Average Error: 29.5 → 1.0
Time: 5.0m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 35.794330612085844:\\ \;\;\;\;\frac{(\left((\left({x}^{3}\right) \cdot \frac{2}{3} + 2)_*\right) \cdot \left((\left({x}^{3}\right) \cdot \frac{2}{3} + 2)_*\right) + \left({x}^{3} \cdot \left(-x\right)\right))_*}{(\left((\frac{2}{3} \cdot x + 1)_*\right) \cdot \left(x \cdot x\right) + 2)_* \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{1}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}} - \frac{\frac{1}{\varepsilon} - 1}{e^{(x \cdot \varepsilon + x)_*}}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 35.794330612085844

    1. Initial program 39.1

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

    2. 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 flip--1.2

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

    5. Applied simplify1.2

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

    6. Applied simplify1.2

      \[\leadsto \frac{\frac{(\left((\left({x}^{3}\right) \cdot \frac{2}{3} + 2)_*\right) \cdot \left((\left({x}^{3}\right) \cdot \frac{2}{3} + 2)_*\right) + \left({x}^{3} \cdot \left(-x\right)\right))_*}{\color{blue}{(\left((\frac{2}{3} \cdot x + 1)_*\right) \cdot \left(x \cdot x\right) + 2)_*}}}{2}\]
    7. Using strategy rm

    8. Applied log1p-expm1-u1.2

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

    if 35.794330612085844 < x

    1. Initial program 0.3

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

    3. Applied pow10.3

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

    4. Applied pow10.3

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

    5. Applied pow-prod-down0.3

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

    6. Applied simplify0.3

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

  5. Applied simplify1.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le 35.794330612085844:\\ \;\;\;\;\frac{(\left((\left({x}^{3}\right) \cdot \frac{2}{3} + 2)_*\right) \cdot \left((\left({x}^{3}\right) \cdot \frac{2}{3} + 2)_*\right) + \left({x}^{3} \cdot \left(-x\right)\right))_*}{(\left((\frac{2}{3} \cdot x + 1)_*\right) \cdot \left(x \cdot x\right) + 2)_* \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{1}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}} - \frac{\frac{1}{\varepsilon} - 1}{e^{(x \cdot \varepsilon + x)_*}}}{2}\\ \end{array}}\]

Runtime

Time bar (total: 5.0m)Debug log

herbie shell --seed '#(1889797285 268396849 4100589100 2067516092 3019009300 3748763710)' +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))