Turing Sugar Cane Hoe

The Turing Sugar Cane Hoe is a to  (if recombobulated) rarity Hoe. It can be crafted from the Mathematical Hoe Blueprint. This hoe can be upgraded, increasing its Sugar Cane harvest yield, and Exp rate. For to  rarities the hoe's bonus EXP rate is directly equal to its rarity (higher rarities deviate from this rule with  giving +5%,  giving +8% and  giving +12%).

Rarity
The Turing Sugar Cane Hoe's rarity increases on two occasions. The first is when the player crafts the recipe to upgrade its tier, and the second is when certain milestones are hit on the hoe's Counter.

Counter
The Counter on the hoe counts how many Sugar Cane have been harvested. When the counter hits certain milestones, the hoe's rarity will increase by one level.

Logarithmic Counter
At the first tier of crafted unlock, the Turing Sugar Cane Hoe gains Logarithmic Counter. Logarithmic Counter uses the Counter stat on the hoe, and grants additional Sugar Cane harvest yield. Its description reads:"Harvest 16% Sugar Cane per digits on the counter, minus 4!"A digit refers to the amount of individual numbers on the counter. This may be better represented as:$$(CounterDigits - 4) \times 16$$

Logarithmic Counter Table
Some of these numbers have not yet been reached, and whether these numbers are actually achievable in-game is yet to be discovered.

Collection Analysis
At the second tier of crafted unlock, the Turing Sugar Cane Hoe gains Collection Analysis. Collection Analysis uses the player's current Sugar Cane collection, and grants additional Sugar Cane harvest yield. Its description reads:"Harvest 8% Sugar Cane per digits of your collection, minus 4!"A digit refers to the amount of individual numbers in the player's collection. This may be better represented as:$$(CounterDigits - 4) \times 8$$

Collection Analysis Table
Some of these numbers have not yet been reached, and whether these numbers are actually achievable in-game is yet to be discovered.

Additional EXP boost:

The Hoe also gives an EXP boost based on its rarity, this EXP boost will stack with the EXP boost given by the Blessed enchant, which is also based on rarity.

Crafted Upgrades
Harvest Gain Farming Exp
 * -|Tier 1 Stats=

Harvest Gain Farming Exp Harvest sugar cane per digits on the Counter, minus 4!
 * -|Tier 2 Stats=

Harvest +50% sugar cane Gain +3% Farming Exp (Up to +12% Exp) Harvest sugar cane per digits on the Counter, minus 4!
 * -|Tier 3 Stats=

Harvest sugar cane per digits of your collection, minus 4!

Formulas
The number of digits on the counter may be calculated using:
 * $$n = \left\lfloor\log_{10} counter\right\rfloor + 1$$

In order to mathematically represent the and  ability scaling, the following formula has to be used:
 * $$\text{Logarithmic Counter Bonus} =

\begin{cases} 0 & n \leq 4 \\ n \times 16% & otherwise \end{cases}$$
 * $$\text{Collection Analysis Bonus} =

\begin{cases} 0 & n \leq 4 \\ n \times 8% & otherwise \end{cases}$$