Average Error: 29.4 → 1.0
Time: 3.6m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\sqrt[3]{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}} \cdot \sqrt[3]{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}\right) \cdot \sqrt[3]{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}}{2} \le 3173368680.1185236:\\ \;\;\;\;\frac{\sqrt[3]{8 + \left(x \cdot x\right) \cdot \left(8 \cdot x - 12\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}} \cdot \left(\frac{1}{\varepsilon} + 1\right)\right) \cdot \sqrt{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

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 (/ (* (* (cbrt (- (+ (* 2/3 (pow x 3)) 2) (pow x 2))) (cbrt (- (+ (* 2/3 (pow x 3)) 2) (pow x 2)))) (cbrt (- (+ (* 2/3 (pow x 3)) 2) (pow x 2)))) 2) < 3173368680.1185236

    1. Initial program 39.0

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

    2. Taylor expanded around 0 1.3

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

    4. Applied add-cbrt-cube1.3

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

    5. Applied simplify1.3

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

    6. Taylor expanded around 0 1.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(8 \cdot {x}^{3} + 8\right) - 12 \cdot {x}^{2}}}}{2}\]

    7. Applied simplify1.3

      \[\leadsto \color{blue}{\frac{\sqrt[3]{8 + \left(x \cdot x\right) \cdot \left(8 \cdot x - 12\right)}}{2}}\]

    if 3173368680.1185236 < (/ (* (* (cbrt (- (+ (* 2/3 (pow x 3)) 2) (pow x 2))) (cbrt (- (+ (* 2/3 (pow x 3)) 2) (pow x 2)))) (cbrt (- (+ (* 2/3 (pow x 3)) 2) (pow x 2)))) 2)

    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

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

    4. Applied associate-*r*0.0

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

    5. Applied simplify0.0

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

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed 2018296 
(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))