579976438849:\\ \;\;\;\;\frac{\left(2 + \left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({\left(e^{x}\right)}^{\left(\varepsilon - 1\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}\right)}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 282.1579976438849

    1. Initial program 38.4

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

    2. Taylor expanded around 0 1.2

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

    4. Applied add-cube-cbrt1.2

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

    if 282.1579976438849 < x

    1. Initial program 0.1

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

    3. Applied add-exp-log0.1

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

    4. Applied simplify0.1

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

Runtime

Time bar (total: 5.8m)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' 
(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))