Average Error: 23.8 → 1.3
Time: 6.1s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 307.626020480098248:\\ \;\;\;\;\frac{\left(e^{\left(\sqrt[3]{\log \left(\left(0.66666666666666674 \cdot x\right) \cdot x\right)} \cdot \sqrt[3]{\log \left(\left(0.66666666666666674 \cdot x\right) \cdot x\right)}\right) \cdot \sqrt[3]{\log \left(\left(0.66666666666666674 \cdot x\right) \cdot x\right)}} \cdot {\left(\sqrt[3]{x}\right)}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 307.626020480098248

    1. Initial program 31.2

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

    2. Taylor expanded around 0 1.6

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

    4. Applied add-cube-cbrt1.6

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

    5. Applied unpow-prod-down1.6

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

    6. Applied associate-*r*1.6

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

    7. Simplified1.6

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

    9. Applied add-exp-log34.9

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

    10. Applied add-exp-log34.9

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

    11. Applied add-exp-log34.9

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

    12. Applied prod-exp34.9

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

    13. Applied prod-exp34.9

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

    14. Simplified1.6

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

    16. Applied add-cube-cbrt1.6

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

    if 307.626020480098248 < 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-sqr-sqrt0.0

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

    4. Applied associate-*r*0.0

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 307.626020480098248:\\ \;\;\;\;\frac{\left(e^{\left(\sqrt[3]{\log \left(\left(0.66666666666666674 \cdot x\right) \cdot x\right)} \cdot \sqrt[3]{\log \left(\left(0.66666666666666674 \cdot x\right) \cdot x\right)}\right) \cdot \sqrt[3]{\log \left(\left(0.66666666666666674 \cdot x\right) \cdot x\right)}} \cdot {\left(\sqrt[3]{x}\right)}^{3} + 2\right) - 1 \cdot {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020157 
(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))