Average Error: 29.4 → 1.0
Time: 3.1m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\frac{e^{x \cdot \varepsilon}}{\varepsilon \cdot e^{x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{(x \cdot \varepsilon + x)_*}}\right) + e^{x \cdot \varepsilon - x}}{2} \le 8.290358806514041 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(\frac{e^{x \cdot \varepsilon}}{\varepsilon \cdot e^{x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{(x \cdot \varepsilon + x)_*}}\right) + e^{x \cdot \varepsilon - x}}{2}\\ \mathbf{if}\;\frac{\left(\frac{e^{x \cdot \varepsilon}}{\varepsilon \cdot e^{x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{(x \cdot \varepsilon + x)_*}}\right) + e^{x \cdot \varepsilon - x}}{2} \le 1.0000000001745346:\\ \;\;\;\;\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\log_* (1 + \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x})} - 1)^* - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ (- (/ (exp (* x eps)) (* eps (exp x))) (/ (- (/ 1 eps) 1) (exp (fma x eps x)))) (exp (- (* x eps) x))) 2) < 8.290358806514041e-25

    1. Initial program 2.1

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

    2. Taylor expanded around inf 2.1

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

    3. Applied simplify2.1

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

    if 8.290358806514041e-25 < (/ (+ (- (/ (exp (* x eps)) (* eps (exp x))) (/ (- (/ 1 eps) 1) (exp (fma x eps x)))) (exp (- (* x eps) x))) 2) < 1.0000000001745346

    1. Initial program 39.5

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

    2. Taylor expanded around 0 0.3

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

    if 1.0000000001745346 < (/ (+ (- (/ (exp (* x eps)) (* eps (exp x))) (/ (- (/ 1 eps) 1) (exp (fma x eps x)))) (exp (- (* x eps) x))) 2)

    1. Initial program 4.5

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

    3. Applied expm1-log1p-u4.6

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

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018193 +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))