Average Error: 29.4 → 0.8
Time: 1.9m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 647.6536497016051:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt[3]{\left(\left(\frac{2}{3} \cdot {x}^{3}\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right)\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right)}} \cdot \left(\sqrt[3]{\sqrt[3]{\left(\left(\frac{2}{3} \cdot {x}^{3}\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right)\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right)}} \cdot \sqrt[3]{\sqrt[3]{\left(\left(\frac{2}{3} \cdot {x}^{3}\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right)\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right)}}\right) + 2\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)} \cdot \sqrt{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)} - e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 647.6536497016051

    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 1.1

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

    4. Applied add-cbrt-cube1.1

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

    6. Applied add-cube-cbrt1.1

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

    if 647.6536497016051 < x

    1. Initial program 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

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 647.6536497016051:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt[3]{\left(\left(\frac{2}{3} \cdot {x}^{3}\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right)\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right)}} \cdot \left(\sqrt[3]{\sqrt[3]{\left(\left(\frac{2}{3} \cdot {x}^{3}\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right)\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right)}} \cdot \sqrt[3]{\sqrt[3]{\left(\left(\frac{2}{3} \cdot {x}^{3}\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right)\right) \cdot \left(\frac{2}{3} \cdot {x}^{3}\right)}}\right) + 2\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)} \cdot \sqrt{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)} - e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018219 +o rules:numerics
(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))