Average Error: 29.4 → 1.0
Time: 4.0m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 125.54754116772546:\\ \;\;\;\;\sqrt[3]{{\left(\frac{(\left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right) + \left(2 - x \cdot x\right))_*}{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left((\left(\frac{1}{\varepsilon}\right) \cdot \left({\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}\right) + \left({\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}\right))_* - \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}\right)}^{3}}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 125.54754116772546

    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

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

    if 125.54754116772546 < 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-cbrt-cube0.3

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

    4. Applied simplify0.3

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

Runtime

Time bar (total: 4.0m)Debug logProfile

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