Average Error: 28.8 → 0.9
Time: 2.8m
Precision: 64
Internal Precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.878033538763582:\\ \;\;\;\;\frac{\frac{\left(4 + \frac{8}{3} \cdot {x}^{3}\right) - {x}^{4}}{\left({x}^{3} \cdot \frac{2}{3} + 2\right) \cdot \left({x}^{3} \cdot \frac{2}{3} + 2\right) - {x}^{2} \cdot {x}^{2}} \cdot \left(\left({x}^{3} \cdot \frac{2}{3} + 2\right) - {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} + e^{\varepsilon \cdot x - x}\right) - e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 2.878033538763582

    1. Initial program 38.4

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

    2. Taylor expanded around 0 1.1

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

    4. Applied flip--1.1

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

    5. Taylor expanded around 0 1.1

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{8}{3} \cdot {x}^{3} + 4\right) - {x}^{4}}}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) + {x}^{2}}}{2}\]
    6. Using strategy rm

    7. Applied flip-+1.1

      \[\leadsto \frac{\frac{\left(\frac{8}{3} \cdot {x}^{3} + 4\right) - {x}^{4}}{\color{blue}{\frac{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}}}}{2}\]

    8. Applied associate-/r/1.1

      \[\leadsto \frac{\color{blue}{\frac{\left(\frac{8}{3} \cdot {x}^{3} + 4\right) - {x}^{4}}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}} \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right)}}{2}\]

    if 2.878033538763582 < x

    1. Initial program 0.5

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

    2. Taylor expanded around -inf 0.5

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.878033538763582:\\ \;\;\;\;\frac{\frac{\left(4 + \frac{8}{3} \cdot {x}^{3}\right) - {x}^{4}}{\left({x}^{3} \cdot \frac{2}{3} + 2\right) \cdot \left({x}^{3} \cdot \frac{2}{3} + 2\right) - {x}^{2} \cdot {x}^{2}} \cdot \left(\left({x}^{3} \cdot \frac{2}{3} + 2\right) - {x}^{2}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} + e^{\varepsilon \cdot x - x}\right) - e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018349 +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))