Average Error: 28.7 → 1.1
Time: 6.7s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.67741126011574404:\\ \;\;\;\;\frac{\left(\left(\left(0.66666666666666674 \cdot x\right) \cdot x\right) \cdot {\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{9} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\left(\frac{e^{x \cdot \varepsilon - 1 \cdot x}}{\varepsilon} + e^{x \cdot \varepsilon - 1 \cdot x}\right) - \frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon}\right) + 1 \cdot e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 1.67741126011574404

    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. Taylor expanded around 0 1.3

      \[\leadsto \frac{\color{blue}{\left(0.66666666666666674 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt1.3

      \[\leadsto \frac{\left(0.66666666666666674 \cdot {\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\]

    5. Applied unpow-prod-down1.3

      \[\leadsto \frac{\left(0.66666666666666674 \cdot \color{blue}{\left({\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3} \cdot {\left(\sqrt[3]{x}\right)}^{3}\right)} + 2\right) - 1 \cdot {x}^{2}}{2}\]

    6. Applied associate-*r*1.3

      \[\leadsto \frac{\left(\color{blue}{\left(0.66666666666666674 \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{3}\right) \cdot {\left(\sqrt[3]{x}\right)}^{3}} + 2\right) - 1 \cdot {x}^{2}}{2}\]

    7. Simplified1.3

      \[\leadsto \frac{\left(\color{blue}{\left(\left(0.66666666666666674 \cdot x\right) \cdot x\right)} \cdot {\left(\sqrt[3]{x}\right)}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt1.3

      \[\leadsto \frac{\left(\left(\left(0.66666666666666674 \cdot x\right) \cdot x\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\right)}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\]

    10. Applied cbrt-prod1.3

      \[\leadsto \frac{\left(\left(\left(0.66666666666666674 \cdot x\right) \cdot x\right) \cdot {\color{blue}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\]
    11. Using strategy rm

    12. Applied cbrt-prod1.3

      \[\leadsto \frac{\left(\left(\left(0.66666666666666674 \cdot x\right) \cdot x\right) \cdot {\left(\color{blue}{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\]

    13. Applied pow31.3

      \[\leadsto \frac{\left(\left(\left(0.66666666666666674 \cdot x\right) \cdot x\right) \cdot {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{3}\right)}}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\]

    14. Applied pow-pow1.3

      \[\leadsto \frac{\left(\left(\left(0.66666666666666674 \cdot x\right) \cdot x\right) \cdot \color{blue}{{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{\left(3 \cdot 3\right)}} + 2\right) - 1 \cdot {x}^{2}}{2}\]

    15. Simplified1.3

      \[\leadsto \frac{\left(\left(\left(0.66666666666666674 \cdot x\right) \cdot x\right) \cdot {\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{\color{blue}{9}} + 2\right) - 1 \cdot {x}^{2}}{2}\]

    if 1.67741126011574404 < x

    1. Initial program 0.7

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

    2. Taylor expanded around inf 0.7

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

    3. Simplified0.6

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.67741126011574404:\\ \;\;\;\;\frac{\left(\left(\left(0.66666666666666674 \cdot x\right) \cdot x\right) \cdot {\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{9} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\left(\frac{e^{x \cdot \varepsilon - 1 \cdot x}}{\varepsilon} + e^{x \cdot \varepsilon - 1 \cdot x}\right) - \frac{e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{\varepsilon}\right) + 1 \cdot e^{-\left(x \cdot \varepsilon + 1 \cdot x\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(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))