Average Error: 29.8 → 0.9
Time: 6.2s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.448332226218201291:\\ \;\;\;\;\frac{0.66666666666666674 \cdot {x}^{3} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\left(\frac{e^{x \cdot \varepsilon - 1 \cdot x}}{\varepsilon} + e^{x \cdot \varepsilon - 1 \cdot x}\right) - \frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon}\right) + 1 \cdot e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 0.448332226218201291

    1. Initial program 39.2

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

    2. Taylor expanded around 0 0.9

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]
    3. Using strategy rm

    4. Applied associate--l+0.9

      \[\leadsto \frac{\color{blue}{0.66666666666666674 \cdot {x}^{3} + \left(2 - 1 \cdot {x}^{2}\right)}}{2}\]

    if 0.448332226218201291 < x

    1. Initial program 0.9

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

    2. Taylor expanded around inf 0.9

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

    3. Simplified0.8

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.448332226218201291:\\ \;\;\;\;\frac{0.66666666666666674 \cdot {x}^{3} + \left(2 - 1 \cdot {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\left(\frac{e^{x \cdot \varepsilon - 1 \cdot x}}{\varepsilon} + e^{x \cdot \varepsilon - 1 \cdot x}\right) - \frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon}\right) + 1 \cdot e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (neg (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (neg (* (+ 1.0 eps) x))))) 2.0))