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== Calculation ==

=== Calculating Move Speed ===
While items often display movement speed bonuses as flat values (e.g., +2 m/s), multiple sources of move speed do not simply add together. Instead, they stack '''multiplicatively''' using a similar formula to [[Damage Resistance|resistance]], meaning each additional source provides diminishing returns.

To find your effective move speed bonus, use the following formula:

<math>\text{Effective Move Bonus} = 12 \times (1 - (1 - \frac{B_1}{12}) \times (1 - \frac{B_2}{12}) \dots)</math>

Where <math>B_1</math>, <math>B_2</math>, etc. are individual move speed bonuses.

Then calculate your total speed:

<math>\text{Total Speed} = \text{Base Move Speed} + \text{Effective Move Bonus} + \text{Base Sprint Speed} + \text{Sprint Bonus}_1 + \text{Sprint Bonus}_2 \dots</math>

'''Example:'''
If you have two sources of move speed, one providing '''+2 m/s''' and another providing '''+3 m/s''':
# Divide each bonus by 12: <math>{\approx}0.167</math> and 0.25.
# Subtract from 1: <math>(1 - 0.167) = \textbf{0.833}</math> and <math>(1 - 0.25) = \textbf{0.75}</math>.
# Multiply the remainders: <math>0.833 \times 0.75 = \textbf{0.625}</math>.
# Subtract from 1: <math>1 - 0.625 = \textbf{0.375}</math>.
# Multiply by 12: <math>0.375 \times 12 = \textbf{4.5 m/s}</math>.
# Your Effective Move Bonus is '''4.5 m/s''' (not 5 m/s).

If your hero has 6.5 base move speed, 1.6 base sprint speed, and no sprint bonuses:
<math>6.5 + 4.5 + 1.6 = \mathbf{12.6 m/s}</math>

=== Sprint Speed Stacking ===
Unlike move speed, sprint speed stacks '''additively''' with no diminishing returns. If you have base 1.6 sprint speed and get items with +2 and +5 sprint speed, you simply have 8.6 sprint speed.

<math>\text{Total Sprint} = \text{Base Sprint} + \text{Bonus}_1 + \text{Bonus}_2 \dots</math>

=== Combined Example ===
A hero with:
* 6.4 base move speed
* 1.6 base sprint speed  
* +2 and +3 move speed bonuses
* +2 and +1.5 sprint speed bonuses

'''Step 1: Calculate Move Speed'''
<math>12 \times (1 - (1 - \frac{2}{12}) \times (1 - \frac{3}{12})) = 4.5 \text{ m/s effective}</math>
<math>6.4 + 4.5 = \mathbf{10.9 \text{ m/s}}</math>

'''Step 2: Calculate Sprint Speed'''
<math>1.6 + 2 + 1.5 = \mathbf{5.1 \text{ m/s}}</math>

'''Step 3: Add Together'''
<math>10.9 + 5.1 = \mathbf{16.0 \text{ m/s}}</math>