Average Error: 29.4 → 0.9
Time: 3.9m
Precision: 64
Internal Precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.288607759181657:\\ \;\;\;\;\frac{(\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right))_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{-\left(\varepsilon \cdot x + x\right)} + e^{\varepsilon \cdot x - x}\right) + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right) - \frac{e^{-\left(\varepsilon \cdot x + x\right)}}{\varepsilon}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 3.288607759181657

    1. Initial program 39.0

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

    2. Taylor expanded around 0 1.0

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

    4. Applied *-un-lft-identity1.0

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

    5. Applied add-cube-cbrt2.6

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

    6. Applied prod-diff2.6

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

    7. Simplified1.0

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

    8. Simplified1.0

      \[\leadsto \frac{(\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right))_* + \color{blue}{0}}{2}\]

    if 3.288607759181657 < 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. Taylor expanded around inf 0.6

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.288607759181657:\\ \;\;\;\;\frac{(\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right))_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{-\left(\varepsilon \cdot x + x\right)} + e^{\varepsilon \cdot x - x}\right) + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right) - \frac{e^{-\left(\varepsilon \cdot x + x\right)}}{\varepsilon}}{2}\\ \end{array}\]

Runtime

Time bar (total: 3.9m)Debug logProfile

herbie shell --seed 2018296 +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))