Average Error: 29.9 → 0.4
Time: 2.5m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + 1\right)} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{e^{(1 \cdot x + x)_*}}}{2} \le 1.0:\\ \;\;\;\;\frac{\left(2 + \log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} \cdot \sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (- (* (+ 1 (/ 1 eps)) (exp (* (+ 1 eps) x))) (* (exp (* (- 1 eps) x)) (- (/ 1 eps) 1))) (exp (fma (- 1 0) x x))) 2) < 1.0

    1. Initial program 40.2

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

    2. Taylor expanded around 0 0.4

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

    4. Applied add-log-exp0.4

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

    if 1.0 < (/ (/ (- (* (+ 1 (/ 1 eps)) (exp (* (+ 1 eps) x))) (* (exp (* (- 1 eps) x)) (- (/ 1 eps) 1))) (exp (fma (- 1 0) x x))) 2)

    1. Initial program 0.3

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

    3. Applied add-sqr-sqrt0.4

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

  4. Applied simplify0.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + 1\right)} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{e^{(1 \cdot x + x)_*}}}{2} \le 1.0:\\ \;\;\;\;\frac{\left(2 + \log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} \cdot \sqrt{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - e^{\left(\varepsilon + 1\right) \cdot \left(-x\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array}}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' +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))