Average Error: 29.8 → 0.8
Time: 2.5m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 12.298818826425293:\\ \;\;\;\;\frac{\left(2 + \left(\log \left(\sqrt[3]{e^{\frac{2}{3} \cdot {x}^{3}}} \cdot \sqrt[3]{e^{\frac{2}{3} \cdot {x}^{3}}}\right) + \log \left(\sqrt[3]{e^{\frac{2}{3} \cdot {x}^{3}}}\right)\right)\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(e^{-\left(1 - \varepsilon\right) \cdot x}\right) + \left(\frac{1 - \frac{1}{\varepsilon}}{e^{(\varepsilon \cdot x + x)_*}}\right))_*}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 12.298818826425293

    1. Initial program 39.3

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

    2. Taylor expanded around 0 1.0

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

    4. Applied add-log-exp1.0

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

    6. Applied add-cube-cbrt1.0

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

    7. Applied log-prod1.0

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

    if 12.298818826425293 < 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 fma-neg0.2

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

    4. Applied simplify0.2

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +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))