Average Error: 29.0 → 0.9
Time: 9.4m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 9.08124565108286:\\ \;\;\;\;\frac{\left(\left(-x\right) \cdot x + 2\right) + \log \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(\sqrt[3]{(\left({x}^{6}\right) \cdot \frac{2}{9} + 1)_*} \cdot \sqrt[3]{(\left({x}^{6}\right) \cdot \frac{2}{9} + 1)_*}\right) \cdot \sqrt[3]{(\left({x}^{6}\right) \cdot \frac{2}{9} + 1)_*}\right))_*\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 9.08124565108286

    1. Initial program 38.3

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

    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 add-log-exp1.2

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

    5. Applied add-log-exp1.2

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

    6. Applied diff-log1.2

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

    7. Taylor expanded around 0 1.2

      \[\leadsto \frac{\log \left(\frac{\color{blue}{e^{2} + \left(\frac{2}{9} \cdot \left(e^{2} \cdot {x}^{6}\right) + \frac{2}{3} \cdot \left(e^{2} \cdot {x}^{3}\right)\right)}}{e^{{x}^{2}}}\right)}{2}\]

    8. Applied simplify1.2

      \[\leadsto \color{blue}{\frac{\left(\left(-x\right) \cdot x + 2\right) + \log \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left((\left({x}^{6}\right) \cdot \frac{2}{9} + 1)_*\right))_*\right)}{2}}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt1.2

      \[\leadsto \frac{\left(\left(-x\right) \cdot x + 2\right) + \log \left((\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \color{blue}{\left(\left(\sqrt[3]{(\left({x}^{6}\right) \cdot \frac{2}{9} + 1)_*} \cdot \sqrt[3]{(\left({x}^{6}\right) \cdot \frac{2}{9} + 1)_*}\right) \cdot \sqrt[3]{(\left({x}^{6}\right) \cdot \frac{2}{9} + 1)_*}\right)})_*\right)}{2}\]

    if 9.08124565108286 < x

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt0.3

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

Runtime

Time bar (total: 9.4m)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' +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))