Average Error: 29.1 → 1.1
Time: 5.8m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{(\left(e^{x}\right) \cdot \left(\frac{(\left(\frac{1}{\varepsilon}\right) \cdot \left(-1\right) + 1)_*}{e^{\varepsilon \cdot x}}\right) + \left((\left(e^{(\varepsilon \cdot x + x)_*}\right) \cdot \left(\frac{1}{\varepsilon}\right) + \left(e^{(\varepsilon \cdot x + x)_*}\right))_*\right))_*}{e^{(1 \cdot x + x)_*}}}{2} \le 1.0:\\ \;\;\;\;\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}} \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}}\right) \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (fma (exp x) (/ (fma (/ 1 eps) (- 1) 1) (exp (* eps x))) (fma (exp (fma eps x x)) (/ 1 eps) (exp (fma eps x x)))) (exp (fma (- 1 0) x x))) 2) < 1.0

    1. Initial program 39.2

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

    2. Taylor expanded around 0 0.3

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

    if 1.0 < (/ (/ (fma (exp x) (/ (fma (/ 1 eps) (- 1) 1) (exp (* eps x))) (fma (exp (fma eps x x)) (/ 1 eps) (exp (fma eps x x)))) (exp (fma (- 1 0) x x))) 2)

    1. Initial program 3.1

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

    3. Applied add-cube-cbrt3.2

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

    4. Applied simplify3.2

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

    5. Applied simplify3.2

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

  4. Applied simplify1.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{(\left(e^{x}\right) \cdot \left(\frac{(\left(\frac{1}{\varepsilon}\right) \cdot \left(-1\right) + 1)_*}{e^{\varepsilon \cdot x}}\right) + \left((\left(e^{(\varepsilon \cdot x + x)_*}\right) \cdot \left(\frac{1}{\varepsilon}\right) + \left(e^{(\varepsilon \cdot x + x)_*}\right))_*\right))_*}{e^{(1 \cdot x + x)_*}}}{2} \le 1.0:\\ \;\;\;\;\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}} \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}}\right) \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon \cdot x + x)_*}}}}{2}\\ \end{array}}\]

Runtime

Time bar (total: 5.8m)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' +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))