Average Error: 29.3 → 1.0
Time: 5.8s
Precision: binary64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 289.9965618814249:\\ \;\;\;\;\frac{{x}^{3} \cdot 0.66666666666666674 + \left(2 - 1 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt[3]{{\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)}} \cdot \sqrt[3]{{\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)}}\right)\right) \cdot \sqrt[3]{e^{x \cdot \left(-\left(1 - \varepsilon\right)\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-\left(1 + \varepsilon\right)\right)}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 289.9965618814249

    1. Initial program 38.9

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

    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. Simplified1.3

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

    if 289.9965618814249 < x

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied associate-*r*0.1

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

    5. Simplified0.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 289.9965618814249:\\ \;\;\;\;\frac{{x}^{3} \cdot 0.66666666666666674 + \left(2 - 1 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt[3]{{\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)}} \cdot \sqrt[3]{{\left(e^{1 - \varepsilon}\right)}^{\left(-x\right)}}\right)\right) \cdot \sqrt[3]{e^{x \cdot \left(-\left(1 - \varepsilon\right)\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{x \cdot \left(-\left(1 + \varepsilon\right)\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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))