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Math behind the rewards contract

LR - Block number when the last rebalancing took place. Certain variables would be rebalanced when the state change is necessary. Those events would include:
adding or modifying staked tokens balance by the user
changing parameters used to calculate boost, such as rewards per block 
ULR - Time of rebalancing that occurrs when the user modifies his position (ipToken, pwToken delegation). It is saved under individual staking position. 
User power-up
User power-up is calculated for each user whose position changes. That includes staking or unstaking ipTokens or modifying the delegated balance of pwIPOR. 
The user's power-up is governed by the below function:  
UPU={10⋅(pwIporipToken)+0.2pwIporipToken<0.014⋅(pwIporipToken)+0.26pwIporipToken<0.023⋅(pwIporipToken)+0.28pwIporipToken<0.032⋅(pwIporipToken)+0.31pwIporipToken<0.041⋅(pwIporipToken)+0.35pwIporipToken<0.05VS+log⁡2(HS+pwIporipToken)pwIporipToken≥0.05UPU = \begin{cases}        

10\cdot \left(\frac{pwIpor}{ipToken} \right) +0.2 & \frac{pwIpor}{ipToken} < 0.01 \\

4\cdot \left(\frac{pwIpor}{ipToken} \right) +0.26 & \frac{pwIpor}{ipToken} < 0.02 \\

3\cdot \left(\frac{pwIpor}{ipToken} \right) +0.28 & \frac{pwIpor}{ipToken} < 0.03 \\

2\cdot \left(\frac{pwIpor}{ipToken} \right) +0.31 & \frac{pwIpor}{ipToken} < 0.04 \\

1\cdot \left(\frac{pwIpor}{ipToken} \right) +0.35 & \frac{pwIpor}{ipToken} < 0.05 \\

 VS + \log_2\left(HS +\frac{pwIpor}{ipToken}\right) & \frac{pwIpor}{ipToken} \ge 0.05 \\ 

\end{cases}UPU=⎩⎨⎧​10⋅(ipTokenpwIpor​)+0.24⋅(ipTokenpwIpor​)+0.263⋅(ipTokenpwIpor​)+0.282⋅(ipTokenpwIpor​)+0.311⋅(ipTokenpwIpor​)+0.35VS+log2​(HS+ipTokenpwIpor​)​ipTokenpwIpor​<0.01ipTokenpwIpor​<0.02ipTokenpwIpor​<0.03ipTokenpwIpor​<0.04ipTokenpwIpor​<0.05ipTokenpwIpor​≥0.05​
VS - Vertical Shift 
HS - Horizontal Shift
pwIpor - User's pwIPOR balance delegated to a particular asset
ipToken - User's staked ipToken balance
The curve can be swapped by the DAO for a different one at a later stage in time, affecting all future rebalancings. The user's power-up value is fixed and can only be changed if the rebalancing is triggered. 
Aggregate Power-up
Aggregate Power-up is a synthetic summary of all power-ups across all users. It's used to calculate individual rewards in relation to the rest of the pool. 
APU=∑#all users whitch stake ipTokens≥1(ipTokens⋅UPU)APU = \sum_{\# \text{all users whitch stake }ipTokens \geq 1} (ipTokens \cdot UPU)APU=#all users whitch stake ipTokens≥1∑​(ipTokens⋅UPU)
APU - Aggregate Power-up
UPU - User Power-up
​
Accrued rewards
Accrued rewards are tracked over time to allow for the calculation of users' rewards between events of rebalancing.
accruedRewards⁡=accruedRewards⁡LR+rewardsPerBlock⁡ ⋅ (blockNumber−blockNumber⁡LR)\operatorname{accruedRewards} = \operatorname{accruedRewards}_{LR} +  \operatorname{rewardsPerBlock} ~\cdot~ \left(\text{blockNumber} - \operatorname{blockNumber}_{LR}\right)accruedRewards=accruedRewardsLR​+rewardsPerBlock ⋅ (blockNumber−blockNumberLR​)
Composite Multiplier
CM=blockRewardsAPUCM = \frac{blockRewards}{APU}CM=APUblockRewards​
The composite multiplier (CM) accounts for the rewards per "boosted unit of liquidity staked" per block. 
CMC=∑#blocksCMBlockCMC = \sum_{\# blocks}CM_{Block}CMC=#blocks∑​CMBlock​
Composite Multiplier Cummulative (CMC) is a sum of Composite multipliers for every block until now. 
​
Account Mining Rewards
User rewards at block n can be calculated using the below formula. Block ULR is the block with the state of compositeMultiplierCumulative (CMC) at the time of the last rebalancing of user rewards. 
userRewardsn​=ipToken∗UPU ∗ (CMC⁡n-1⁡​−CMCULR−1​)userRewards_{n}​=ipToken * UPU~*~\left(\operatorname{CMC}_{\operatorname{n-1}}​−CMC_{ULR-1}​\right)userRewardsn​​=ipToken∗UPU ∗ (CMCn-1​​−CMCULR−1​​)
The CMCULR−1CMC_{ULR-1}CMCULR−1​
 is saved when the user modifies his position. It is calculated to reflect the state without accounting for the current block.
​
CMCULR−1=CMCLR−1+(URL−LR)⋅CMLRCMC_{ULR-1} = CMC_{LR-1} + \left(URL-LR \right) \cdot CM_{LR}CMCULR−1​=CMCLR−1​+(URL−LR)⋅CMLR​
When the user modifies their position, the most recent CMC is saved per the user's record.
​
Basic assumptions
VerticalShift ⇒ VS ⇒10−4 10^{-4}10−4
 ≤\leq≤
 VS ≤\leq≤
 3
HorizontalShift ⇒ HS ⇒ 1 ≤\leq≤
 HS ≤\leq≤
 10310^{3}103
​
ipToken ⇒ 1 ≤\leq≤
 ipToken
pwIpor ⇒ 0 ≤\leq≤
 pwIpor ≤\leq≤
 25 ⋅\cdot⋅
 10610^{6}106
​
$blockRewards ≤\leq≤
 100
$totalRewards = 25⋅\cdot⋅
​10610^6106
​
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Last modified 2mo ago