ALTBCEquations

Git Source Author: @oscarsernarosero @mpetersoCode55 @cirsteve @palmerg4

State Variables

FLOAT_2

packedFloat constant FLOAT_2 = packedFloat.wrap(0x7f6c00000000000000000000000000000f0bdc21abb48db201e86d4000000000);

FLOAT_NEG_1

packedFloat constant FLOAT_NEG_1 = packedFloat.wrap(0x7f6d00000000000000000000000000000785ee10d5da46d900f436a000000000);

FLOAT_0

packedFloat constant FLOAT_0 = packedFloat.wrap(0);

FLOAT_1

packedFloat constant FLOAT_1 = packedFloat.wrap(0x7f6c00000000000000000000000000000785ee10d5da46d900f436a000000000);

FLOAT_WAD

packedFloat constant FLOAT_WAD =
    packedFloat.wrap(57507338264406853159277167054180511853162945875507645848942038639672188469248);

Functions

calculateBn

The result will be a packedFloat Bn is equal to V / (Xn + C) in the spec This function calculates B(n) and stores it in the tbc definition as b.

function calculateBn(ALTBCDef storage altbc, packedFloat Xn) internal;

Parameters

Name	Type	Description
`altbc`	`ALTBCDef`	the tbc definition
`Xn`	`packedFloat`	the X value at n

calculatefx

The result for f(x) will be a packedFloat. This equation is used to calculate the spot price of the x token and is equal to (bn * x) + cn This function calculates f(x) at n.

function calculatefx(ALTBCDef storage altbc, packedFloat x) internal view returns (packedFloat result);

Parameters

Name	Type	Description
`altbc`	`ALTBCDef`	the tbc definition
`x`	`packedFloat`	value for x at n

Returns

Name	Type	Description
`result`	`packedFloat`	the calculated f(x), this value will be a packedFloat

calculateDn

The result for Dn will be a packedFloat This equation is used to calculate the area under the curve at n and is equal to (1/2)(bnx^2) + cnx This function calculates D at n.

function calculateDn(ALTBCDef storage altbc, packedFloat x) internal view returns (packedFloat result);

Parameters

Name	Type	Description
`altbc`	`ALTBCDef`	the tbc definition
`x`	`packedFloat`	value for x at n

Returns

Name	Type	Description
`result`	`packedFloat`	result

calculateH

This method is implemented using packedFloats and the float128 library This equation in the spec is equal to (Ln + Zn) / (Wn - wInactive) + phi This function calculates h at n which is the total revenue per unit of liquidity at time n.

function calculateH(ALTBCDef storage altbc, packedFloat L, packedFloat W, packedFloat wInactive, packedFloat phi)
    internal
    view
    returns (packedFloat result);

Parameters

Name	Type	Description
`altbc`	`ALTBCDef`
`L`	`packedFloat`	the x coordinate
`W`	`packedFloat`	the total amount of units of liquidity in circulation
`wInactive`	`packedFloat`
`phi`	`packedFloat`	the total amount of units of liquidity in circulation

Returns

Name	Type	Description
`result`	`packedFloat`	the calculated h

calculateXofNPlus1

This method is implemented using packedFloats and the float128 library This equation in the spec is equal to 2Dn / (c + sqrt(c^2 + 2bDn)) This function calculates the value of Xn+1.

function calculateXofNPlus1(ALTBCDef storage altbc, packedFloat Dn) internal view returns (packedFloat newX);

Parameters

Name	Type	Description
`altbc`	`ALTBCDef`	the tbc definition
`Dn`	`packedFloat`	the area under the curve.

Returns

Name	Type	Description
`newX`	`packedFloat`	the calculated Xn+1

calculateC

This function calculates the parameter c and stores it in the tbc definition.

function calculateC(ALTBCDef storage altbc, packedFloat Xn, packedFloat oldBn) internal;

Parameters

Name	Type	Description
`altbc`	`ALTBCDef`	the tbc definition.
`Xn`	`packedFloat`	the x coordinate
`oldBn`	`packedFloat`	the previous state of b

calculateLastRevenueClaim

The result for last revenue claim will be a Float. This function calculates the last revenue claim to be stored in the associated LPToken variable rj. The result will be a WAD value.

function calculateLastRevenueClaim(packedFloat hn, packedFloat wj, packedFloat r_hat, packedFloat w_hat)
    internal
    pure
    returns (packedFloat);

Parameters

Name	Type	Description
`hn`	`packedFloat`	The revenue parameter. Expected to be a Float.
`wj`	`packedFloat`	The share of the pool’s liquidity the associated LPToken represents. Expected to be a Float.
`r_hat`	`packedFloat`	The current last revenue claim value of the associated LPToken. Expected to be a Float.
`w_hat`	`packedFloat`	The current liquidity amount of the associated LPToken. Expected to be a Float.

calculateL

This function calculates the parameter L.

function calculateL(ALTBCDef storage altbc, packedFloat Xn) internal view returns (packedFloat result);

Parameters

Name	Type	Description
`altbc`	`ALTBCDef`	the tbc definition.
`Xn`	`packedFloat`	the x coordinate

Returns

Name	Type	Description
`result`	`packedFloat`	the calculate L parameter.

calculateZ

This function calculates the parameter Z, which is a balancing quantity used to ensure fair LP accounting.

function calculateZ(
    ALTBCDef storage altbc,
    packedFloat Ln,
    packedFloat Wn,
    packedFloat WIn,
    packedFloat q,
    bool withdrawal
) internal;

Parameters

Name	Type	Description
`altbc`	`ALTBCDef`	the tbc definition.
`Ln`	`packedFloat`	the liquidity parameter.
`Wn`	`packedFloat`	the total amount of units of liquidity in circulation.
`WIn`	`packedFloat`	the total amount of units of liquidity in circulation.
`q`	`packedFloat`	the liquidity units to receive in exchange for A and B
`withdrawal`	`bool`	the boolean value for withdrawal

calculateQ

This function calculates q.

function calculateQ(
    ALTBCDef storage altbc,
    packedFloat Xn,
    packedFloat _A,
    packedFloat _B,
    packedFloat L,
    packedFloat Dn
) internal view returns (packedFloat A, packedFloat B, packedFloat q);

Parameters

Name	Type	Description
`altbc`	`ALTBCDef`	the tbc definition.
`Xn`	`packedFloat`	the x coordinate
`_A`	`packedFloat`	The amount of incoming X Token.
`_B`	`packedFloat`	The amount of incoming collateral.
`L`	`packedFloat`	the liquidity parameter.
`Dn`	`packedFloat`	The current area under the curve.

Returns

Name	Type	Description
`A`	`packedFloat`	the actual amount to take for token x
`B`	`packedFloat`	the actual amount to take for token y
`q`	`packedFloat`	the liquidity units to receive in exchange for A and B

calculateRevenueAvailable

This function calculates the revenue available for a given LPToken.

function calculateRevenueAvailable(packedFloat wj, packedFloat hn, packedFloat rj)
    internal
    pure
    returns (packedFloat result);

Parameters

Name	Type	Description
`wj`	`packedFloat`	The share of the pool’s liquidity the associated LPToken represents.
`hn`	`packedFloat`	The revenue parameter.
`rj`	`packedFloat`	The last revenue claim for the associated LPToken.

Returns

Name	Type	Description
`result`	`packedFloat`	The calculated revenue available for the LPToken.

_liquidityUpdateHelper

This function updates related tbc variables when a liquidity deposit or withdrawal is made

function _liquidityUpdateHelper(ALTBCDef storage altbc, packedFloat Xn, packedFloat multiplier)
    internal
    returns (packedFloat x);

Parameters

Name	Type	Description
`altbc`	`ALTBCDef`	the tbc definition.
`Xn`	`packedFloat`	the x coordinate.
`multiplier`	`packedFloat`	The value for multiplier for pool state

Returns

Name	Type	Description
`x`	`packedFloat`	The updated x value.

Reference

Core Concepts

State Variables

FLOAT_2

FLOAT_NEG_1

FLOAT_0

FLOAT_1

FLOAT_WAD

Functions

calculateBn

calculatefx

calculateDn

calculateH

calculateXofNPlus1

calculateC

calculateLastRevenueClaim

calculateL

calculateZ

calculateQ

calculateRevenueAvailable

_liquidityUpdateHelper

​State Variables

​FLOAT_2

​FLOAT_NEG_1

​FLOAT_0

​FLOAT_1

​FLOAT_WAD

​Functions

​calculateBn

​calculatefx

​calculateDn

​calculateH

​calculateXofNPlus1

​calculateC

​calculateLastRevenueClaim

​calculateL

​calculateZ

​calculateQ

​calculateRevenueAvailable

​_liquidityUpdateHelper

State Variables

FLOAT_2

FLOAT_NEG_1

FLOAT_0

FLOAT_1

FLOAT_WAD

Functions

calculateBn

calculatefx

calculateDn

calculateH

calculateXofNPlus1

calculateC

calculateLastRevenueClaim

calculateL

calculateZ

calculateQ

calculateRevenueAvailable

_liquidityUpdateHelper