Average Error: 29.5 → 1.0
Time: 4.7m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 162.0127110200184:\\ \;\;\;\;\frac{\frac{{\left(x \cdot (\left(-2 + x\right) \cdot x + 2)_*\right)}^{3} + {\left(e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x - x}\right)}^{3}}{\left(e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x - x}\right) \cdot \left(e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x - x}\right) + \left(\left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) \cdot \left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) - \left(e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x - x}\right) \cdot \left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x \cdot \left(\varepsilon + -1\right)} + \frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon}\right) - (\left(e^{x \cdot \left(-1 - \varepsilon\right)}\right) \cdot \left(\frac{1}{\varepsilon}\right) + \left(\left(\sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}} \cdot \sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}}\right) \cdot \left(-\sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}}\right)\right))_*}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 162.0127110200184

    1. Initial program 39.2

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

    2. Initial simplification39.2

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

    4. Applied add-sqr-sqrt61.0

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

    5. Applied *-un-lft-identity61.0

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

    6. Applied *-un-lft-identity61.0

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

    7. Applied distribute-lft-out61.0

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

    8. Applied prod-diff61.3

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

    9. Simplified61.3

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

    10. Simplified1.3

      \[\leadsto \frac{\left(\left(e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) + \left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)\right) + \color{blue}{0}}{2}\]
    11. Using strategy rm

    12. Applied flip3-+1.3

      \[\leadsto \frac{\color{blue}{\frac{{\left(e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right)}^{3} + {\left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}^{3}}{\left(e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \cdot \left(e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) + \left(\left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) \cdot \left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) - \left(e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \cdot \left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)\right)}} + 0}{2}\]

    13. Taylor expanded around 0 1.3

      \[\leadsto \frac{\frac{{\left(e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right)}^{3} + {\color{blue}{\left(\left(2 \cdot x + {x}^{3}\right) - 2 \cdot {x}^{2}\right)}}^{3}}{\left(e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \cdot \left(e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) + \left(\left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) \cdot \left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) - \left(e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \cdot \left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)\right)} + 0}{2}\]

    14. Simplified1.3

      \[\leadsto \frac{\frac{{\left(e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right)}^{3} + {\color{blue}{\left(x \cdot (\left(-2 + x\right) \cdot x + 2)_*\right)}}^{3}}{\left(e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \cdot \left(e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) + \left(\left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) \cdot \left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) - \left(e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right) \cdot \left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)\right)} + 0}{2}\]

    if 162.0127110200184 < x

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied div-inv0.1

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

    6. Applied prod-diff0.1

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

    7. Applied associate--r+0.1

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

    8. Simplified0.1

      \[\leadsto \frac{\left(\left(\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon} + e^{x \cdot \left(\varepsilon + -1\right)}\right) - (\left(e^{x \cdot \left(-1 - \varepsilon\right)}\right) \cdot \left(\frac{1}{\varepsilon}\right) + \left(-\sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}} \cdot \left(\sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}} \cdot \sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}}\right)\right))_*\right) - \color{blue}{0}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 162.0127110200184:\\ \;\;\;\;\frac{\frac{{\left(x \cdot (\left(-2 + x\right) \cdot x + 2)_*\right)}^{3} + {\left(e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x - x}\right)}^{3}}{\left(e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x - x}\right) \cdot \left(e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x - x}\right) + \left(\left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) \cdot \left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right) - \left(e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x - x}\right) \cdot \left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} - \frac{e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x \cdot \left(\varepsilon + -1\right)} + \frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon}\right) - (\left(e^{x \cdot \left(-1 - \varepsilon\right)}\right) \cdot \left(\frac{1}{\varepsilon}\right) + \left(\left(\sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}} \cdot \sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}}\right) \cdot \left(-\sqrt[3]{e^{x \cdot \left(-1 - \varepsilon\right)}}\right)\right))_*}{2}\\ \end{array}\]

Runtime

Time bar (total: 4.7m)Debug logProfile

herbie shell --seed 2018234 +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))