Average Error: 28.5 → 1.1
Time: 6.3s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.35384572610823661:\\ \;\;\;\;0.33333333333333337 \cdot {x}^{3} + \sqrt[3]{{\left(1 - 0.5 \cdot {x}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 1.35384572610823661

    1. Initial program 38.6

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

    2. Simplified38.6

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

    3. Taylor expanded around 0 1.3

      \[\leadsto \color{blue}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\]
    4. Using strategy rm

    5. Applied associate--l+1.3

      \[\leadsto \color{blue}{0.33333333333333337 \cdot {x}^{3} + \left(1 - 0.5 \cdot {x}^{2}\right)}\]
    6. Using strategy rm

    7. Applied add-cbrt-cube1.3

      \[\leadsto 0.33333333333333337 \cdot {x}^{3} + \color{blue}{\sqrt[3]{\left(\left(1 - 0.5 \cdot {x}^{2}\right) \cdot \left(1 - 0.5 \cdot {x}^{2}\right)\right) \cdot \left(1 - 0.5 \cdot {x}^{2}\right)}}\]

    8. Simplified1.3

      \[\leadsto 0.33333333333333337 \cdot {x}^{3} + \sqrt[3]{\color{blue}{{\left(1 - 0.5 \cdot {x}^{2}\right)}^{3}}}\]

    if 1.35384572610823661 < x

    1. Initial program 0.6

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

    2. Simplified0.6

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

    4. Applied div-sub0.6

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

    5. Applied div-sub0.6

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

    6. Applied associate--r-0.5

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.35384572610823661:\\ \;\;\;\;0.33333333333333337 \cdot {x}^{3} + \sqrt[3]{{\left(1 - 0.5 \cdot {x}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(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))