Average Error: 29.1 → 1.0
Time: 3.2m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 10.163623375656135:\\ \;\;\;\;\frac{\left(2 + \log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right) \cdot \left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 10.163623375656135

    1. Initial program 38.6

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

    2. Taylor expanded around 0 1.2

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

    4. Applied add-log-exp1.2

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

    if 10.163623375656135 < 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 add-cube-cbrt0.3

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

    4. Applied associate-*l*0.3

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

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed '#(1063154770 1824007522 645063331 41291047 494775821 1237684644)' 
(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))