Average Error: 29.8 → 1.0
Time: 1.9m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 461.14583916801655:\\ \;\;\;\;\frac{\left(2 + (e^{\log_* (1 + \frac{2}{3} \cdot {x}^{3})} - 1)^*\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\log_* (1 + \left(\frac{1 + \frac{1}{\varepsilon}}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}} - \frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}\right))} - 1)^*}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 461.14583916801655

    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 1.2

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

    4. Applied expm1-log1p-u1.3

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

    if 461.14583916801655 < 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 expm1-log1p-u0.1

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

    4. Applied simplify0.1

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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))