Average Error: 29.4 → 1.0
Time: 9.5m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 150.54096709715435:\\ \;\;\;\;\frac{\frac{{\left(\frac{8 + {\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)}^{3}}{e^{\log \left(\left(\left(\frac{2}{3} \cdot {x}^{3}\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right) - \left(\frac{2}{3} \cdot {x}^{3}\right) \cdot 2\right) + 4\right)}}\right)}^{3} - {\left({x}^{2}\right)}^{3}}{\left(\left(2 + x \cdot x\right) + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(x \cdot x\right) + \left(2 + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(2 + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{{\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}} \cdot \sqrt[3]{{\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)\right) \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 150.54096709715435

    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-exp-log1.2

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

    6. Applied flip3--1.2

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

    7. Applied simplify1.2

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

    9. Applied flip3-+1.2

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

    10. Applied log-div3.1

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

    11. Applied exp-diff3.9

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

    12. Applied simplify1.2

      \[\leadsto \frac{\frac{{\left(\frac{\color{blue}{8 + {\left(\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right)}^{3}}}{e^{\log \left(2 \cdot 2 + \left(\left(\frac{2}{3} \cdot {x}^{3}\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right) - 2 \cdot \left(\frac{2}{3} \cdot {x}^{3}\right)\right)\right)}}\right)}^{3} - {\left({x}^{2}\right)}^{3}}{\left(\left(x \cdot x + 2\right) + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(x \cdot x\right) + \left(2 + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(2 + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)}}{2}\]

    if 150.54096709715435 < 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

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

    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

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

  4. Applied simplify1.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le 150.54096709715435:\\ \;\;\;\;\frac{\frac{{\left(\frac{8 + {\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)}^{3}}{e^{\log \left(\left(\left(\frac{2}{3} \cdot {x}^{3}\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right) - \left(\frac{2}{3} \cdot {x}^{3}\right) \cdot 2\right) + 4\right)}}\right)}^{3} - {\left({x}^{2}\right)}^{3}}{\left(\left(2 + x \cdot x\right) + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(x \cdot x\right) + \left(2 + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(2 + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{{\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}} \cdot \sqrt[3]{{\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)\right) \cdot \sqrt[3]{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-x\right) \cdot \left(\varepsilon + 1\right)}}{2}\\ \end{array}}\]

Runtime

Time bar (total: 9.5m)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))