Average Error: 29.5 → 1.1
Time: 2.1m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 42.01964098399155:\\ \;\;\;\;\frac{\log \left(\sqrt[3]{(\left(e^{2} \cdot x\right) \cdot \left((x \cdot \left(x \cdot \frac{2}{3}\right) + \left(-x\right))_*\right) + \left(e^{2}\right))_*} \cdot \left(\sqrt[3]{(\left(e^{2} \cdot x\right) \cdot \left((x \cdot \left(x \cdot \frac{2}{3}\right) + \left(-x\right))_*\right) + \left(e^{2}\right))_*} \cdot \sqrt[3]{(\left(e^{2} \cdot x\right) \cdot \left((x \cdot \left(x \cdot \frac{2}{3}\right) + \left(-x\right))_*\right) + \left(e^{2}\right))_*}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}} \cdot \left(\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}}\right)}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 42.01964098399155

    1. Initial program 39.1

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

    2. 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 add-log-exp1.3

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

    5. Applied add-log-exp1.3

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

    6. Applied diff-log1.3

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

    7. Taylor expanded around 0 1.3

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

    8. Simplified1.3

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

    10. Applied add-cube-cbrt1.3

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

    if 42.01964098399155 < 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-cube-cbrt0.3

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 42.01964098399155:\\ \;\;\;\;\frac{\log \left(\sqrt[3]{(\left(e^{2} \cdot x\right) \cdot \left((x \cdot \left(x \cdot \frac{2}{3}\right) + \left(-x\right))_*\right) + \left(e^{2}\right))_*} \cdot \left(\sqrt[3]{(\left(e^{2} \cdot x\right) \cdot \left((x \cdot \left(x \cdot \frac{2}{3}\right) + \left(-x\right))_*\right) + \left(e^{2}\right))_*} \cdot \sqrt[3]{(\left(e^{2} \cdot x\right) \cdot \left((x \cdot \left(x \cdot \frac{2}{3}\right) + \left(-x\right))_*\right) + \left(e^{2}\right))_*}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}} \cdot \left(\sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}}\right)}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018221 +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))