Average Error: 29.7 → 0.8
Time: 2.9m
Precision: 64
Internal Precision: 128
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) \le -5.894359694351766 \cdot 10^{+17} \lor \neg \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) \le 0.9999889407307174\right):\\ \;\;\;\;\frac{\frac{\sqrt{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}} \cdot \sqrt{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}}}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) + {x}^{2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(\sqrt[3]{e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}} \cdot \sqrt[3]{e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}}\right) \cdot \sqrt[3]{e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}}\right)}{2}\\ \end{array}\]

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 (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) < -5.894359694351766e+17 or 0.9999889407307174 < (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x))))

    1. Initial program 40.3

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

    2. Taylor expanded around 0 0.9

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

    4. Applied flip--0.9

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

    6. Applied add-sqr-sqrt0.9

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

    if -5.894359694351766e+17 < (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) < 0.9999889407307174

    1. Initial program 0.4

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

    3. Applied add-cube-cbrt0.5

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) \le -5.894359694351766 \cdot 10^{+17} \lor \neg \left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) \le 0.9999889407307174\right):\\ \;\;\;\;\frac{\frac{\sqrt{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}} \cdot \sqrt{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) \cdot \left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2} \cdot {x}^{2}}}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) + {x}^{2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(\sqrt[3]{e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}} \cdot \sqrt[3]{e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}}\right) \cdot \sqrt[3]{e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)}}\right)}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018277 +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))