\(a\) \(b\)

◑ Average
\(f(a,b) = {a + b \over 2}\)

◑ Interpolation (Pegtop)
\(f(a,b) = {2-cos(a\pi)-cos(b\pi) \over 4}\)

○ Multiply
\(f(a,b) = ab\)

● Screen
\(f(a,b) = 1-(1-a)(1-b)\)

○ Geometric Mean
\(f(a,b) = \sqrt {ab}\)

● Geometric-1
\(f(a,b) = 1 - \sqrt {(1-a)(1-b)}\)

○ Heronian
\(f(a,b) = {a + \sqrt {ab} +b \over 3}\)

● Heronian-1
\(f(a,b) = 1 - {2 - a + \sqrt {(1-a)(1-b)} - b \over 3}\)

○ Pythagorean-1
\(f(a,b) = 1 - \sqrt {(1-a)^2+(1-b)^2 \over 2}\)

● Pythagorean
\(f(a,b) = \sqrt {a^2+b^2 \over 2}\)

○ Haze (Glare-1)
\(f(a,b) = a + b + ab - a^2 - b^2\)

● Glare
\(f(a,b) = a^2 + b^2 - ab\)

○ Absorb
\(f(a,b) = \begin{cases} a, & if\ a = b \\ | {1-b \over 1-a} - (1-b) \% (1-a) |, & if\ a < b \\ | {1-a \over 1-b} - (1-a) \% (1-b) |, & otherwise \end{cases}\)

● Emit
\(f(a,b) = \begin{cases} a, & if\ a = b \\ 1 - | {a \over b} - a \% b |, & if\ a < b \\ 1 - | {b \over a} - b \% a |, & otherwise \end{cases}\)

○ Darken
\(f(a,b) = \begin{cases} a, & if\ a < b \\ b, & otherwise \end{cases}\)

● Lighten
\(f(a,b) = \begin{cases} a, & if\ a > b \\ b, & otherwise \end{cases}\)

○ Root-1
\(f(a,b) = 1 - \sqrt {2-a-b \over 2}\)

● Root
\(f(a,b) = \sqrt {a+b \over 2}\)

○ Linear Burn
\(f(a,b) = a + b - 1\)

● Linear Dodge (Add)
\(f(a,b) = a + b\)

○ Color Burn
\(f(a,b) = \begin{cases} 0, & if\ b = 0 \\ 1 - {(1 - a)\over b}, & otherwise \end{cases}\)

● Color Dodge
\(f(a,b) = \begin{cases} 1, & if\ b = 1 \\ {a\over(1 - b)}, & otherwise \end{cases}\)

○ Soft Burn
\(f(a,b) = 1-{1-a \over | 1-a+b | }\)

● Soft Dodge
\(f(a,b) = {a \over | 1+a-b | }\)

○ Gamma Dark
\(f(a,b) = a^{1 \over b}\)

● Gamma Light
\(f(a,b) = 1-(1-a)^{1 \over 1-b}\)

○ Freeze
\(f(a,b) = 1 - {(1-a)^2 \over b}\)
\(f_{Freeze}(a,b) = f_{Heat}(b,a)\)

● Reflect
\(f(a,b) = {a^2 \over (1-b)}\)
\(f_{Reflect}(a,b) = f_{Glow}(b,a)\)

○ Heat
\(f(a,b) = 1 - {(1-b)^2 \over a}\)
\(f_{Heat}(a,b) = f_{Freeze}(b,a)\)

● Glow
\(f(a,b) = {b^2 \over (1-a)}\)
\(f_{Glow}(a,b) = f_{Reflect}(b,a)\)

◑ Overlay
\(f_{Overlay}(a,b) = f_{Hard Light}(b,a)\)

◑ Hard Light
\(f(a,b) = \begin{cases} 2ab, & if\ b \lt 0.5 \\ 1 - 2(1 - a)(1 - b), & otherwise \end{cases}\)
\(f(a,b) = \begin{cases} f_{Multiply}(a,2b), & if\ b \lt 0.5 \\ f_{Screen}(a,2b-1), & otherwise \end{cases}\)

◑ Soft Light (Photoshop)
\(f(a,b) = \begin{cases} 2ab+a^2(1-2b), & if\ b \lt 0.5 \\ 2a(1-b)+ \sqrt a (2b-1), & otherwise \end{cases}\)

◑ Soft Light
\(f(a,b) = 2ab + a^2(1 - 2b)\)
\(f(a,b) = (1-a)\times f_{Multiply}(a,b) + a\times f_{Screen}(a,b)\)

◑ Pin Light
\(f(a,b) = \begin{cases} f_{Darken}(a,2b), & if\ b \lt 0.5 \\ f_{Lighten}(a,2b-1), & otherwise \end{cases}\)

◑ Extrapolate (Kai’s Power Tools)
\(f(a,b) = 2a-b\)

◑ Vivid Light
\(f(a,b) = \begin{cases} f_{Burn}(a,2b), & if\ b \lt 0.5 \\ f_{Dodge}(a,2b-1), & otherwise \end{cases}\)

◑ Linear Light
\(f(a,b) = \begin{cases} f_{LinearBurn}(a,ab), & if\ b \lt 0.5 \\ f_{LinearDodge}(a,2b-1), & otherwise \end{cases}\)
\(f(a,b) = a+2b-1\)

◑ Quadratic Light
\(f(a,b) = (1-b)\times f_{Freeze}(a,b)+b\times f_{Reflect}(a,b)\)

◑ Modulated Light
\(f(a,b) = \begin{cases} a, & if\ a=1-b \\ |{b \over 1-a} - b\% (1-a) |, & if\ a\lt 1-b \\1-|{1-b \over a} - (1-b)\%a|, & otherwise \end{cases}\)

◑ Hard Mix
\(f(a,b) = \begin{cases} 0, & if\ {a+b \over 2} \lt 0.5 \\ 1, & otherwise \end{cases}\)

◐ Difference
\(f(a,b) = | a-b |\)

◐ Exclusion
\(f(a,b) = a+b-2ab\)

◐ Erosion
\(f(a,b) = (a-b)^2\)

◐ Solarization
\(f(a,b) = 1 - \sqrt {(2a-1)^2+(2b-1)^2 \over 2}\)

◐ Phoenix
\(f(a,b) = 1-|a-b|\)

◐ Negation
\(f(a,b) = 1-|1-a-b|\)

○ Subtract
\(f(a,b) = a - b\)

● Divide
\(f(a,b) = {a \over b}\)