Average Error: 29.5 → 1.1
Time: 2.7m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{(e^{\log_* (1 + \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x})} - 1)^* - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \le -3.5012930550927 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(2 + \left(\log_* (1 + (e^{x \cdot \frac{2}{3}} - 1)^*) \cdot x\right) \cdot \log_* (1 + (e^{x} - 1)^*)\right) - {x}^{2}}{2}\\ \mathbf{elif}\;\frac{(e^{\log_* (1 + \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x})} - 1)^* - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \le 1.0197246061422032:\\ \;\;\;\;\frac{(e^{\log_* (1 + \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x})} - 1)^* - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 + \left(\log_* (1 + (e^{x \cdot \frac{2}{3}} - 1)^*) \cdot x\right) \cdot \log_* (1 + (e^{x} - 1)^*)\right) - {x}^{2}}{2}\\ \end{array}\]

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 (/ (- (expm1 (log1p (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2) < -3.5012930550927e-310 or 1.0197246061422032 < (/ (- (expm1 (log1p (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2)

    1. Initial program 59.8

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

    2. Taylor expanded around 0 1.3

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

    4. Applied add-cube-cbrt1.3

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

    5. Applied associate-*r*1.3

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

    6. Applied simplify1.3

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

    8. Applied log1p-expm1-u1.3

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

    9. Applied simplify1.3

      \[\leadsto \frac{\left(2 + \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot \log_* (1 + \color{blue}{(e^{x} - 1)^*})\right) - {x}^{2}}{2}\]
    10. Using strategy rm

    11. Applied log1p-expm1-u1.3

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

    if -3.5012930550927e-310 < (/ (- (expm1 (log1p (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2) < 1.0197246061422032

    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 expm1-log1p-u0.9

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018208 +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))