Average Error: 29.8 → 1.0
Time: 2.2m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\sqrt[3]{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}} \cdot \sqrt[3]{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}\right) \cdot \sqrt[3]{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}}{2} \le 159114574.3600766:\\ \;\;\;\;\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\varepsilon \cdot x - x} + e^{\left(-x\right) + \left(-x\right) \cdot \varepsilon}\right) + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right) - \frac{e^{\left(-x\right) + \left(-x\right) \cdot \varepsilon}}{\varepsilon}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (* (cbrt (- (+ 2 (* 2/3 (pow x 3))) (pow x 2))) (cbrt (- (+ 2 (* 2/3 (pow x 3))) (pow x 2)))) (cbrt (- (+ 2 (* 2/3 (pow x 3))) (pow x 2)))) 2) < 159114574.3600766

    1. Initial program 39.5

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

    2. Taylor expanded around 0 1.4

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

    if 159114574.3600766 < (/ (* (* (cbrt (- (+ 2 (* 2/3 (pow x 3))) (pow x 2))) (cbrt (- (+ 2 (* 2/3 (pow x 3))) (pow x 2)))) (cbrt (- (+ 2 (* 2/3 (pow x 3))) (pow x 2)))) 2)

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\sqrt[3]{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}} \cdot \sqrt[3]{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}\right) \cdot \sqrt[3]{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}}{2} \le 159114574.3600766:\\ \;\;\;\;\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{\varepsilon \cdot x - x} + e^{\left(-x\right) + \left(-x\right) \cdot \varepsilon}\right) + \frac{e^{\varepsilon \cdot x - x}}{\varepsilon}\right) - \frac{e^{\left(-x\right) + \left(-x\right) \cdot \varepsilon}}{\varepsilon}}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018215 +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))