Average Error: 33.5 → 0.8
Time: 7.6s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.302816513994034:\\ \;\;\;\;\left(\sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \log \left(e^{0.5 \cdot {x}^{2}}\right)} \cdot \sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \log \left(e^{0.5 \cdot {x}^{2}}\right)}\right) \cdot \sqrt[3]{\left(\sqrt{0.33333333333333337 \cdot {x}^{3} + 1} + \sqrt{\log \left(e^{0.5 \cdot {x}^{2}}\right)}\right) \cdot \left(\sqrt{0.33333333333333337 \cdot {x}^{3} + 1} - \sqrt{\log \left(e^{0.5 \cdot {x}^{2}}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 1.302816513994034

    1. Initial program 47.0

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

    2. Simplified47.0

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

    3. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - 0.5 \cdot {x}^{2}}\]
    4. Using strategy rm

    5. Applied add-log-exp0.9

      \[\leadsto \left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \color{blue}{\log \left(e^{0.5 \cdot {x}^{2}}\right)}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt0.9

      \[\leadsto \color{blue}{\left(\sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \log \left(e^{0.5 \cdot {x}^{2}}\right)} \cdot \sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \log \left(e^{0.5 \cdot {x}^{2}}\right)}\right) \cdot \sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \log \left(e^{0.5 \cdot {x}^{2}}\right)}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt0.9

      \[\leadsto \left(\sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \log \left(e^{0.5 \cdot {x}^{2}}\right)} \cdot \sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \log \left(e^{0.5 \cdot {x}^{2}}\right)}\right) \cdot \sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \color{blue}{\sqrt{\log \left(e^{0.5 \cdot {x}^{2}}\right)} \cdot \sqrt{\log \left(e^{0.5 \cdot {x}^{2}}\right)}}}\]

    10. Applied add-sqr-sqrt0.9

      \[\leadsto \left(\sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \log \left(e^{0.5 \cdot {x}^{2}}\right)} \cdot \sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \log \left(e^{0.5 \cdot {x}^{2}}\right)}\right) \cdot \sqrt[3]{\color{blue}{\sqrt{0.33333333333333337 \cdot {x}^{3} + 1} \cdot \sqrt{0.33333333333333337 \cdot {x}^{3} + 1}} - \sqrt{\log \left(e^{0.5 \cdot {x}^{2}}\right)} \cdot \sqrt{\log \left(e^{0.5 \cdot {x}^{2}}\right)}}\]

    11. Applied difference-of-squares0.9

      \[\leadsto \left(\sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \log \left(e^{0.5 \cdot {x}^{2}}\right)} \cdot \sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \log \left(e^{0.5 \cdot {x}^{2}}\right)}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt{0.33333333333333337 \cdot {x}^{3} + 1} + \sqrt{\log \left(e^{0.5 \cdot {x}^{2}}\right)}\right) \cdot \left(\sqrt{0.33333333333333337 \cdot {x}^{3} + 1} - \sqrt{\log \left(e^{0.5 \cdot {x}^{2}}\right)}\right)}}\]

    if 1.302816513994034 < x

    1. Initial program 0.4

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

    2. Simplified0.4

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

    4. Applied div-sub0.4

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

    5. Applied div-sub0.4

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

    6. Applied associate--r-0.4

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.302816513994034:\\ \;\;\;\;\left(\sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \log \left(e^{0.5 \cdot {x}^{2}}\right)} \cdot \sqrt[3]{\left(0.33333333333333337 \cdot {x}^{3} + 1\right) - \log \left(e^{0.5 \cdot {x}^{2}}\right)}\right) \cdot \sqrt[3]{\left(\sqrt{0.33333333333333337 \cdot {x}^{3} + 1} + \sqrt{\log \left(e^{0.5 \cdot {x}^{2}}\right)}\right) \cdot \left(\sqrt{0.33333333333333337 \cdot {x}^{3} + 1} - \sqrt{\log \left(e^{0.5 \cdot {x}^{2}}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}}{2} - \frac{\frac{\frac{1}{\varepsilon}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\right) + \frac{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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))