Average Error: 30.0 → 1.1
Time: 1.5m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 395.5913263784546:\\ \;\;\;\;\frac{\left(2 + (e^{\log_* (1 + \left(x \cdot \frac{2}{3}\right) \cdot x)} - 1)^* \cdot \sqrt[3]{{x}^{3}}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(e^{-\left(1 - \varepsilon\right) \cdot x}\right) + \left(-\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}\right))_*}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 395.5913263784546

    1. Initial program 39.6

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

    2. Taylor expanded around 0 1.4

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

    4. Applied add-cube-cbrt1.4

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

    5. Applied associate-*r*1.4

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

    6. Applied simplify1.4

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

    8. Applied expm1-log1p-u1.4

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

    if 395.5913263784546 < x

    1. Initial program 0.2

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

    3. Applied fma-neg0.2

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

    4. Applied simplify0.2

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +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))