Average Error: 29.5 → 1.1
Time: 3.0m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{(\left(\frac{1}{\varepsilon}\right) \cdot \left(e^{(\varepsilon \cdot x + x)_*}\right) + \left((\left(1 - \frac{1}{\varepsilon}\right) \cdot \left({\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}\right) + \left(e^{(\varepsilon \cdot x + x)_*}\right))_*\right))_*}{e^{(1 \cdot x + x)_*}}}{2} \le 0.4922027587890625:\\ \;\;\;\;\frac{(\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right))_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}\right)}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (fma (/ 1 eps) (exp (fma eps x x)) (fma (- 1 (/ 1 eps)) (pow (exp x) (- 1 eps)) (exp (fma eps x x)))) (exp (fma (- 1 0) x x))) 2) < 0.4922027587890625

    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 0.6

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

    4. Applied add-cube-cbrt0.6

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

    5. Applied add-cube-cbrt2.1

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

    6. Applied prod-diff2.1

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

    7. Applied simplify0.6

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

    8. Applied simplify0.6

      \[\leadsto \frac{(\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right))_* + \color{blue}{0}}{2}\]

    if 0.4922027587890625 < (/ (/ (fma (/ 1 eps) (exp (fma eps x x)) (fma (- 1 (/ 1 eps)) (pow (exp x) (- 1 eps)) (exp (fma eps x x)))) (exp (fma (- 1 0) x x))) 2)

    1. Initial program 1.5

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

    3. Applied add-exp-log1.5

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

  4. Applied simplify1.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{(\left(\frac{1}{\varepsilon}\right) \cdot \left(e^{(\varepsilon \cdot x + x)_*}\right) + \left((\left(1 - \frac{1}{\varepsilon}\right) \cdot \left({\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}\right) + \left(e^{(\varepsilon \cdot x + x)_*}\right))_*\right))_*}{e^{(1 \cdot x + x)_*}}}{2} \le 0.4922027587890625:\\ \;\;\;\;\frac{(\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right))_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}\right)}}{2}\\ \end{array}}\]

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed 2018198 +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))