Average Error: 29.6 → 1.0
Time: 2.7m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 30.08968739359558:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot x + 2\right) + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(2 - x \cdot x\right) + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right)}{{\left(\left({x}^{2} + \left({x}^{3} \cdot \frac{2}{3} + 2\right)\right) \cdot \left(\left({x}^{2} + \left({x}^{3} \cdot \frac{2}{3} + 2\right)\right) \cdot \left({x}^{2} + \left({x}^{3} \cdot \frac{2}{3} + 2\right)\right)\right)\right)}^{\frac{1}{3}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{\sqrt[3]{\left(-1 - \varepsilon\right) \cdot x} \cdot \sqrt[3]{\left(-1 - \varepsilon\right) \cdot x}}\right)}^{\left(\sqrt[3]{\left(1 + \varepsilon\right) \cdot \left(-x\right)}\right)}}{2}\\ \end{array}\]

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 < 30.08968739359558

    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 1.2

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

    4. Applied add-cbrt-cube1.2

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

    6. Applied flip--1.2

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

    7. Applied flip--1.2

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

    8. Applied flip--1.2

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

    9. Applied frac-times1.2

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

    10. Applied frac-times1.2

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

    11. Applied cbrt-div1.2

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

    12. Simplified1.2

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

    14. Applied pow1/31.2

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

    if 30.08968739359558 < x

    1. Initial program 0.4

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

    3. Applied add-cube-cbrt0.4

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

    4. Applied exp-prod0.4

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

    5. Simplified0.4

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 30.08968739359558:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot x + 2\right) + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(2 - x \cdot x\right) + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right)}{{\left(\left({x}^{2} + \left({x}^{3} \cdot \frac{2}{3} + 2\right)\right) \cdot \left(\left({x}^{2} + \left({x}^{3} \cdot \frac{2}{3} + 2\right)\right) \cdot \left({x}^{2} + \left({x}^{3} \cdot \frac{2}{3} + 2\right)\right)\right)\right)}^{\frac{1}{3}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\left(e^{\sqrt[3]{\left(-1 - \varepsilon\right) \cdot x} \cdot \sqrt[3]{\left(-1 - \varepsilon\right) \cdot x}}\right)}^{\left(\sqrt[3]{\left(1 + \varepsilon\right) \cdot \left(-x\right)}\right)}}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018254 
(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))