NMSE Section 6.1 mentioned, A

Average Error: 29.2 → 1.1

Time: 14.8min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 97.0065926893151271:\\ \;\;\;\;\frac{\left(1 \cdot \left(x + x \cdot \varepsilon\right) + 1\right) + \frac{1}{\left(\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\frac{1}{\varepsilon} + 1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}} \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} + 1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}\right) \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} + 1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if x < 97.0065926893151271

   1. Initial program 38.8

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

   2. Simplified 38.8

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

   4. Applied div-sub 38.8

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

   5. Applied associate--r- 32.6

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

   6. Taylor expanded around 0 1.3

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

   7. Simplified 1.3

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

   9. Applied add-cube-cbrt 1.3

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

   if 97.0065926893151271 < x

   1. Initial program 0.2

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

   2. Simplified 0.2

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

   4. Applied add-cube-cbrt 0.2

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 97.0065926893151271:\\ \;\;\;\;\frac{\left(1 \cdot \left(x + x \cdot \varepsilon\right) + 1\right) + \frac{1}{\left(\sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\frac{1}{\varepsilon} + 1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}} \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} + 1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}\right) \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} + 1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (neg (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (neg (* (+ 1.0 eps) x))))) 2.0))