Average Error: 29.6 → 1.1
Time: 2.4m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\log \left(\sqrt[3]{\left(1 - x \cdot x\right) + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \sqrt[3]{\left(1 - x \cdot x\right) + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)}\right) + \log \left(\sqrt[3]{\left(1 - x \cdot x\right) + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)}\right)\right) + 2}{2} \le 1.0:\\ \;\;\;\;\frac{\log \left(\frac{e^{2 + \frac{2}{3} \cdot {x}^{3}}}{e^{{x}^{2}}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\sqrt{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}} \cdot \sqrt{{\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (+ (log (* (cbrt (+ (- 1 (* x x)) (* (* 2/3 x) (* x x)))) (cbrt (+ (- 1 (* x x)) (* (* 2/3 x) (* x x)))))) (log (cbrt (+ (- 1 (* x x)) (* (* 2/3 x) (* x x)))))) 2) 2) < 1.0

    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 1.0

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

    4. Applied add-log-exp1.0

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

    5. Applied add-log-exp1.0

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

    6. Applied diff-log1.0

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

    if 1.0 < (/ (+ (+ (log (* (cbrt (+ (- 1 (* x x)) (* (* 2/3 x) (* x x)))) (cbrt (+ (- 1 (* x x)) (* (* 2/3 x) (* x x)))))) (log (cbrt (+ (- 1 (* x x)) (* (* 2/3 x) (* x x)))))) 2) 2)

    1. Initial program 1.3

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

    3. Applied add-sqr-sqrt1.3

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

    4. Applied simplify1.3

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

    5. Applied simplify1.3

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' 
(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))