Average Error: 29.9 → 0.4
Time: 2.5m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + 1\right)} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{{\left(e^{x}\right)}^{\left(1 + 1\right)}}}{2} \le 0.24962851659980406:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot e^{\sqrt[3]{{\left(\log \left(\sqrt[3]{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right)} \cdot \sqrt[3]{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right)}\right)\right)}^{3}}} + 2\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\left(\sqrt[3]{e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}} \cdot \sqrt[3]{e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}}\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right) \cdot \sqrt[3]{e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (- (* (+ 1 (/ 1 eps)) (exp (* (+ 1 eps) x))) (* (exp (* (- 1 eps) x)) (- (/ 1 eps) 1))) (pow (exp x) (+ (+ 1 1) 0))) 2) < 0.24962851659980406

    1. Initial program 62.0

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

    2. Taylor expanded around 0 0.5

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

    4. Applied add-cube-cbrt0.5

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

    6. Applied add-exp-log0.5

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

    8. Applied add-cbrt-cube0.5

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

    9. Applied simplify0.5

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

    if 0.24962851659980406 < (/ (/ (- (* (+ 1 (/ 1 eps)) (exp (* (+ 1 eps) x))) (* (exp (* (- 1 eps) x)) (- (/ 1 eps) 1))) (pow (exp x) (+ (+ 1 1) 0))) 2)

    1. Initial program 0.2

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

    3. Applied add-cube-cbrt0.2

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

    4. Applied associate-*r*0.2

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

  4. Applied simplify0.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + 1\right)} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{{\left(e^{x}\right)}^{\left(1 + 1\right)}}}{2} \le 0.24962851659980406:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot e^{\sqrt[3]{{\left(\log \left(\sqrt[3]{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right)} \cdot \sqrt[3]{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right)}\right)\right)}^{3}}} + 2\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\left(\sqrt[3]{e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}} \cdot \sqrt[3]{e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}}\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right) \cdot \sqrt[3]{e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}}}{2}\\ \end{array}}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' 
(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))