Average Error: 29.4 → 1.0
Time: 6.3m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 217.19010290896483:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right) - \left(x \cdot x - 2\right)}{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt[3]{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}} \cdot \sqrt[3]{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}}\right)\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 217.19010290896483

    1. Initial program 38.7

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

    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-cbrt-cube1.2

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

    5. Applied simplify1.2

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

    if 217.19010290896483 < x

    1. Initial program 0.2

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

    3. Applied add-cube-cbrt0.3

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

    4. Applied associate-*r*0.3

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

    5. Applied simplify0.3

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

Runtime

Time bar (total: 6.3m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(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))