Average Error: 29.5 → 0.6
Time: 2.7m
Precision: 64
Internal Precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right) \le -15798539845785176.0:\\ \;\;\;\;\frac{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}{2}\\ \mathbf{elif}\;e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right) \le -0.0:\\ \;\;\;\;\frac{(\left(e^{\left(\varepsilon + -1\right) \cdot x}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(\frac{1 + \frac{-1}{\varepsilon}}{e^{(x \cdot \varepsilon + x)_*}}\right))_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x)))) < -15798539845785176.0 or -0.0 < (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))

    1. Initial program 60.0

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

    2. Taylor expanded around 0 1.0

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

    if -15798539845785176.0 < (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x)))) < -0.0

    1. Initial program 0.1

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

    3. Applied prod-diff0.1

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

    4. Simplified0.1

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

    5. Simplified0.1

      \[\leadsto \frac{(\left(e^{x \cdot \left(-1 + \varepsilon\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(\frac{1 + \frac{-1}{\varepsilon}}{e^{(x \cdot \varepsilon + x)_*}}\right))_* + \color{blue}{0}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right) \le -15798539845785176.0:\\ \;\;\;\;\frac{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}{2}\\ \mathbf{elif}\;e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right) \le -0.0:\\ \;\;\;\;\frac{(\left(e^{\left(\varepsilon + -1\right) \cdot x}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(\frac{1 + \frac{-1}{\varepsilon}}{e^{(x \cdot \varepsilon + x)_*}}\right))_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018274 +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))