Average Error: 29.6 → 1.0
Time: 9.2m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 416.4778212389762:\\ \;\;\;\;\left(\sqrt[3]{\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}} \cdot \sqrt[3]{\left(\sqrt[3]{\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}} \cdot \sqrt[3]{\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}}\right) \cdot \sqrt[3]{\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}} \cdot \sqrt[3]{\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}}\right) \cdot \sqrt[3]{\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}}\right)\right) \cdot \sqrt[3]{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 416.4778212389762

    1. Initial program 39.4

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

    2. Taylor expanded around 0 1.3

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

    4. Applied add-cube-cbrt1.3

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

    6. Applied add-cube-cbrt1.3

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

    8. Applied add-cube-cbrt1.3

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

    if 416.4778212389762 < x

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

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

    4. Applied associate-*r*0.0

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

Runtime

Time bar (total: 9.2m)Debug log

herbie shell --seed '#(991339738 1419949195 2842012120 4157638069 1320221275 2092628673)' 
(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))