Average Error: 29.8 → 1.0
Time: 6.4m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 285.18028128811636:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(\frac{2}{3} \cdot x\right) - \left(x \cdot x - 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}} \cdot \sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{\frac{2}{\sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}} \cdot \left(\frac{1}{\varepsilon} + 1\right)}} - \frac{\frac{1}{\varepsilon} - 1}{2} \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 285.18028128811636

    1. Initial program 39.4

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

    2. Taylor expanded around 0 1.4

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

    4. Applied add-cbrt-cube1.4

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

    5. Applied simplify1.4

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

    if 285.18028128811636 < 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-cube-cbrt0.0

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

    4. Applied associate-*r*0.0

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

  5. Applied simplify1.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le 285.18028128811636:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left(\frac{2}{3} \cdot x\right) - \left(x \cdot x - 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}} \cdot \sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}}}{\frac{2}{\sqrt[3]{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)}} \cdot \left(\frac{1}{\varepsilon} + 1\right)}} - \frac{\frac{1}{\varepsilon} - 1}{2} \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}\\ \end{array}}\]

Runtime

Time bar (total: 6.4m)Debug log

herbie shell --seed '#(3622638036 3041702260 3649696288 21285302 1742518495 296600799)' 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))