Average Error: 29.8 → 0.9
Time: 1.9m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 27.937358935982928:\\ \;\;\;\;\frac{\left(2 + (e^{(e^{\log_* (1 + \log_* (1 + \frac{2}{3} \cdot {x}^{3}))} - 1)^*} - 1)^*\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}\right) + \left(e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(-\left(\frac{1}{\varepsilon} - 1\right)\right)\right))_*}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 27.937358935982928

    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 1.0

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

    4. Applied expm1-log1p-u1.1

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

    6. Applied expm1-log1p-u1.1

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

    if 27.937358935982928 < x

    1. Initial program 0.3

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

    3. Applied fma-neg0.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 27.937358935982928:\\ \;\;\;\;\frac{\left(2 + (e^{(e^{\log_* (1 + \log_* (1 + \frac{2}{3} \cdot {x}^{3}))} - 1)^*} - 1)^*\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}\right) + \left(e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(-\left(\frac{1}{\varepsilon} - 1\right)\right)\right))_*}{2}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018216 +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))