Compo Comparisons

Unfortunately, there are a lot of decisions to make in Stick Ranger. Don't worry. If you have to choose between two compo items to equip on a weapon, I'll help you with some of them.

The following tables give formulas for you to check which compo item produces the better output. All formulas are in inequality form for convenience. As a reference, if the content in the cell becomes true after plugging in the appropriate variables, then the row's compo item is a better choice than the column's. If it is false, the opposite is true.

Physical AT

Variables:
$a$-minimum AT gain from a Red Crystal
$b$-maximum AT gain from a Red Crystal
$c$-AT gain (in decimal form) from a Yellow Crystal
$d$-AT gain (in decimal form) from a Critical's Card
$i$-SP invested into a stat as a result of equipping the respective Stone
$p$-chance of critical activation (in decimal form) from a Critical's Card
$x$-minimum base AT
$y$-maximum base AT

	Red Crystal	Yellow Crystal	Critical's Card
Boxer STR	$a+b < i$	$c(x+y) < i$	$\frac{(x+y)(d+\frac{1}{p}-1)}{2} < i$
Gladiator STR	$a+b < i$	$c(x+y) < i$	$(x+y)(d+\frac{1}{p}-1) < i$
Gladiator DEX	$a+b < i$	$c(x+y) < i$	$(x+y)(d+\frac{1}{p}-1) < i$
Sniper DEX	$\frac{12(a+b)}{7} < i$	$\frac{12c(x+y)}{7} < i$	$\frac{12(x+y)(d+\frac{1}{p}-1)}{7} < i$
Gunner STR	$\frac{50(a+b)}{x+y} < i$	$50c < i$	$50(d+\frac{1}{p}-1) < i$
Whipper STR	$a+b < i$	$c(x+y) < i$	$\frac{(x+y)(d+\frac{1}{p}-1)}{2} < i$
Angel STR	$\frac{12(a+b)}{7} < i$	$\frac{12c(x+y)}{7} < i$	$\frac{12(x+y)(d+\frac{1}{p}-1)}{7} < i$
Red Crystal	-	$a+b > c(x+y)$	$a+b > dp(x+y)$
Yellow Crystal	-	-	$c > dp$

Fire AT

Variables:
$a$-minimum Fire AT gain from a Ruby
$b$-maximum Fire AT gain from a Ruby
$c$-length (in seconds) of Fire effect Garnet adds
$x$-minimum base Fire AT
$y$-maximum base Fire AT
$z$-base length (in seconds) of Fire effect

	Garnet
Ruby	$z > \frac{c(x+y)}{a+b}$

Poison AT

Variables:
$a$-minimum Poison AT gain from an Emerald
$b$-maximum Poison AT gain from an Emerald
$c$-length (in seconds) of Poison effect Peridot adds
$x$-minimum Poison base AT
$y$-maximum Poison base AT
$z$-base length (in seconds) of Poison effect

	Peridot
Emerald	$z > \frac{c(x+y)}{a+b}$

Defense

Variables:
$a$-defense gain from a Silver Crystal
$b$-defense gain from a Black Crystal
$i$-SP invested into a stat as a result of equipping the respective Stone
$x$-enemy attack damage

	Silver Crystal	Black Crystal
Priest DEX (Aura DF)	$5a > i$	$5bx > i$
Silver Crystal	-	$\frac{b}{a} > x$

AGI

Variables:
$a$-AGI reduction from a Quick's Card
$i$-SP invested into a stat as a result of equipping the respective Stone
$x$-minimum base AGI
$y$-maximum base AGI

	Quick's Card
Boxer DEX	$\frac{a(x+y)}{2} > i$
Magician DEX	$\frac{a(x+y)}{2} > i$
Gunner DEX	$\frac{50}{\frac{1}{a}-1} < i$

Range

Variables:
$a$-range gain from a Catapult's Card
$i$-SP invested into a stat as a result of equipping the respective Stone

	Catapult's Card
Sniper STR	$\frac{a}{2} < i$
Magician STR	$\frac{a}{2} < i$
Priest MAG	$\frac{a}{2} < i$

Bullets

Variables:
$a$-bullet gain from a Bullet's Card that increases bullets by a fixed amount
$b$-bullet gain from a Bullet's Card that increases bullets by a percentage
$i$-SP invested into a stat as a result of equipping the respective Stone
$x$-base amount of bullets

	Bullet's Card (fixed)	Bullet's Card (percentage)
Whipper DEX	$5a < i$	$5bx < i$
Bullet's Card (fixed)	-	$a > bx$

Notes

To find these formulas:

1. Set the formula for the "unaffected" situation equal to itself, simulating two unique situations that will be affected in different ways.
2. Add how both compo items affect their respective sides.
3. Because we're trying to find out when one situation will produce the more desired output, we can use an inequality. To keep track of the direction (in some instances, it may change), make one situation greater than the other.
4. Use algebra to obtain a convenient form. The forms above are entirely optional.

Here is an example comparing the effects of a Red Stone and a Red Crystal on a Boxer's Glove:

1. $\frac{x+y}{2} = \frac{x+y}{2}$ The average damage of the unaffected Glove.
2. $\frac{(x+\frac{i}{2})+(y+\frac{i}{2})}{2} = \frac{(x+a)+(y+b)}{2}$ The effect of the Red Stone was applied to the left side, while the effect of the Red Crystal was applied to the right.
3. $\frac{(x+\frac{i}{2})+(y+\frac{i}{2})}{2} > \frac{(x+a)+(y+b)}{2}$ For no particular reason, let's make the left side greater than the right. This means if the inequality is true, then the Red Stone is a better choice.
4. $\frac{(x+\frac{i}{2})+(y+\frac{i}{2})}{2} > \frac{(x+a)+(y+b)}{2}$

$x+y+i > x+y+a+b$

$i > a+b$ Note that the final product is identical to the corresponding equation above: $a+b < i$.

If you're creating compo items with mods, you can use this technique to find a good balance of effectiveness!

If there are any errors in my math and you happen to find any, please leave a comment, preferably with a proof.