Average Error: 29.7 → 0.9
Time: 1.8m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 264.18730392737905:\\ \;\;\;\;\frac{\log \left(e^{2} \cdot \left(\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 - x \cdot x\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \sqrt{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) \cdot \sqrt{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 264.18730392737905

    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 1.2

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

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

    8. Applied simplify1.2

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

    if 264.18730392737905 < 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 add-sqr-sqrt0.1

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

    4. Applied associate-*r*0.1

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(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))