Average Error: 29.7 → 1.0
Time: 2.8m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 391.2622826264862:\\ \;\;\;\;\frac{\frac{{\left(\frac{2}{3} \cdot {x}^{3} + 2\right)}^{3} - {\left({x}^{2}\right)}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right) + \left(x \cdot x + 2\right)\right) \cdot \left(2 + \left(x \cdot x\right) \cdot \left(\left(\sqrt[3]{x \cdot \frac{2}{3}} \cdot \sqrt[3]{x \cdot \frac{2}{3}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x \cdot \frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{x \cdot \frac{2}{3}}}\right) \cdot \sqrt[3]{\sqrt[3]{x \cdot \frac{2}{3}}}\right)\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {e}^{\left(-\left(1 + \varepsilon\right) \cdot x\right)}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 391.2622826264862

    1. Initial program 39.4

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

    2. Taylor expanded around 0 1.4

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

    4. Applied flip3--1.4

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

    5. Applied simplify1.4

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

    7. Applied add-cube-cbrt1.4

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

    9. Applied add-cube-cbrt1.4

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

    if 391.2622826264862 < 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. Using strategy rm

    3. Applied *-un-lft-identity0.1

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

    4. Applied exp-prod0.1

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

    5. Applied simplify0.1

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

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2020178 
(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))