Average Error: 29.3 → 0.9
Time: 6.0m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\sqrt[3]{{\left(2 + \frac{2}{3} \cdot {x}^{3}\right)}^{3} - {\left({x}^{2}\right)}^{3}} \cdot \sqrt[3]{{\left(2 + \frac{2}{3} \cdot {x}^{3}\right)}^{3} - {\left({x}^{2}\right)}^{3}}}{\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) + \left({x}^{2} \cdot {x}^{2} + \left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot {x}^{2}\right)}{\sqrt[3]{{\left(2 + \frac{2}{3} \cdot {x}^{3}\right)}^{3} - {\left({x}^{2}\right)}^{3}}}}}{2} \le 1.0198730801337745:\\ \;\;\;\;\frac{\frac{\sqrt[3]{{\left(2 + \frac{2}{3} \cdot {x}^{3}\right)}^{3} - {\left({x}^{2}\right)}^{3}} \cdot \sqrt[3]{{\left(2 + \frac{2}{3} \cdot {x}^{3}\right)}^{3} - {\left({x}^{2}\right)}^{3}}}{\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) + \left({x}^{2} \cdot {x}^{2} + \left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot {x}^{2}\right)}{\sqrt[3]{{\left(2 + \frac{2}{3} \cdot {x}^{3}\right)}^{3} - {\left({x}^{2}\right)}^{3}}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(\sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}{2}\\ \end{array}\]

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 (- (pow (+ 2 (* 2/3 (pow x 3))) 3) (pow (pow x 2) 3))) (cbrt (- (pow (+ 2 (* 2/3 (pow x 3))) 3) (pow (pow x 2) 3)))) (/ (+ (* (+ 2 (* 2/3 (pow x 3))) (+ 2 (* 2/3 (pow x 3)))) (+ (* (pow x 2) (pow x 2)) (* (+ 2 (* 2/3 (pow x 3))) (pow x 2)))) (cbrt (- (pow (+ 2 (* 2/3 (pow x 3))) 3) (pow (pow x 2) 3))))) 2) < 1.0198730801337745

    1. Initial program 39.1

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

    2. Taylor expanded around 0 1.1

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

    4. Applied flip3--1.1

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

    6. Applied add-cube-cbrt1.1

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

    7. Applied associate-/l*1.1

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

    if 1.0198730801337745 < (/ (/ (* (cbrt (- (pow (+ 2 (* 2/3 (pow x 3))) 3) (pow (pow x 2) 3))) (cbrt (- (pow (+ 2 (* 2/3 (pow x 3))) 3) (pow (pow x 2) 3)))) (/ (+ (* (+ 2 (* 2/3 (pow x 3))) (+ 2 (* 2/3 (pow x 3)))) (+ (* (pow x 2) (pow x 2)) (* (+ 2 (* 2/3 (pow x 3))) (pow x 2)))) (cbrt (- (pow (+ 2 (* 2/3 (pow x 3))) 3) (pow (pow x 2) 3))))) 2)

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt0.3

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

Runtime

Time bar (total: 6.0m)Debug logProfile

herbie shell --seed 2018178 
(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))