Average Error: 29.5 → 1.1
Time: 7.0m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{8 + \left(8 \cdot {x}^{3} + \frac{5}{3} \cdot {x}^{6}\right)}{(\left((\left({x}^{3}\right) \cdot \frac{2}{3} + 2)_*\right) \cdot \left((\left((x \cdot \frac{2}{3} + 1)_*\right) \cdot \left(x \cdot x\right) + 2)_*\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right))_*}}{2} \le 1.856984050831293:\\ \;\;\;\;\frac{\left(\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) - (\frac{1}{4} \cdot \left({x}^{6}\right) + \left(x \cdot x\right))_*\right) + (\frac{1}{6} \cdot \left({x}^{9}\right) + \left((\frac{1}{8} \cdot \left({x}^{8}\right) + 2)_*\right))_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}} \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}}\right) \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (+ 8 (+ (* 8 (pow x 3)) (* 5/3 (pow x 6)))) (fma (fma (pow x 3) 2/3 2) (fma (fma x 2/3 1) (* x x) 2) (* (* x x) (* x x)))) 2) < 1.856984050831293

    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 1.2

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

    4. Applied flip3--1.2

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

    5. Applied simplify1.2

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

    6. Taylor expanded around 0 1.2

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{8} \cdot {x}^{8} + \left(\frac{1}{6} \cdot {x}^{9} + \left(\frac{2}{3} \cdot {x}^{3} + 2\right)\right)\right) - \left({x}^{2} + \frac{1}{4} \cdot {x}^{6}\right)}}{2}\]

    7. Applied simplify1.2

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) - (\frac{1}{4} \cdot \left({x}^{6}\right) + \left(x \cdot x\right))_*\right) + (\frac{1}{6} \cdot \left({x}^{9}\right) + \left((\frac{1}{8} \cdot \left({x}^{8}\right) + 2)_*\right))_*}{2}}\]

    if 1.856984050831293 < (/ (/ (+ 8 (+ (* 8 (pow x 3)) (* 5/3 (pow x 6)))) (fma (fma (pow x 3) 2/3 2) (fma (fma x 2/3 1) (* x x) 2) (* (* x x) (* x x)))) 2)

    1. Initial program 0.7

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

    3. Applied add-cube-cbrt0.7

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

    4. Applied simplify0.7

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

    5. Applied simplify0.7

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

Runtime

Time bar (total: 7.0m)Debug logProfile

herbie shell --seed 2018199 +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))