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color dodge mode
Description:
This mode is some kind of inverted multiply mode (the multiplication is replaced by a division of the "inverse").

Formula:
f(a,b) = a / (1 - b)



Disadvantage:
The formula can result in values greater than 1, so some truncation can occur.

Code:
inverse color dodge mode
Description:
I've never seen this mode anywhere, probably because it is the same than color dodge mode, but base color and blend color are swapped (diagram 2 equals "flipped multiply mode").

Formula:
f(a,b) = b / (1 - a)



Disadvantage:
The formula can result in values greater than 1, so some truncation can occur.

Code:
soft dodge mode
Description:
This mode is a combination of color dodge and inverse color burn mode, but a lot smoother than both of them. The base image is darkened a bit, but very bright blend colors are "dodged" in. I chose this name, because my aim was to make the color dodge mode softer.

Formula:
f(a,b) =  ½a / (1 - b) (for a + b < 1)
1 - ½(1 - b) / a (else)



Disadvantages:
The formula can result in values smaller than 0 or greater than 255, so some truncation can occur (but not as extreme as for color dodge).
There are two separate formulas (but no real discontinuance).

Code:
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