Average Error: 29.7 → 1.0
Time: 2.7m
Precision: 64
Internal Precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 391.6745205327363:\\ \;\;\;\;\frac{\frac{\sqrt{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2} \cdot {x}^{2}}}{\left(1 + \frac{1}{8} \cdot {x}^{4}\right) + \frac{1}{2} \cdot {x}^{2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 391.6745205327363

    1. Initial program 39.1

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

    2. Taylor expanded around 0 1.3

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

    4. Applied flip--1.3

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

    6. Applied add-sqr-sqrt1.3

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

    7. Applied associate-/l*1.3

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

    8. Taylor expanded around 0 1.3

      \[\leadsto \frac{\frac{\sqrt{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}}}{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{8} \cdot {x}^{4} + 1\right)}}}{2}\]

    if 391.6745205327363 < 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 neg-mul-10.1

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

    4. Applied exp-prod0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 391.6745205327363:\\ \;\;\;\;\frac{\frac{\sqrt{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2} \cdot {x}^{2}}}{\left(1 + \frac{1}{8} \cdot {x}^{4}\right) + \frac{1}{2} \cdot {x}^{2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + 1\right) - {\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018277 
(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))