Average Error: 30.2 → 1.0
Time: 2.4m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{2}{3} \cdot {x}^{3} - {x}^{2} \le 94771532.36338314:\\ \;\;\;\;\frac{2 + \left(\frac{2}{3} \cdot {x}^{3} - {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left({\left(e^{x}\right)}^{\left(\varepsilon - 1\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}\right)}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (- (* 2/3 (pow x 3)) (pow x 2)) < 94771532.36338314

    1. Initial program 39.6

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

    2. Taylor expanded around 0 1.3

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

    4. Applied associate--l+1.3

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

    if 94771532.36338314 < (- (* 2/3 (pow x 3)) (pow x 2))

    1. Initial program 0.0

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

    3. Applied add-exp-log0.0

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

    4. Applied simplify0.0

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' 
(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))