I've forgotten the formal proof for this, but hopefully this is correct:

Consider a D6 (for the sake of concrete language).

When you roll a 1, you reroll the die and keep the result. This produces an average value of 3.5, and happens 1/6 of the time.

When you roll a 2, you reroll the die and keep the result (even if it's lower). This produces an average value of 3.5, and happens 1/6 of the time.

When you roll a 3, you keep the result. This produces an average value of 3, and happens 1/6 of the time.

And so on.

This gives the following formula for the average of the D6: \$ (3.5 + 3.5 + 3 + 4 + 5 + 6) / 6 = 4.1\bar{6}\$.

Working similar formulas for the other dice, we get this table:

\begin{array}{lccc}
\hline
\text{Die} & \text{(standard) Avg.} & \text{GWF Avg.} & \Delta \\
\hline
\text{d4} & 2.5 & 3.00 & 0.50 \\
\text{d6} & 3.5 & 4.1\bar{6} & 0.6\bar{6} \\
\text{d8} & 4.5 & 5.25 & 0.75 \\
\text{d10} & 5.5 & 6.30 & 0.80 \\
\text{d12} & 6.5 & 7.3\bar{3} & 0.8\bar{3} \\
\hline
\end{array}

Dice are independent. 2D6 will have an average value of \$2 \cdot 4.1\bar{6} = 8.3\bar{3}\$.

Common weapon average damage (Great Weapon Fighting):

\begin{array}{lcc}
\hline
\text{Weapon} & \text{Avg. GWF dmg} & \text{improvement w/ GWF}\\ \hline
\text{Greatsword (2d6)} & 8.3\bar{3} & 1.3\bar{3} \\
\text{Greataxe (1d12)} & 7.3\bar{3} & 0.8\bar{3} \\
\text{Longsword (1d10)} & 6.30 & 0.80 \\
\text{Double-bladed Scimitar (2d4)} & 6 & 1 \\
\text{Smite (level 1, 2d8)} & 10.50 & 1.50 \\
\qquad \text{(+ weapon damage)} \\ \hline
\end{array}

Observations:

The ability works out to about a +1 to damage.

It scales to almost a +3 when smiting. The more dice you add (high level smite, for example), the better the ability.^{See errata, below}

The bonus is "swingy." It can range from a -2 to a +10 on 2D6, for example.

# Errata

In April of 2016, Jeremy Crawford ruled that additional dice from abilities like smite can not be re-rolled by Great Weapon Fighting.

If you are using a versatile weapon, you can only gain the benefit of the Great Weapon Fighting Style if you you are using it in both hands. Similarly, you can only gain the benefit of the Dueling Fighting Style if you are using it in one hand. (And holding no other weapons.)

So, let's take a look at how this breaks down with a longsword (or battleaxe, they're equivalent.)

- 2-handed: 1d10 + Str, reroll 1s and 2s. Average damage = 6.3 + Str
- 1-handed: 1d8 + Str + 2. Average damage = 6.5 + Str.

Obligatory anydice link: http://anydice.com/program/5b1e.

As well as this, wielding your weapon 1-handed allows you to use your other hand for something. (Like a **shield**!) This can be a huge benefit.

So for versatile weapons, the Dueling Fighting Style is actually strictly superior to the Great Weapon Fighting Style.

However, if you want to use a two-handed weapon, you should probably use an actual two-handed weapon rather than a versatile one, at which point you can achieve much higher damage.

## Best Answer

From Sage Advice — Rules Answers: April 2016:

Also: As pointed out by Doval, Jeremy Crawford allows all weapon dices to be rerolled. This includes, for instance, a Frost Brand sword's additional cold damage, since that damage is part of the weapon's damage itself (not from an additional feature).