Average Error: 29.8 → 1.2
Time: 1.9m
Precision: 64
Internal Precision: 1408
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\frac{\frac{\frac{2}{3}}{{x}^{3}} \cdot \left(\frac{\frac{27}{4}}{x} - \frac{9}{2}\right)}{\frac{\frac{x}{\frac{2}{3}} \cdot {x}^{3}}{\frac{\frac{2}{3}}{{x}^{3}}}} + {\left(\frac{\frac{2}{3}}{{x}^{3}}\right)}^{3}}}{2} \le -2.0703859980849916 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}\\ \mathbf{if}\;\frac{\sqrt[3]{\frac{\frac{\frac{2}{3}}{{x}^{3}} \cdot \left(\frac{\frac{27}{4}}{x} - \frac{9}{2}\right)}{\frac{\frac{x}{\frac{2}{3}} \cdot {x}^{3}}{\frac{\frac{2}{3}}{{x}^{3}}}} + {\left(\frac{\frac{2}{3}}{{x}^{3}}\right)}^{3}}}{2} \le 5.842364092482766 \cdot 10^{-05}:\\ \;\;\;\;\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \sqrt{e^{-\left(1 - \varepsilon\right) \cdot x}}\right) \cdot \sqrt{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (/ (cbrt (+ (/ (* (/ 2/3 (pow x 3)) (- (/ 27/4 x) 9/2)) (/ (* (/ x 2/3) (pow x 3)) (/ 2/3 (pow x 3)))) (pow (/ 2/3 (pow x 3)) 3))) 2) < -2.0703859980849916e+22 or 5.842364092482766e-05 < (/ (cbrt (+ (/ (* (/ 2/3 (pow x 3)) (- (/ 27/4 x) 9/2)) (/ (* (/ x 2/3) (pow x 3)) (/ 2/3 (pow x 3)))) (pow (/ 2/3 (pow x 3)) 3))) 2)

    1. Initial program 39.1

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

    2. Taylor expanded around 0 0.9

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

    if -2.0703859980849916e+22 < (/ (cbrt (+ (/ (* (/ 2/3 (pow x 3)) (- (/ 27/4 x) 9/2)) (/ (* (/ x 2/3) (pow x 3)) (/ 2/3 (pow x 3)))) (pow (/ 2/3 (pow x 3)) 3))) 2) < 5.842364092482766e-05

    1. Initial program 2.4

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

    3. Applied add-sqr-sqrt2.4

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

    4. Applied associate-*r*2.4

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' 
(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))