Average Error: 29.5 → 0.9
Time: 2.4m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 292.3339702384316:\\ \;\;\;\;\frac{\left(2 + \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot \sqrt[3]{{x}^{3}}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \sqrt{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

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 < 292.3339702384316

    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 1.2

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

    4. Applied add-cube-cbrt1.2

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

    5. Applied associate-*r*1.2

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

    6. Applied simplify1.2

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

    if 292.3339702384316 < 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 add-sqr-sqrt0.1

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

    4. Applied associate-*r*0.1

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

    5. Applied simplify0.1

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018208 
(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))