Average Error: 29.3 → 1.0
Time: 4.2m
Precision: 64
Internal Precision: 4096
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 268.51895894372376:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2} \cdot {x}^{2}} \cdot \sqrt[3]{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) \cdot \left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2} \cdot {x}^{2}}}{\sqrt[3]{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) + {x}^{2}} \cdot \sqrt[3]{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) + {x}^{2}}} \cdot \sqrt[3]{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\log \left(e^{\left(1 - \varepsilon\right) \cdot x}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 268.51895894372376

    1. Initial program 38.9

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

    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-cbrt2.8

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

    6. Applied flip--2.8

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

    7. Applied cbrt-div2.3

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

    8. Applied flip--2.3

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

    9. Applied cbrt-div2.3

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

    10. Applied frac-times1.3

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

    if 268.51895894372376 < x

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' 
(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))