Average Error: 28.7 → 1.1
Time: 2.9m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{(e^{\log_* (1 + \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x})} - 1)^* - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \le -3.62733960507604 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(2 + \left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) - {x}^{2}}{2}\\ \mathbf{if}\;\frac{(e^{\log_* (1 + \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x})} - 1)^* - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \le 1.0202275613144554:\\ \;\;\;\;\frac{(e^{\log_* (1 + \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x})} - 1)^* - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 + \left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) - {x}^{2}}{2}\\ \end{array}\]

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) < -3.62733960507604e-310 or 1.0202275613144554 < (/ (- (expm1 (log1p (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2)

    1. Initial program 59.4

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

    2. Taylor expanded around 0 1.5

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

    4. Applied add-cube-cbrt1.5

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

    if -3.62733960507604e-310 < (/ (- (expm1 (log1p (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2) < 1.0202275613144554

    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-u0.7

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

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018167 +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))