Average Error: 29.7 → 17.5
Time: 6.9m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{(e^{\log \left((\left({\left(e^{x}\right)}^{\left(\varepsilon - 1\right)}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + 1)_* - \frac{\frac{1}{\varepsilon} - 1}{e^{(x \cdot \varepsilon + x)_*}}\right)} - 1)^*}{2} \le -1.4935731958935078 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{\varepsilon} + \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(e^{-\left(1 - \varepsilon\right) \cdot x}\right) + \left(\sqrt{\frac{1 - \frac{1}{\varepsilon}}{e^{(x \cdot \varepsilon + x)_*}}} \cdot \sqrt{\frac{1 - \frac{1}{\varepsilon}}{e^{(x \cdot \varepsilon + x)_*}}}\right))_*}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if (/ (expm1 (log (- (fma (pow (exp x) (- eps 1)) (+ 1 (/ 1 eps)) 1) (/ (- (/ 1 eps) 1) (exp (fma x eps x)))))) 2) < -1.4935731958935078e-17

    1. Initial program 62.0

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

    3. Applied fma-neg62.0

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

    4. Applied simplify62.0

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

    5. Taylor expanded around 0 34.1

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

    6. Applied simplify34.1

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

    if -1.4935731958935078e-17 < (/ (expm1 (log (- (fma (pow (exp x) (- eps 1)) (+ 1 (/ 1 eps)) 1) (/ (- (/ 1 eps) 1) (exp (fma x eps x)))))) 2)

    1. Initial program 1.0

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

    3. Applied fma-neg1.0

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

    4. Applied simplify1.0

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

    6. Applied add-sqr-sqrt2.7

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

Runtime

Time bar (total: 6.9m)Debug logProfile

herbie shell --seed 2020178 +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))