Average Error: 29.5 → 1.1
Time: 2.1m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 10.078479018093358:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left(\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*) + 2\right) - {x}^{2}\right) \cdot \left(\left(\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*) + 2\right) - {x}^{2}\right)\right) \cdot \left(\left(\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*) + 2\right) - {x}^{2}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot (e^{\log_* (1 + e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)})} - 1)^*}{2}\\ \end{array}\]

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 < 10.078479018093358

    1. Initial program 39.2

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

    2. 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 log1p-expm1-u1.3

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

    6. Applied add-cbrt-cube1.3

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

    if 10.078479018093358 < x

    1. Initial program 0.4

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

    3. Applied expm1-log1p-u0.4

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 10.078479018093358:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left(\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*) + 2\right) - {x}^{2}\right) \cdot \left(\left(\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*) + 2\right) - {x}^{2}\right)\right) \cdot \left(\left(\log_* (1 + (e^{\frac{2}{3} \cdot {x}^{3}} - 1)^*) + 2\right) - {x}^{2}\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot (e^{\log_* (1 + e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)})} - 1)^*}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018234 +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))