Average Error: 29.4 → 0.9
Time: 5.7m
Precision: 64
Internal Precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 356.0101387754159:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\left(\left({x}^{3} \cdot 8 + \left(8 + \frac{5}{3} \cdot {x}^{6}\right)\right) \cdot \left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}\right)\right) \cdot \left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2} \cdot {x}^{2}\right)}}{\sqrt[3]{\left(\left({x}^{2} \cdot {x}^{2} + {x}^{2} \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right)\right) + \left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right)\right) \cdot \left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) + {x}^{2}\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{e^{x \cdot \varepsilon - x}}{\varepsilon} + \left(e^{-\left(x + x \cdot \varepsilon\right)} + e^{x \cdot \varepsilon - x}\right)\right) - \frac{e^{-\left(x + x \cdot \varepsilon\right)}}{\varepsilon}}{2}\\ \end{array}\]

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 < 356.0101387754159

    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. 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 flip3--1.2

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

    8. Applied associate-*r/1.2

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

    9. Applied frac-times1.2

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

    10. Applied cbrt-div1.3

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

    11. Taylor expanded around 0 1.3

      \[\leadsto \frac{\frac{\sqrt[3]{\left(\left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right) \cdot \color{blue}{\left(8 \cdot {x}^{3} + \left(\frac{5}{3} \cdot {x}^{6} + 8\right)\right)}\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}\right)}}{\sqrt[3]{\left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) + \left({x}^{2} \cdot {x}^{2} + \left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot {x}^{2}\right)\right) \cdot \left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) + {x}^{2}\right)}}}{2}\]

    if 356.0101387754159 < x

    1. Initial program 0.0

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

    2. Taylor expanded around -inf 0.0

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 356.0101387754159:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\left(\left({x}^{3} \cdot 8 + \left(8 + \frac{5}{3} \cdot {x}^{6}\right)\right) \cdot \left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}\right)\right) \cdot \left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2} \cdot {x}^{2}\right)}}{\sqrt[3]{\left(\left({x}^{2} \cdot {x}^{2} + {x}^{2} \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right)\right) + \left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right)\right) \cdot \left(\left(2 + \frac{2}{3} \cdot {x}^{3}\right) + {x}^{2}\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{e^{x \cdot \varepsilon - x}}{\varepsilon} + \left(e^{-\left(x + x \cdot \varepsilon\right)} + e^{x \cdot \varepsilon - x}\right)\right) - \frac{e^{-\left(x + x \cdot \varepsilon\right)}}{\varepsilon}}{2}\\ \end{array}\]

Runtime

Time bar (total: 5.7m)Debug logProfile

herbie shell --seed 2018336 +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))