Average Error: 29.7 → 1.1
Time: 2.3m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 448.3054261514617:\\ \;\;\;\;\frac{\left(2 + x \cdot \left(\frac{2}{3} \cdot \left(\sqrt[3]{{x}^{3}} \cdot \log_* (1 + (e^{\sqrt[3]{{x}^{3}}} - 1)^*)\right)\right)\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(e^{x \cdot \left(-1 + \varepsilon\right)}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right) + \left(\frac{\frac{-1}{\varepsilon} + 1}{e^{(x \cdot \varepsilon + x)_*}}\right))_*}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 448.3054261514617

    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 1.4

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

    4. Applied add-cube-cbrt1.4

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

    5. Applied associate-*r*1.4

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

    6. Simplified1.4

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

    8. Applied log1p-expm1-u1.4

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

    if 448.3054261514617 < 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 prod-diff0.1

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

    4. Simplified0.1

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

    5. Simplified0.1

      \[\leadsto \frac{(\left(e^{x \cdot \left(-1 + \varepsilon\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(\frac{1 + \frac{-1}{\varepsilon}}{e^{(x \cdot \varepsilon + x)_*}}\right))_* + \color{blue}{0}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 448.3054261514617:\\ \;\;\;\;\frac{\left(2 + x \cdot \left(\frac{2}{3} \cdot \left(\sqrt[3]{{x}^{3}} \cdot \log_* (1 + (e^{\sqrt[3]{{x}^{3}}} - 1)^*)\right)\right)\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(e^{x \cdot \left(-1 + \varepsilon\right)}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right) + \left(\frac{\frac{-1}{\varepsilon} + 1}{e^{(x \cdot \varepsilon + x)_*}}\right))_*}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018235 +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))