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TwoCrypto Pool

A Twocrypto-NG pool consists of two non-pegged assets. The LP token is a ERC-20 token integrated directly into the liquidity pool.

Liquidity Pool (LP) Token

The LP token is directly integrated into the exchange contract. Pool and LP token share the same address.

The token has the regular ERC-20 methods, which will not be further documented.

In Twocrypto-NG pools, price scaling and fee parameters are bundled and stored as a single unsigned integer. This consolidation reduces storage read and write operations, leading to more cost-efficient calls.

pack

This internal function packs two or three integers into a single uint256.

```vyper 
@pure
@internal
def _pack_2(p1: uint256, p2: uint256) -> uint256:
    return p1 | (p2 << 128)

@internal
@pure
def _pack_3(x: uint256[3]) -> uint256:
    """
    @notice Packs 3 integers with values <= 10**18 into a uint256
    @param x The uint256[3] to pack
    @return uint256 Integer with packed values
    """
    return (x[0] << 128) | (x[1] << 64) | x[2]
```
unpack

This internal function unpacks a single uin256 into two or three integers.

```vyper
@pure
@internal
def _unpack_2(packed: uint256) -> uint256[2]:
    return [packed & (2**128 - 1), packed >> 128]

@internal
@pure
def _unpack_3(_packed: uint256) -> uint256[3]:
    """
    @notice Unpacks a uint256 into 3 integers (values must be <= 10**18)
    @param val The uint256 to unpack
    @return uint256[3] A list of length 3 with unpacked integers
    """
    return [
        (_packed >> 128) & 18446744073709551615,
        (_packed >> 64) & 18446744073709551615,
        _packed & 18446744073709551615,
    ]
```

The AMM contract utilizes two internal functions to transfer coins in and out of the pool e.g. when exchanging tokens or adding/removing liquidity:

Token transfer into the AMM:

_transfer_in(_coin_idx: uint256, _dx: uint256, sender: address, expect_optimistic_transfer: bool) -> uint256:

Internal function to transfer tokens into the AMM, called by exchange, exchange_received or add_liquidity.

Input Type Description
_coin_idx int128 Index of the token to transfer in.
_dx uint256 Amount to transfer in.
sender address Address to transfer coins from.
expect_optimistic_transfer bool True if the contract expects an optimistic coin transfer.

expect_optimistic_transfer is only True when using the exchange_received function.

balances: public(uint256[N_COINS])

@internal
def _transfer_in(
    _coin_idx: uint256,
    _dx: uint256,
    sender: address,
    expect_optimistic_transfer: bool,
) -> uint256:
    """
    @notice Transfers `_coin` from `sender` to `self` and calls `callback_sig`
            if it is not empty.
    @params _coin_idx uint256 Index of the coin to transfer in.
    @params dx amount of `_coin` to transfer into the pool.
    @params sender address to transfer `_coin` from.
    @params expect_optimistic_transfer bool True if pool expects user to transfer.
            This is only enabled for exchange_received.
    @return The amount of tokens received.
    """
    coin_balance: uint256 = ERC20(coins[_coin_idx]).balanceOf(self)

    if expect_optimistic_transfer:  # Only enabled in exchange_received:
        # it expects the caller of exchange_received to have sent tokens to
        # the pool before calling this method.

        # If someone donates extra tokens to the contract: do not acknowledge.
        # We only want to know if there are dx amount of tokens. Anything extra,
        # we ignore. This is why we need to check if received_amounts (which
        # accounts for coin balances of the contract) is atleast dx.
        # If we checked for received_amounts == dx, an extra transfer without a
        # call to exchange_received will break the method.
        dx: uint256 = coin_balance - self.balances[_coin_idx]
        assert dx >= _dx  # dev: user didn't give us coins

        # Adjust balances
        self.balances[_coin_idx] += dx

        return dx

    # ----------------------------------------------- ERC20 transferFrom flow.

    # EXTERNAL CALL
    assert ERC20(coins[_coin_idx]).transferFrom(
        sender,
        self,
        _dx,
        default_return_value=True
    )

    dx: uint256 = ERC20(coins[_coin_idx]).balanceOf(self) - coin_balance
    self.balances[_coin_idx] += dx
    return dx

Token transfer out of the AMM:

_transfer_out(_coin_idx: int128, _amount: uint256, receiver: address):

Internal function to transfer tokens out of the AMM, called by the remove_liquidity, remove_liquidity_one, _claim_admin_fees, and _exchange methods.

Input Type Description
_coin_idx int128 Index of the token to transfer out.
_amount uint256 Amount to transfer out.
receiver address Address to send the tokens to.
balances: public(uint256[N_COINS])

@internal
def _transfer_out(_coin_idx: uint256, _amount: uint256, receiver: address):
    """
    @notice Transfer a single token from the pool to receiver.
    @dev This function is called by `remove_liquidity` and
        `remove_liquidity_one`, `_claim_admin_fees` and `_exchange` methods.
    @params _coin_idx uint256 Index of the token to transfer out
    @params _amount Amount of token to transfer out
    @params receiver Address to send the tokens to
    """

    # Adjust balances before handling transfers:
    self.balances[_coin_idx] -= _amount

    # EXTERNAL CALL
    assert ERC20(coins[_coin_idx]).transfer(
        receiver,
        _amount,
        default_return_value=True
    )

Exchange Methods

The contract offers two different ways to exchange tokens:

  • A regular exchange method.
  • A novel exchange_received method, which swaps tokens based on the "internal balances" of the pool. This method is of great use for aggregators, as it does not require token approval of the pool, which eliminates certain smart contract risks and can remove one redundant ERC-20 transfer. More here.

exchange

TwoCrypto.exchange(i: uint256, j: uint256, dx: uint256, min_dy: uint256, receiver: address = msg.sender) -> uint256:

Function to exchange dx amount of coin i for coin j and receive a minimum amount of min_dy. Charged fee at current states is Pool.fee().

Returns: amount of output coin j received (uint256).

Emits: TokenExchange

Input Type Description
i uint256 Index value for the input coin.
j uint256 Index value for the output coin.
dx uint256 Amount of input coin being swapped in.
min_dy uint256 Minimum amount of output coin to receive.
receiver address Address to send output coin to. Defaults to msg.sender.
Source code
event TokenExchange:
    buyer: indexed(address)
    sold_id: uint256
    tokens_sold: uint256
    bought_id: uint256
    tokens_bought: uint256
    fee: uint256
    packed_price_scale: uint256

@external
@nonreentrant("lock")
def exchange(
    i: uint256,
    j: uint256,
    dx: uint256,
    min_dy: uint256,
    receiver: address = msg.sender
) -> uint256:
    """
    @notice Exchange using wrapped native token by default
    @param i Index value for the input coin
    @param j Index value for the output coin
    @param dx Amount of input coin being swapped in
    @param min_dy Minimum amount of output coin to receive
    @param receiver Address to send the output coin to. Default is msg.sender
    @return uint256 Amount of tokens at index j received by the `receiver
    """
    # _transfer_in updates self.balances here:
    dx_received: uint256 = self._transfer_in(
        i,
        dx,
        msg.sender,
        False
    )

    # No ERC20 token transfers occur here:
    out: uint256[3] = self._exchange(
        i,
        j,
        dx_received,
        min_dy,
    )

    # _transfer_out updates self.balances here. Update to state occurs before
    # external calls:
    self._transfer_out(j, out[0], receiver)

    # log:
    log TokenExchange(msg.sender, i, dx_received, j, out[0], out[1], out[2])

    return out[0]

@internal
def _exchange(
    i: uint256,
    j: uint256,
    dx_received: uint256,
    min_dy: uint256,
) -> uint256[3]:

    assert i != j  # dev: coin index out of range
    assert dx_received > 0  # dev: do not exchange 0 coins

    A_gamma: uint256[2] = self._A_gamma()
    xp: uint256[N_COINS] = self.balances
    dy: uint256 = 0

    y: uint256 = xp[j]
    x0: uint256 = xp[i] - dx_received  # old xp[i]

    price_scale: uint256 = self.cached_price_scale
    xp = [
        xp[0] * PRECISIONS[0],
        unsafe_div(xp[1] * price_scale * PRECISIONS[1], PRECISION)
    ]

    # ----------- Update invariant if A, gamma are undergoing ramps ---------

    t: uint256 = self.future_A_gamma_time
    if t > block.timestamp:

        x0 *= PRECISIONS[i]

        if i > 0:
            x0 = unsafe_div(x0 * price_scale, PRECISION)

        x1: uint256 = xp[i]  # <------------------ Back up old value in xp ...
        xp[i] = x0                                                         # |
        self.D = MATH.newton_D(A_gamma[0], A_gamma[1], xp, 0)              # |
        xp[i] = x1  # <-------------------------------------- ... and restore.

    # ----------------------- Calculate dy and fees --------------------------

    D: uint256 = self.D
    y_out: uint256[2] = MATH.get_y(A_gamma[0], A_gamma[1], xp, D, j)
    dy = xp[j] - y_out[0]
    xp[j] -= dy
    dy -= 1

    if j > 0:
        dy = dy * PRECISION / price_scale
    dy /= PRECISIONS[j]

    fee: uint256 = unsafe_div(self._fee(xp) * dy, 10**10)
    dy -= fee  # <--------------------- Subtract fee from the outgoing amount.
    assert dy >= min_dy, "Slippage"
    y -= dy

    y *= PRECISIONS[j]
    if j > 0:
        y = unsafe_div(y * price_scale, PRECISION)
    xp[j] = y  # <------------------------------------------------- Update xp.

    # ------ Tweak price_scale with good initial guess for newton_D ----------

    price_scale = self.tweak_price(A_gamma, xp, 0, y_out[1])

    return [dy, fee, price_scale]
@external
@view
def newton_D(ANN: uint256, gamma: uint256, x_unsorted: uint256[N_COINS], K0_prev: uint256 = 0) -> uint256:
    """
    Finding the invariant using Newton method.
    ANN is higher by the factor A_MULTIPLIER
    ANN is already A * N**N
    """

    # Safety checks
    assert ANN > MIN_A - 1 and ANN < MAX_A + 1  # dev: unsafe values A
    assert gamma > MIN_GAMMA - 1 and gamma < MAX_GAMMA + 1  # dev: unsafe values gamma

    # Initial value of invariant D is that for constant-product invariant
    x: uint256[N_COINS] = x_unsorted
    if x[0] < x[1]:
        x = [x_unsorted[1], x_unsorted[0]]

    assert x[0] > 10**9 - 1 and x[0] < 10**15 * 10**18 + 1  # dev: unsafe values x[0]
    assert unsafe_div(x[1] * 10**18, x[0]) > 10**14 - 1  # dev: unsafe values x[i] (input)

    S: uint256 = unsafe_add(x[0], x[1])  # can unsafe add here because we checked x[0] bounds

    D: uint256 = 0
    if K0_prev == 0:
        D = N_COINS * isqrt(unsafe_mul(x[0], x[1]))
    else:
        # D = isqrt(x[0] * x[1] * 4 / K0_prev * 10**18)
        D = isqrt(unsafe_mul(unsafe_div(unsafe_mul(unsafe_mul(4, x[0]), x[1]), K0_prev), 10**18))
        if S < D:
            D = S

    __g1k0: uint256 = gamma + 10**18
    diff: uint256 = 0

    for i in range(255):
        D_prev: uint256 = D
        assert D > 0
        # Unsafe division by D and D_prev is now safe

        # K0: uint256 = 10**18
        # for _x in x:
        #     K0 = K0 * _x * N_COINS / D
        # collapsed for 2 coins
        K0: uint256 = unsafe_div(unsafe_div((10**18 * N_COINS**2) * x[0], D) * x[1], D)

        _g1k0: uint256 = __g1k0
        if _g1k0 > K0:
            _g1k0 = unsafe_add(unsafe_sub(_g1k0, K0), 1)  # > 0
        else:
            _g1k0 = unsafe_add(unsafe_sub(K0, _g1k0), 1)  # > 0

        # D / (A * N**N) * _g1k0**2 / gamma**2
        mul1: uint256 = unsafe_div(unsafe_div(unsafe_div(10**18 * D, gamma) * _g1k0, gamma) * _g1k0 * A_MULTIPLIER, ANN)

        # 2*N*K0 / _g1k0
        mul2: uint256 = unsafe_div(((2 * 10**18) * N_COINS) * K0, _g1k0)

        # calculate neg_fprime. here K0 > 0 is being validated (safediv).
        neg_fprime: uint256 = (S + unsafe_div(S * mul2, 10**18)) + mul1 * N_COINS / K0 - unsafe_div(mul2 * D, 10**18)

        # D -= f / fprime; neg_fprime safediv being validated
        D_plus: uint256 = D * (neg_fprime + S) / neg_fprime
        D_minus: uint256 = unsafe_div(D * D,  neg_fprime)
        if 10**18 > K0:
            D_minus += unsafe_div(unsafe_div(D * unsafe_div(mul1, neg_fprime), 10**18) * unsafe_sub(10**18, K0), K0)
        else:
            D_minus -= unsafe_div(unsafe_div(D * unsafe_div(mul1, neg_fprime), 10**18) * unsafe_sub(K0, 10**18), K0)

        if D_plus > D_minus:
            D = unsafe_sub(D_plus, D_minus)
        else:
            D = unsafe_div(unsafe_sub(D_minus, D_plus), 2)

        if D > D_prev:
            diff = unsafe_sub(D, D_prev)
        else:
            diff = unsafe_sub(D_prev, D)

        if diff * 10**14 < max(10**16, D):  # Could reduce precision for gas efficiency here

            for _x in x:
                frac: uint256 = _x * 10**18 / D
                assert (frac >= 10**16 - 1) and (frac < 10**20 + 1)  # dev: unsafe values x[i]
            return D

    raise "Did not converge"

@external
@pure
def get_y(
    _ANN: uint256,
    _gamma: uint256,
    _x: uint256[N_COINS],
    _D: uint256,
    i: uint256
) -> uint256[2]:

    # Safety checks
    assert _ANN > MIN_A - 1 and _ANN < MAX_A + 1  # dev: unsafe values A
    assert _gamma > MIN_GAMMA - 1 and _gamma < MAX_GAMMA + 1  # dev: unsafe values gamma
    assert _D > 10**17 - 1 and _D < 10**15 * 10**18 + 1 # dev: unsafe values D

    ANN: int256 = convert(_ANN, int256)
    gamma: int256 = convert(_gamma, int256)
    D: int256 = convert(_D, int256)
    x_j: int256 = convert(_x[1 - i], int256)
    gamma2: int256 = unsafe_mul(gamma, gamma)

    # savediv by x_j done here:
    y: int256 = D**2 / (x_j * N_COINS**2)

    # K0_i: int256 = (10**18 * N_COINS) * x_j / D
    K0_i: int256 = unsafe_div(10**18 * N_COINS * x_j, D)
    assert (K0_i > 10**16 * N_COINS - 1) and (K0_i < 10**20 * N_COINS + 1)  # dev: unsafe values x[i]

    ann_gamma2: int256 = ANN * gamma2

    # a = 10**36 / N_COINS**2
    a: int256 = 10**32

    # b = ANN*D*gamma2/4/10000/x_j/10**4 - 10**32*3 - 2*gamma*10**14
    b: int256 = (
        D*ann_gamma2/400000000/x_j
        - convert(unsafe_mul(10**32, 3), int256)
        - unsafe_mul(unsafe_mul(2, gamma), 10**14)
    )

    # c = 10**32*3 + 4*gamma*10**14 + gamma2/10**4 + 4*ANN*gamma2*x_j/D/10000/4/10**4 - 4*ANN*gamma2/10000/4/10**4
    c: int256 = (
        unsafe_mul(10**32, convert(3, int256))
        + unsafe_mul(unsafe_mul(4, gamma), 10**14)
        + unsafe_div(gamma2, 10**4)
        + unsafe_div(unsafe_div(unsafe_mul(4, ann_gamma2), 400000000) * x_j, D)
        - unsafe_div(unsafe_mul(4, ann_gamma2), 400000000)
    )

    # d = -(10**18+gamma)**2 / 10**4
    d: int256 = -unsafe_div(unsafe_add(10**18, gamma) ** 2, 10**4)

    # delta0: int256 = 3*a*c/b - b
    delta0: int256 = 3 * a * c / b - b  # safediv by b

    # delta1: int256 = 9*a*c/b - 2*b - 27*a**2/b*d/b
    delta1: int256 = 3 * delta0 + b - 27*a**2/b*d/b

    divider: int256 = 1
    threshold: int256 = min(min(abs(delta0), abs(delta1)), a)
    if threshold > 10**48:
        divider = 10**30
    elif threshold > 10**46:
        divider = 10**28
    elif threshold > 10**44:
        divider = 10**26
    elif threshold > 10**42:
        divider = 10**24
    elif threshold > 10**40:
        divider = 10**22
    elif threshold > 10**38:
        divider = 10**20
    elif threshold > 10**36:
        divider = 10**18
    elif threshold > 10**34:
        divider = 10**16
    elif threshold > 10**32:
        divider = 10**14
    elif threshold > 10**30:
        divider = 10**12
    elif threshold > 10**28:
        divider = 10**10
    elif threshold > 10**26:
        divider = 10**8
    elif threshold > 10**24:
        divider = 10**6
    elif threshold > 10**20:
        divider = 10**2

    a = unsafe_div(a, divider)
    b = unsafe_div(b, divider)
    c = unsafe_div(c, divider)
    d = unsafe_div(d, divider)

    # delta0 = 3*a*c/b - b: here we can do more unsafe ops now:
    delta0 = unsafe_div(unsafe_mul(unsafe_mul(3, a), c), b) - b

    # delta1 = 9*a*c/b - 2*b - 27*a**2/b*d/b
    delta1 = 3 * delta0 + b - unsafe_div(unsafe_mul(unsafe_div(unsafe_mul(27, a**2), b), d), b)

    # sqrt_arg: int256 = delta1**2 + 4*delta0**2/b*delta0
    sqrt_arg: int256 = delta1**2 + unsafe_mul(unsafe_div(4*delta0**2, b), delta0)
    sqrt_val: int256 = 0
    if sqrt_arg > 0:
        sqrt_val = convert(isqrt(convert(sqrt_arg, uint256)), int256)
    else:
        return [
            self._newton_y(_ANN, _gamma, _x, _D, i),
            0
        ]

    b_cbrt: int256 = 0
    if b > 0:
        b_cbrt = convert(self._cbrt(convert(b, uint256)), int256)
    else:
        b_cbrt = -convert(self._cbrt(convert(-b, uint256)), int256)

    second_cbrt: int256 = 0
    if delta1 > 0:
        # second_cbrt = convert(self._cbrt(convert((delta1 + sqrt_val), uint256) / 2), int256)
        second_cbrt = convert(self._cbrt(convert(unsafe_add(delta1, sqrt_val), uint256) / 2), int256)
    else:
        # second_cbrt = -convert(self._cbrt(convert(unsafe_sub(sqrt_val, delta1), uint256) / 2), int256)
        second_cbrt = -convert(self._cbrt(unsafe_div(convert(unsafe_sub(sqrt_val, delta1), uint256), 2)), int256)

    # C1: int256 = b_cbrt**2/10**18*second_cbrt/10**18
    C1: int256 = unsafe_div(unsafe_mul(unsafe_div(b_cbrt**2, 10**18), second_cbrt), 10**18)

    # root: int256 = (10**18*C1 - 10**18*b - 10**18*b*delta0/C1)/(3*a), keep 2 safe ops here.
    root: int256 = (unsafe_mul(10**18, C1) - unsafe_mul(10**18, b) - unsafe_mul(10**18, b)/C1*delta0)/unsafe_mul(3, a)

    # y_out: uint256[2] =  [
    #     convert(D**2/x_j*root/4/10**18, uint256),   # <--- y
    #     convert(root, uint256)  # <----------------------- K0Prev
    # ]
    y_out: uint256[2] = [convert(unsafe_div(unsafe_div(unsafe_mul(unsafe_div(D**2, x_j), root), 4), 10**18), uint256), convert(root, uint256)]

    frac: uint256 = unsafe_div(y_out[0] * 10**18, _D)
    assert (frac >= 10**16 - 1) and (frac < 10**20 + 1)  # dev: unsafe value for y

    return y_out
>>> soon  

exchange_received

TwoCrypto.exchange_received(i: uint256, j: uint256, dx: uint256, min_dy: uint256, receiver: address = msg.sender) -> uint256:

Warning

The transfer of coins into the pool and then calling exchange_received is highly advised to be done in the same transaction. If not, other users or MEV bots may frontrun exchange_received, potentially stealing the coins.

Function to exchange dx amount of coin i for coin j and receive a minimum amount of min_dy. This function requires a transfer of dx amount of coin i to the pool prior to calling this function, as this exchange is based on the change of token balances in the pool. The pool will not call transferFrom and will only check if a surplus of coins[i] is greater than or equal to dx. Charged fee at current states is Pool.fee().

Returns: amount of output coin j received (uint256).

Emits: TokenExchange

Input Type Description
i uint256 Index value for the input coin.
j uint256 Index value for the output coin.
dx uint256 Amount of input coin being swapped in.
min_dy uint256 Minimum amount of output coin to receive.
receiver address Address to send output coin to. Defaults to msg.sender.
Source code
event TokenExchange:
    buyer: indexed(address)
    sold_id: uint256
    tokens_sold: uint256
    bought_id: uint256
    tokens_bought: uint256
    fee: uint256
    packed_price_scale: uint256

@external
@nonreentrant('lock')
def exchange_received(
    i: uint256,
    j: uint256,
    dx: uint256,
    min_dy: uint256,
    receiver: address = msg.sender,
) -> uint256:
    """
    @notice Exchange: but user must transfer dx amount of coin[i] tokens to pool first.
            Pool will not call transferFrom and will only check if a surplus of
            coins[i] is greater than or equal to `dx`.
    @dev Use-case is to reduce the number of redundant ERC20 token
        transfers in zaps. Primarily for dex-aggregators/arbitrageurs/searchers.
        Note for users: please transfer + exchange_received in 1 tx.
    @param i Index value for the input coin
    @param j Index value for the output coin
    @param dx Amount of input coin being swapped in
    @param min_dy Minimum amount of output coin to receive
    @param receiver Address to send the output coin to
    @return uint256 Amount of tokens at index j received by the `receiver`
    """
    # _transfer_in updates self.balances here:
    dx_received: uint256 = self._transfer_in(
        i,
        dx,
        msg.sender,
        True  # <---- expect_optimistic_transfer is set to True here.
    )

    # No ERC20 token transfers occur here:
    out: uint256[3] = self._exchange(
        i,
        j,
        dx_received,
        min_dy,
    )

    # _transfer_out updates self.balances here. Update to state occurs before
    # external calls:
    self._transfer_out(j, out[0], receiver)

    # log:
    log TokenExchange(msg.sender, i, dx_received, j, out[0], out[1], out[2])

    return out[0]

@internal
def _exchange(
    i: uint256,
    j: uint256,
    dx_received: uint256,
    min_dy: uint256,
) -> uint256[3]:

    assert i != j  # dev: coin index out of range
    assert dx_received > 0  # dev: do not exchange 0 coins

    A_gamma: uint256[2] = self._A_gamma()
    xp: uint256[N_COINS] = self.balances
    dy: uint256 = 0

    y: uint256 = xp[j]
    x0: uint256 = xp[i] - dx_received  # old xp[i]

    price_scale: uint256 = self.cached_price_scale
    xp = [
        xp[0] * PRECISIONS[0],
        unsafe_div(xp[1] * price_scale * PRECISIONS[1], PRECISION)
    ]

    # ----------- Update invariant if A, gamma are undergoing ramps ---------

    t: uint256 = self.future_A_gamma_time
    if t > block.timestamp:

        x0 *= PRECISIONS[i]

        if i > 0:
            x0 = unsafe_div(x0 * price_scale, PRECISION)

        x1: uint256 = xp[i]  # <------------------ Back up old value in xp ...
        xp[i] = x0                                                         # |
        self.D = MATH.newton_D(A_gamma[0], A_gamma[1], xp, 0)              # |
        xp[i] = x1  # <-------------------------------------- ... and restore.

    # ----------------------- Calculate dy and fees --------------------------

    D: uint256 = self.D
    y_out: uint256[2] = MATH.get_y(A_gamma[0], A_gamma[1], xp, D, j)
    dy = xp[j] - y_out[0]
    xp[j] -= dy
    dy -= 1

    if j > 0:
        dy = dy * PRECISION / price_scale
    dy /= PRECISIONS[j]

    fee: uint256 = unsafe_div(self._fee(xp) * dy, 10**10)
    dy -= fee  # <--------------------- Subtract fee from the outgoing amount.
    assert dy >= min_dy, "Slippage"
    y -= dy

    y *= PRECISIONS[j]
    if j > 0:
        y = unsafe_div(y * price_scale, PRECISION)
    xp[j] = y  # <------------------------------------------------- Update xp.

    # ------ Tweak price_scale with good initial guess for newton_D ----------

    price_scale = self.tweak_price(A_gamma, xp, 0, y_out[1])

    return [dy, fee, price_scale]
@external
@view
def newton_D(ANN: uint256, gamma: uint256, x_unsorted: uint256[N_COINS], K0_prev: uint256 = 0) -> uint256:
    """
    Finding the invariant using Newton method.
    ANN is higher by the factor A_MULTIPLIER
    ANN is already A * N**N
    """

    # Safety checks
    assert ANN > MIN_A - 1 and ANN < MAX_A + 1  # dev: unsafe values A
    assert gamma > MIN_GAMMA - 1 and gamma < MAX_GAMMA + 1  # dev: unsafe values gamma

    # Initial value of invariant D is that for constant-product invariant
    x: uint256[N_COINS] = x_unsorted
    if x[0] < x[1]:
        x = [x_unsorted[1], x_unsorted[0]]

    assert x[0] > 10**9 - 1 and x[0] < 10**15 * 10**18 + 1  # dev: unsafe values x[0]
    assert unsafe_div(x[1] * 10**18, x[0]) > 10**14 - 1  # dev: unsafe values x[i] (input)

    S: uint256 = unsafe_add(x[0], x[1])  # can unsafe add here because we checked x[0] bounds

    D: uint256 = 0
    if K0_prev == 0:
        D = N_COINS * isqrt(unsafe_mul(x[0], x[1]))
    else:
        # D = isqrt(x[0] * x[1] * 4 / K0_prev * 10**18)
        D = isqrt(unsafe_mul(unsafe_div(unsafe_mul(unsafe_mul(4, x[0]), x[1]), K0_prev), 10**18))
        if S < D:
            D = S

    __g1k0: uint256 = gamma + 10**18
    diff: uint256 = 0

    for i in range(255):
        D_prev: uint256 = D
        assert D > 0
        # Unsafe division by D and D_prev is now safe

        # K0: uint256 = 10**18
        # for _x in x:
        #     K0 = K0 * _x * N_COINS / D
        # collapsed for 2 coins
        K0: uint256 = unsafe_div(unsafe_div((10**18 * N_COINS**2) * x[0], D) * x[1], D)

        _g1k0: uint256 = __g1k0
        if _g1k0 > K0:
            _g1k0 = unsafe_add(unsafe_sub(_g1k0, K0), 1)  # > 0
        else:
            _g1k0 = unsafe_add(unsafe_sub(K0, _g1k0), 1)  # > 0

        # D / (A * N**N) * _g1k0**2 / gamma**2
        mul1: uint256 = unsafe_div(unsafe_div(unsafe_div(10**18 * D, gamma) * _g1k0, gamma) * _g1k0 * A_MULTIPLIER, ANN)

        # 2*N*K0 / _g1k0
        mul2: uint256 = unsafe_div(((2 * 10**18) * N_COINS) * K0, _g1k0)

        # calculate neg_fprime. here K0 > 0 is being validated (safediv).
        neg_fprime: uint256 = (S + unsafe_div(S * mul2, 10**18)) + mul1 * N_COINS / K0 - unsafe_div(mul2 * D, 10**18)

        # D -= f / fprime; neg_fprime safediv being validated
        D_plus: uint256 = D * (neg_fprime + S) / neg_fprime
        D_minus: uint256 = unsafe_div(D * D,  neg_fprime)
        if 10**18 > K0:
            D_minus += unsafe_div(unsafe_div(D * unsafe_div(mul1, neg_fprime), 10**18) * unsafe_sub(10**18, K0), K0)
        else:
            D_minus -= unsafe_div(unsafe_div(D * unsafe_div(mul1, neg_fprime), 10**18) * unsafe_sub(K0, 10**18), K0)

        if D_plus > D_minus:
            D = unsafe_sub(D_plus, D_minus)
        else:
            D = unsafe_div(unsafe_sub(D_minus, D_plus), 2)

        if D > D_prev:
            diff = unsafe_sub(D, D_prev)
        else:
            diff = unsafe_sub(D_prev, D)

        if diff * 10**14 < max(10**16, D):  # Could reduce precision for gas efficiency here

            for _x in x:
                frac: uint256 = _x * 10**18 / D
                assert (frac >= 10**16 - 1) and (frac < 10**20 + 1)  # dev: unsafe values x[i]
            return D

    raise "Did not converge"

@external
@pure
def get_y(
    _ANN: uint256,
    _gamma: uint256,
    _x: uint256[N_COINS],
    _D: uint256,
    i: uint256
) -> uint256[2]:

    # Safety checks
    assert _ANN > MIN_A - 1 and _ANN < MAX_A + 1  # dev: unsafe values A
    assert _gamma > MIN_GAMMA - 1 and _gamma < MAX_GAMMA + 1  # dev: unsafe values gamma
    assert _D > 10**17 - 1 and _D < 10**15 * 10**18 + 1 # dev: unsafe values D

    ANN: int256 = convert(_ANN, int256)
    gamma: int256 = convert(_gamma, int256)
    D: int256 = convert(_D, int256)
    x_j: int256 = convert(_x[1 - i], int256)
    gamma2: int256 = unsafe_mul(gamma, gamma)

    # savediv by x_j done here:
    y: int256 = D**2 / (x_j * N_COINS**2)

    # K0_i: int256 = (10**18 * N_COINS) * x_j / D
    K0_i: int256 = unsafe_div(10**18 * N_COINS * x_j, D)
    assert (K0_i > 10**16 * N_COINS - 1) and (K0_i < 10**20 * N_COINS + 1)  # dev: unsafe values x[i]

    ann_gamma2: int256 = ANN * gamma2

    # a = 10**36 / N_COINS**2
    a: int256 = 10**32

    # b = ANN*D*gamma2/4/10000/x_j/10**4 - 10**32*3 - 2*gamma*10**14
    b: int256 = (
        D*ann_gamma2/400000000/x_j
        - convert(unsafe_mul(10**32, 3), int256)
        - unsafe_mul(unsafe_mul(2, gamma), 10**14)
    )

    # c = 10**32*3 + 4*gamma*10**14 + gamma2/10**4 + 4*ANN*gamma2*x_j/D/10000/4/10**4 - 4*ANN*gamma2/10000/4/10**4
    c: int256 = (
        unsafe_mul(10**32, convert(3, int256))
        + unsafe_mul(unsafe_mul(4, gamma), 10**14)
        + unsafe_div(gamma2, 10**4)
        + unsafe_div(unsafe_div(unsafe_mul(4, ann_gamma2), 400000000) * x_j, D)
        - unsafe_div(unsafe_mul(4, ann_gamma2), 400000000)
    )

    # d = -(10**18+gamma)**2 / 10**4
    d: int256 = -unsafe_div(unsafe_add(10**18, gamma) ** 2, 10**4)

    # delta0: int256 = 3*a*c/b - b
    delta0: int256 = 3 * a * c / b - b  # safediv by b

    # delta1: int256 = 9*a*c/b - 2*b - 27*a**2/b*d/b
    delta1: int256 = 3 * delta0 + b - 27*a**2/b*d/b

    divider: int256 = 1
    threshold: int256 = min(min(abs(delta0), abs(delta1)), a)
    if threshold > 10**48:
        divider = 10**30
    elif threshold > 10**46:
        divider = 10**28
    elif threshold > 10**44:
        divider = 10**26
    elif threshold > 10**42:
        divider = 10**24
    elif threshold > 10**40:
        divider = 10**22
    elif threshold > 10**38:
        divider = 10**20
    elif threshold > 10**36:
        divider = 10**18
    elif threshold > 10**34:
        divider = 10**16
    elif threshold > 10**32:
        divider = 10**14
    elif threshold > 10**30:
        divider = 10**12
    elif threshold > 10**28:
        divider = 10**10
    elif threshold > 10**26:
        divider = 10**8
    elif threshold > 10**24:
        divider = 10**6
    elif threshold > 10**20:
        divider = 10**2

    a = unsafe_div(a, divider)
    b = unsafe_div(b, divider)
    c = unsafe_div(c, divider)
    d = unsafe_div(d, divider)

    # delta0 = 3*a*c/b - b: here we can do more unsafe ops now:
    delta0 = unsafe_div(unsafe_mul(unsafe_mul(3, a), c), b) - b

    # delta1 = 9*a*c/b - 2*b - 27*a**2/b*d/b
    delta1 = 3 * delta0 + b - unsafe_div(unsafe_mul(unsafe_div(unsafe_mul(27, a**2), b), d), b)

    # sqrt_arg: int256 = delta1**2 + 4*delta0**2/b*delta0
    sqrt_arg: int256 = delta1**2 + unsafe_mul(unsafe_div(4*delta0**2, b), delta0)
    sqrt_val: int256 = 0
    if sqrt_arg > 0:
        sqrt_val = convert(isqrt(convert(sqrt_arg, uint256)), int256)
    else:
        return [
            self._newton_y(_ANN, _gamma, _x, _D, i),
            0
        ]

    b_cbrt: int256 = 0
    if b > 0:
        b_cbrt = convert(self._cbrt(convert(b, uint256)), int256)
    else:
        b_cbrt = -convert(self._cbrt(convert(-b, uint256)), int256)

    second_cbrt: int256 = 0
    if delta1 > 0:
        # second_cbrt = convert(self._cbrt(convert((delta1 + sqrt_val), uint256) / 2), int256)
        second_cbrt = convert(self._cbrt(convert(unsafe_add(delta1, sqrt_val), uint256) / 2), int256)
    else:
        # second_cbrt = -convert(self._cbrt(convert(unsafe_sub(sqrt_val, delta1), uint256) / 2), int256)
        second_cbrt = -convert(self._cbrt(unsafe_div(convert(unsafe_sub(sqrt_val, delta1), uint256), 2)), int256)

    # C1: int256 = b_cbrt**2/10**18*second_cbrt/10**18
    C1: int256 = unsafe_div(unsafe_mul(unsafe_div(b_cbrt**2, 10**18), second_cbrt), 10**18)

    # root: int256 = (10**18*C1 - 10**18*b - 10**18*b*delta0/C1)/(3*a), keep 2 safe ops here.
    root: int256 = (unsafe_mul(10**18, C1) - unsafe_mul(10**18, b) - unsafe_mul(10**18, b)/C1*delta0)/unsafe_mul(3, a)

    # y_out: uint256[2] =  [
    #     convert(D**2/x_j*root/4/10**18, uint256),   # <--- y
    #     convert(root, uint256)  # <----------------------- K0Prev
    # ]
    y_out: uint256[2] = [convert(unsafe_div(unsafe_div(unsafe_mul(unsafe_div(D**2, x_j), root), 4), 10**18), uint256), convert(root, uint256)]

    frac: uint256 = unsafe_div(y_out[0] * 10**18, _D)
    assert (frac >= 10**16 - 1) and (frac < 10**20 + 1)  # dev: unsafe value for y

    return y_out
>>> soon  

get_dy

TwoCrypto.get_dy(i: uint256, j: uint256, dx: uint256) -> uint256:

Getter for the received amount of coin j for swapping in dx amount of coin i. This method includes fees.

Returns: exact amount of output coin j (uint256).

Input Type Description
i uint256 Index of input token.
j uint256 Index of output token.
dx uint256 Amount of input tokens.
Source code
@external
@view
def get_dy(i: uint256, j: uint256, dx: uint256) -> uint256:
    """
    @notice Get amount of coin[j] tokens received for swapping in dx amount of coin[i]
    @dev Includes fee.
    @param i index of input token. Check pool.coins(i) to get coin address at ith index
    @param j index of output token
    @param dx amount of input coin[i] tokens
    @return uint256 Exact amount of output j tokens for dx amount of i input tokens.
    """
    view_contract: address = factory.views_implementation()
    return Views(view_contract).get_dy(i, j, dx, self)
@external
@view
def get_dy(
    i: uint256, j: uint256, dx: uint256, swap: address
) -> uint256:

    dy: uint256 = 0
    xp: uint256[N_COINS] = empty(uint256[N_COINS])

    # dy = (get_y(x + dx) - y) * (1 - fee)
    dy, xp = self._get_dy_nofee(i, j, dx, swap)
    dy -= Curve(swap).fee_calc(xp) * dy / 10**10

    return dy

@internal
@view
def _get_dy_nofee(
    i: uint256, j: uint256, dx: uint256, swap: address
) -> (uint256, uint256[N_COINS]):

    assert i != j and i < N_COINS and j < N_COINS, "coin index out of range"
    assert dx > 0, "do not exchange 0 coins"

    math: Math = Curve(swap).MATH()

    xp: uint256[N_COINS] = empty(uint256[N_COINS])
    precisions: uint256[N_COINS] = empty(uint256[N_COINS])
    price_scale: uint256 = 0
    D: uint256 = 0
    token_supply: uint256 = 0
    A: uint256 = 0
    gamma: uint256 = 0

    xp, D, token_supply, price_scale, A, gamma, precisions = self._prep_calc(swap)

    # adjust xp with input dx
    xp[i] += dx
    xp = [
        xp[0] * precisions[0],
        xp[1] * price_scale * precisions[1] / PRECISION
    ]

    y_out: uint256[2] = math.get_y(A, gamma, xp, D, j)

    dy: uint256 = xp[j] - y_out[0] - 1
    xp[j] = y_out[0]
    if j > 0:
        dy = dy * PRECISION / price_scale
    dy /= precisions[j]

    return dy, xp
@external
@pure
def get_y(
    _ANN: uint256,
    _gamma: uint256,
    _x: uint256[N_COINS],
    _D: uint256,
    i: uint256
) -> uint256[2]:

    # Safety checks
    assert _ANN > MIN_A - 1 and _ANN < MAX_A + 1  # dev: unsafe values A
    assert _gamma > MIN_GAMMA - 1 and _gamma < MAX_GAMMA + 1  # dev: unsafe values gamma
    assert _D > 10**17 - 1 and _D < 10**15 * 10**18 + 1 # dev: unsafe values D

    ANN: int256 = convert(_ANN, int256)
    gamma: int256 = convert(_gamma, int256)
    D: int256 = convert(_D, int256)
    x_j: int256 = convert(_x[1 - i], int256)
    gamma2: int256 = unsafe_mul(gamma, gamma)

    # savediv by x_j done here:
    y: int256 = D**2 / (x_j * N_COINS**2)

    # K0_i: int256 = (10**18 * N_COINS) * x_j / D
    K0_i: int256 = unsafe_div(10**18 * N_COINS * x_j, D)
    assert (K0_i > 10**16 * N_COINS - 1) and (K0_i < 10**20 * N_COINS + 1)  # dev: unsafe values x[i]

    ann_gamma2: int256 = ANN * gamma2

    # a = 10**36 / N_COINS**2
    a: int256 = 10**32

    # b = ANN*D*gamma2/4/10000/x_j/10**4 - 10**32*3 - 2*gamma*10**14
    b: int256 = (
        D*ann_gamma2/400000000/x_j
        - convert(unsafe_mul(10**32, 3), int256)
        - unsafe_mul(unsafe_mul(2, gamma), 10**14)
    )

    # c = 10**32*3 + 4*gamma*10**14 + gamma2/10**4 + 4*ANN*gamma2*x_j/D/10000/4/10**4 - 4*ANN*gamma2/10000/4/10**4
    c: int256 = (
        unsafe_mul(10**32, convert(3, int256))
        + unsafe_mul(unsafe_mul(4, gamma), 10**14)
        + unsafe_div(gamma2, 10**4)
        + unsafe_div(unsafe_div(unsafe_mul(4, ann_gamma2), 400000000) * x_j, D)
        - unsafe_div(unsafe_mul(4, ann_gamma2), 400000000)
    )

    # d = -(10**18+gamma)**2 / 10**4
    d: int256 = -unsafe_div(unsafe_add(10**18, gamma) ** 2, 10**4)

    # delta0: int256 = 3*a*c/b - b
    delta0: int256 = 3 * a * c / b - b  # safediv by b

    # delta1: int256 = 9*a*c/b - 2*b - 27*a**2/b*d/b
    delta1: int256 = 3 * delta0 + b - 27*a**2/b*d/b

    divider: int256 = 1
    threshold: int256 = min(min(abs(delta0), abs(delta1)), a)
    if threshold > 10**48:
        divider = 10**30
    elif threshold > 10**46:
        divider = 10**28
    elif threshold > 10**44:
        divider = 10**26
    elif threshold > 10**42:
        divider = 10**24
    elif threshold > 10**40:
        divider = 10**22
    elif threshold > 10**38:
        divider = 10**20
    elif threshold > 10**36:
        divider = 10**18
    elif threshold > 10**34:
        divider = 10**16
    elif threshold > 10**32:
        divider = 10**14
    elif threshold > 10**30:
        divider = 10**12
    elif threshold > 10**28:
        divider = 10**10
    elif threshold > 10**26:
        divider = 10**8
    elif threshold > 10**24:
        divider = 10**6
    elif threshold > 10**20:
        divider = 10**2

    a = unsafe_div(a, divider)
    b = unsafe_div(b, divider)
    c = unsafe_div(c, divider)
    d = unsafe_div(d, divider)

    # delta0 = 3*a*c/b - b: here we can do more unsafe ops now:
    delta0 = unsafe_div(unsafe_mul(unsafe_mul(3, a), c), b) - b

    # delta1 = 9*a*c/b - 2*b - 27*a**2/b*d/b
    delta1 = 3 * delta0 + b - unsafe_div(unsafe_mul(unsafe_div(unsafe_mul(27, a**2), b), d), b)

    # sqrt_arg: int256 = delta1**2 + 4*delta0**2/b*delta0
    sqrt_arg: int256 = delta1**2 + unsafe_mul(unsafe_div(4*delta0**2, b), delta0)
    sqrt_val: int256 = 0
    if sqrt_arg > 0:
        sqrt_val = convert(isqrt(convert(sqrt_arg, uint256)), int256)
    else:
        return [
            self._newton_y(_ANN, _gamma, _x, _D, i),
            0
        ]

    b_cbrt: int256 = 0
    if b > 0:
        b_cbrt = convert(self._cbrt(convert(b, uint256)), int256)
    else:
        b_cbrt = -convert(self._cbrt(convert(-b, uint256)), int256)

    second_cbrt: int256 = 0
    if delta1 > 0:
        # second_cbrt = convert(self._cbrt(convert((delta1 + sqrt_val), uint256) / 2), int256)
        second_cbrt = convert(self._cbrt(convert(unsafe_add(delta1, sqrt_val), uint256) / 2), int256)
    else:
        # second_cbrt = -convert(self._cbrt(convert(unsafe_sub(sqrt_val, delta1), uint256) / 2), int256)
        second_cbrt = -convert(self._cbrt(unsafe_div(convert(unsafe_sub(sqrt_val, delta1), uint256), 2)), int256)

    # C1: int256 = b_cbrt**2/10**18*second_cbrt/10**18
    C1: int256 = unsafe_div(unsafe_mul(unsafe_div(b_cbrt**2, 10**18), second_cbrt), 10**18)

    # root: int256 = (10**18*C1 - 10**18*b - 10**18*b*delta0/C1)/(3*a), keep 2 safe ops here.
    root: int256 = (unsafe_mul(10**18, C1) - unsafe_mul(10**18, b) - unsafe_mul(10**18, b)/C1*delta0)/unsafe_mul(3, a)

    # y_out: uint256[2] =  [
    #     convert(D**2/x_j*root/4/10**18, uint256),   # <--- y
    #     convert(root, uint256)  # <----------------------- K0Prev
    # ]
    y_out: uint256[2] = [convert(unsafe_div(unsafe_div(unsafe_mul(unsafe_div(D**2, x_j), root), 4), 10**18), uint256), convert(root, uint256)]

    frac: uint256 = unsafe_div(y_out[0] * 10**18, _D)
    assert (frac >= 10**16 - 1) and (frac < 10**20 + 1)  # dev: unsafe value for y

    return y_out
>>> soon  

get_dx

TwoCrypto.get_dx(i: uint256, j: uint256, dy: uint256) -> uint256:

Getter for the required amount of coin i to input for swapping out dy amount of token j.

Returns: amount of input coin i needed (uint256).

Input Type Description
i uint256 Index of input token.
j uint256 Index of output token.
dy uint256 Amount of output tokens.
Source code
@external
@view
def get_dx(i: uint256, j: uint256, dy: uint256) -> uint256:
    """
    @notice Get amount of coin[i] tokens to input for swapping out dy amount
            of coin[j]
    @dev This is an approximate method, and returns estimates close to the input
        amount. Expensive to call on-chain.
    @param i index of input token. Check pool.coins(i) to get coin address at
        ith index
    @param j index of output token
    @param dy amount of input coin[j] tokens received
    @return uint256 Approximate amount of input i tokens to get dy amount of j tokens.
    """
    view_contract: address = factory.views_implementation()
    return Views(view_contract).get_dx(i, j, dy, self)
@view
@external
def get_dx(
    i: uint256, j: uint256, dy: uint256, swap: address
) -> uint256:

    dx: uint256 = 0
    xp: uint256[N_COINS] = empty(uint256[N_COINS])
    fee_dy: uint256 = 0
    _dy: uint256 = dy

    # for more precise dx (but never exact), increase num loops
    for k in range(5):
        dx, xp = self._get_dx_fee(i, j, _dy, swap)
        fee_dy = Curve(swap).fee_calc(xp) * _dy / 10**10
        _dy = dy + fee_dy + 1

    return dx

@internal
@view
def _get_dx_fee(
    i: uint256, j: uint256, dy: uint256, swap: address
) -> (uint256, uint256[N_COINS]):

    # here, dy must include fees (and 1 wei offset)

    assert i != j and i < N_COINS and j < N_COINS, "coin index out of range"
    assert dy > 0, "do not exchange out 0 coins"

    math: Math = Curve(swap).MATH()

    xp: uint256[N_COINS] = empty(uint256[N_COINS])
    precisions: uint256[N_COINS] = empty(uint256[N_COINS])
    price_scale: uint256 = 0
    D: uint256 = 0
    token_supply: uint256 = 0
    A: uint256 = 0
    gamma: uint256 = 0

    xp, D, token_supply, price_scale, A, gamma, precisions = self._prep_calc(swap)

    # adjust xp with output dy. dy contains fee element, which we handle later
    # (hence this internal method is called _get_dx_fee)
    xp[j] -= dy
    xp = [xp[0] * precisions[0], xp[1] * price_scale * precisions[1] / PRECISION]

    x_out: uint256[2] = math.get_y(A, gamma, xp, D, i)
    dx: uint256 = x_out[0] - xp[i]
    xp[i] = x_out[0]

    if i > 0:
        dx = dx * PRECISION / price_scale
    dx /= precisions[i]

    return dx, xp
@external
@pure
def get_y(
    _ANN: uint256,
    _gamma: uint256,
    _x: uint256[N_COINS],
    _D: uint256,
    i: uint256
) -> uint256[2]:

    # Safety checks
    assert _ANN > MIN_A - 1 and _ANN < MAX_A + 1  # dev: unsafe values A
    assert _gamma > MIN_GAMMA - 1 and _gamma < MAX_GAMMA + 1  # dev: unsafe values gamma
    assert _D > 10**17 - 1 and _D < 10**15 * 10**18 + 1 # dev: unsafe values D

    ANN: int256 = convert(_ANN, int256)
    gamma: int256 = convert(_gamma, int256)
    D: int256 = convert(_D, int256)
    x_j: int256 = convert(_x[1 - i], int256)
    gamma2: int256 = unsafe_mul(gamma, gamma)

    # savediv by x_j done here:
    y: int256 = D**2 / (x_j * N_COINS**2)

    # K0_i: int256 = (10**18 * N_COINS) * x_j / D
    K0_i: int256 = unsafe_div(10**18 * N_COINS * x_j, D)
    assert (K0_i > 10**16 * N_COINS - 1) and (K0_i < 10**20 * N_COINS + 1)  # dev: unsafe values x[i]

    ann_gamma2: int256 = ANN * gamma2

    # a = 10**36 / N_COINS**2
    a: int256 = 10**32

    # b = ANN*D*gamma2/4/10000/x_j/10**4 - 10**32*3 - 2*gamma*10**14
    b: int256 = (
        D*ann_gamma2/400000000/x_j
        - convert(unsafe_mul(10**32, 3), int256)
        - unsafe_mul(unsafe_mul(2, gamma), 10**14)
    )

    # c = 10**32*3 + 4*gamma*10**14 + gamma2/10**4 + 4*ANN*gamma2*x_j/D/10000/4/10**4 - 4*ANN*gamma2/10000/4/10**4
    c: int256 = (
        unsafe_mul(10**32, convert(3, int256))
        + unsafe_mul(unsafe_mul(4, gamma), 10**14)
        + unsafe_div(gamma2, 10**4)
        + unsafe_div(unsafe_div(unsafe_mul(4, ann_gamma2), 400000000) * x_j, D)
        - unsafe_div(unsafe_mul(4, ann_gamma2), 400000000)
    )

    # d = -(10**18+gamma)**2 / 10**4
    d: int256 = -unsafe_div(unsafe_add(10**18, gamma) ** 2, 10**4)

    # delta0: int256 = 3*a*c/b - b
    delta0: int256 = 3 * a * c / b - b  # safediv by b

    # delta1: int256 = 9*a*c/b - 2*b - 27*a**2/b*d/b
    delta1: int256 = 3 * delta0 + b - 27*a**2/b*d/b

    divider: int256 = 1
    threshold: int256 = min(min(abs(delta0), abs(delta1)), a)
    if threshold > 10**48:
        divider = 10**30
    elif threshold > 10**46:
        divider = 10**28
    elif threshold > 10**44:
        divider = 10**26
    elif threshold > 10**42:
        divider = 10**24
    elif threshold > 10**40:
        divider = 10**22
    elif threshold > 10**38:
        divider = 10**20
    elif threshold > 10**36:
        divider = 10**18
    elif threshold > 10**34:
        divider = 10**16
    elif threshold > 10**32:
        divider = 10**14
    elif threshold > 10**30:
        divider = 10**12
    elif threshold > 10**28:
        divider = 10**10
    elif threshold > 10**26:
        divider = 10**8
    elif threshold > 10**24:
        divider = 10**6
    elif threshold > 10**20:
        divider = 10**2

    a = unsafe_div(a, divider)
    b = unsafe_div(b, divider)
    c = unsafe_div(c, divider)
    d = unsafe_div(d, divider)

    # delta0 = 3*a*c/b - b: here we can do more unsafe ops now:
    delta0 = unsafe_div(unsafe_mul(unsafe_mul(3, a), c), b) - b

    # delta1 = 9*a*c/b - 2*b - 27*a**2/b*d/b
    delta1 = 3 * delta0 + b - unsafe_div(unsafe_mul(unsafe_div(unsafe_mul(27, a**2), b), d), b)

    # sqrt_arg: int256 = delta1**2 + 4*delta0**2/b*delta0
    sqrt_arg: int256 = delta1**2 + unsafe_mul(unsafe_div(4*delta0**2, b), delta0)
    sqrt_val: int256 = 0
    if sqrt_arg > 0:
        sqrt_val = convert(isqrt(convert(sqrt_arg, uint256)), int256)
    else:
        return [
            self._newton_y(_ANN, _gamma, _x, _D, i),
            0
        ]

    b_cbrt: int256 = 0
    if b > 0:
        b_cbrt = convert(self._cbrt(convert(b, uint256)), int256)
    else:
        b_cbrt = -convert(self._cbrt(convert(-b, uint256)), int256)

    second_cbrt: int256 = 0
    if delta1 > 0:
        # second_cbrt = convert(self._cbrt(convert((delta1 + sqrt_val), uint256) / 2), int256)
        second_cbrt = convert(self._cbrt(convert(unsafe_add(delta1, sqrt_val), uint256) / 2), int256)
    else:
        # second_cbrt = -convert(self._cbrt(convert(unsafe_sub(sqrt_val, delta1), uint256) / 2), int256)
        second_cbrt = -convert(self._cbrt(unsafe_div(convert(unsafe_sub(sqrt_val, delta1), uint256), 2)), int256)

    # C1: int256 = b_cbrt**2/10**18*second_cbrt/10**18
    C1: int256 = unsafe_div(unsafe_mul(unsafe_div(b_cbrt**2, 10**18), second_cbrt), 10**18)

    # root: int256 = (10**18*C1 - 10**18*b - 10**18*b*delta0/C1)/(3*a), keep 2 safe ops here.
    root: int256 = (unsafe_mul(10**18, C1) - unsafe_mul(10**18, b) - unsafe_mul(10**18, b)/C1*delta0)/unsafe_mul(3, a)

    # y_out: uint256[2] =  [
    #     convert(D**2/x_j*root/4/10**18, uint256),   # <--- y
    #     convert(root, uint256)  # <----------------------- K0Prev
    # ]
    y_out: uint256[2] = [convert(unsafe_div(unsafe_div(unsafe_mul(unsafe_div(D**2, x_j), root), 4), 10**18), uint256), convert(root, uint256)]

    frac: uint256 = unsafe_div(y_out[0] * 10**18, _D)
    assert (frac >= 10**16 - 1) and (frac < 10**20 + 1)  # dev: unsafe value for y

    return y_out
>>> soon

fee_calc

TwoCrypto.fee_calc(xp: uint256[N_COINS]) -> uint256:

Getter for the charged exchange fee by the pool at the current state.

Returns: fee (uint256).

Input Type Description
xp uint256[N_COINS] Pool balances multiplied by the coin precisions.
Source code
@external
@view
def fee_calc(xp: uint256[N_COINS]) -> uint256:  # <----- For by view contract.
    """
    @notice Returns the fee charged by the pool at current state.
    @param xp The current balances of the pool multiplied by coin precisions.
    @return uint256 Fee value.
    """
    return self._fee(xp)

@internal
@view
def _fee(xp: uint256[N_COINS]) -> uint256:

    fee_params: uint256[3] = self._unpack_3(self.packed_fee_params)
    f: uint256 = xp[0] + xp[1]
    f = fee_params[2] * 10**18 / (
        fee_params[2] + 10**18 -
        (10**18 * N_COINS**N_COINS) * xp[0] / f * xp[1] / f
    )

    return unsafe_div(
        fee_params[0] * f + fee_params[1] * (10**18 - f),
        10**18
    )
>>> soon  

Adding and Removing Liquidity

The twocrypto-ng implementation utilizes the usual methods to add and remove liquidity.

Adding liquidity can be done via the add_liquidity method. The code uses a list of unsigned integers uint256[N_COINS] as input for the pools underlying tokens to add. Any proportion is possible. For example, adding fully single-sided can be done using [0, 1e18] or [1e18, 0], but again, any variation is possible, e.g., [1e18, 1e19].

Removing liquidity can be done in two different ways. Either withdraw the underlying assets in a balanced proportion using the remove_liquidity method or fully single-sided in a single underlying token using remove_liquidity_one_coin.

add_liquidity

TwoCrypto.add_liquidity(amounts: uint256[N_COINS], min_mint_amount: uint256, receiver: address = msg.sender) -> uint256:

Function to add liquidity to the pool and mint the corresponding LP tokens.

Returns: amount of LP tokens received (uint256).

Emits: AddLiquidity

Input Type Description
amounts uint256[N_COINS] Amount of each coin to add.
min_mint_amount uint256 Minimum amount of LP tokens to mint.
receiver address Receiver of the LP tokens; defaults to msg.sender.
Source code
event AddLiquidity:
    provider: indexed(address)
    token_amounts: uint256[N_COINS]
    fee: uint256
    token_supply: uint256
    packed_price_scale: uint256

@external
@nonreentrant("lock")
def add_liquidity(
    amounts: uint256[N_COINS],
    min_mint_amount: uint256,
    receiver: address = msg.sender
) -> uint256:
    """
    @notice Adds liquidity into the pool.
    @param amounts Amounts of each coin to add.
    @param min_mint_amount Minimum amount of LP to mint.
    @param receiver Address to send the LP tokens to. Default is msg.sender
    @return uint256 Amount of LP tokens received by the `receiver
    """

    A_gamma: uint256[2] = self._A_gamma()
    xp: uint256[N_COINS] = self.balances
    amountsp: uint256[N_COINS] = empty(uint256[N_COINS])
    d_token: uint256 = 0
    d_token_fee: uint256 = 0
    old_D: uint256 = 0

    assert amounts[0] + amounts[1] > 0  # dev: no coins to add

    # --------------------- Get prices, balances -----------------------------

    price_scale: uint256 = self.cached_price_scale

    # -------------------------------------- Update balances and calculate xp.
    xp_old: uint256[N_COINS] = xp
    amounts_received: uint256[N_COINS] = empty(uint256[N_COINS])

    ########################## TRANSFER IN <-------

    for i in range(N_COINS):
        if amounts[i] > 0:
            # Updates self.balances here:
            amounts_received[i] = self._transfer_in(
                i,
                amounts[i],
                msg.sender,
                False,  # <--------------------- Disable optimistic transfers.
            )
            xp[i] = xp[i] + amounts_received[i]

    xp = [
        xp[0] * PRECISIONS[0],
        unsafe_div(xp[1] * price_scale * PRECISIONS[1], PRECISION)
    ]
    xp_old = [
        xp_old[0] * PRECISIONS[0],
        unsafe_div(xp_old[1] * price_scale * PRECISIONS[1], PRECISION)
    ]

    for i in range(N_COINS):
        if amounts_received[i] > 0:
            amountsp[i] = xp[i] - xp_old[i]

    # -------------------- Calculate LP tokens to mint -----------------------

    if self.future_A_gamma_time > block.timestamp:  # <--- A_gamma is ramping.

        # ----- Recalculate the invariant if A or gamma are undergoing a ramp.
        old_D = MATH.newton_D(A_gamma[0], A_gamma[1], xp_old, 0)

    else:

        old_D = self.D

    D: uint256 = MATH.newton_D(A_gamma[0], A_gamma[1], xp, 0)

    token_supply: uint256 = self.totalSupply
    if old_D > 0:
        d_token = token_supply * D / old_D - token_supply
    else:
        d_token = self.get_xcp(D, price_scale)  # <----- Making initial virtual price equal to 1.

    assert d_token > 0  # dev: nothing minted

    if old_D > 0:

        d_token_fee = (
            self._calc_token_fee(amountsp, xp) * d_token / 10**10 + 1
        )

        d_token -= d_token_fee
        token_supply += d_token
        self.mint(receiver, d_token)
        self.admin_lp_virtual_balance += unsafe_div(ADMIN_FEE * d_token_fee, 10**10)

        price_scale = self.tweak_price(A_gamma, xp, D, 0)

    else:

        # (re)instatiating an empty pool:

        self.D = D
        self.virtual_price = 10**18
        self.xcp_profit = 10**18
        self.xcp_profit_a = 10**18

        # Initialise xcp oracle here:
        self.cached_xcp_oracle = d_token  # <--- virtual_price * totalSupply / 10**18

        self.mint(receiver, d_token)

    assert d_token >= min_mint_amount, "Slippage"

    # ---------------------------------------------- Log and claim admin fees.

    log AddLiquidity(
        receiver,
        amounts_received,
        d_token_fee,
        token_supply,
        price_scale
    )

    return d_token
@external
@view
def newton_D(ANN: uint256, gamma: uint256, x_unsorted: uint256[N_COINS], K0_prev: uint256 = 0) -> uint256:
    """
    Finding the invariant using Newton method.
    ANN is higher by the factor A_MULTIPLIER
    ANN is already A * N**N
    """

    # Safety checks
    assert ANN > MIN_A - 1 and ANN < MAX_A + 1  # dev: unsafe values A
    assert gamma > MIN_GAMMA - 1 and gamma < MAX_GAMMA + 1  # dev: unsafe values gamma

    # Initial value of invariant D is that for constant-product invariant
    x: uint256[N_COINS] = x_unsorted
    if x[0] < x[1]:
        x = [x_unsorted[1], x_unsorted[0]]

    assert x[0] > 10**9 - 1 and x[0] < 10**15 * 10**18 + 1  # dev: unsafe values x[0]
    assert unsafe_div(x[1] * 10**18, x[0]) > 10**14 - 1  # dev: unsafe values x[i] (input)

    S: uint256 = unsafe_add(x[0], x[1])  # can unsafe add here because we checked x[0] bounds

    D: uint256 = 0
    if K0_prev == 0:
        D = N_COINS * isqrt(unsafe_mul(x[0], x[1]))
    else:
        # D = isqrt(x[0] * x[1] * 4 / K0_prev * 10**18)
        D = isqrt(unsafe_mul(unsafe_div(unsafe_mul(unsafe_mul(4, x[0]), x[1]), K0_prev), 10**18))
        if S < D:
            D = S

    __g1k0: uint256 = gamma + 10**18
    diff: uint256 = 0

    for i in range(255):
        D_prev: uint256 = D
        assert D > 0
        # Unsafe division by D and D_prev is now safe

        # K0: uint256 = 10**18
        # for _x in x:
        #     K0 = K0 * _x * N_COINS / D
        # collapsed for 2 coins
        K0: uint256 = unsafe_div(unsafe_div((10**18 * N_COINS**2) * x[0], D) * x[1], D)

        _g1k0: uint256 = __g1k0
        if _g1k0 > K0:
            _g1k0 = unsafe_add(unsafe_sub(_g1k0, K0), 1)  # > 0
        else:
            _g1k0 = unsafe_add(unsafe_sub(K0, _g1k0), 1)  # > 0

        # D / (A * N**N) * _g1k0**2 / gamma**2
        mul1: uint256 = unsafe_div(unsafe_div(unsafe_div(10**18 * D, gamma) * _g1k0, gamma) * _g1k0 * A_MULTIPLIER, ANN)

        # 2*N*K0 / _g1k0
        mul2: uint256 = unsafe_div(((2 * 10**18) * N_COINS) * K0, _g1k0)

        # calculate neg_fprime. here K0 > 0 is being validated (safediv).
        neg_fprime: uint256 = (S + unsafe_div(S * mul2, 10**18)) + mul1 * N_COINS / K0 - unsafe_div(mul2 * D, 10**18)

        # D -= f / fprime; neg_fprime safediv being validated
        D_plus: uint256 = D * (neg_fprime + S) / neg_fprime
        D_minus: uint256 = unsafe_div(D * D,  neg_fprime)
        if 10**18 > K0:
            D_minus += unsafe_div(unsafe_div(D * unsafe_div(mul1, neg_fprime), 10**18) * unsafe_sub(10**18, K0), K0)
        else:
            D_minus -= unsafe_div(unsafe_div(D * unsafe_div(mul1, neg_fprime), 10**18) * unsafe_sub(K0, 10**18), K0)

        if D_plus > D_minus:
            D = unsafe_sub(D_plus, D_minus)
        else:
            D = unsafe_div(unsafe_sub(D_minus, D_plus), 2)

        if D > D_prev:
            diff = unsafe_sub(D, D_prev)
        else:
            diff = unsafe_sub(D_prev, D)

        if diff * 10**14 < max(10**16, D):  # Could reduce precision for gas efficiency here

            for _x in x:
                frac: uint256 = _x * 10**18 / D
                assert (frac >= 10**16 - 1) and (frac < 10**20 + 1)  # dev: unsafe values x[i]
            return D

    raise "Did not converge"
>>> soon  

calc_token_fee

TwoCrypto.calc_token_fee(amounts: uint256[N_COINS], xp: uint256[N_COINS]) -> uint256:

Function to calculate the charged fee on amounts when adding liquidity.

Returns: fee (uint256).

Input Type Description
amounts uint256[N_COINS] Amount of coins added to the pool.
xp uint256[N_COINS] Pool balances multiplied by the coin precisions.
Source code
@external
@view
def calc_token_fee(
    amounts: uint256[N_COINS], xp: uint256[N_COINS]
) -> uint256:
    """
    @notice Returns the fee charged on the given amounts for add_liquidity.
    @param amounts The amounts of coins being added to the pool.
    @param xp The current balances of the pool multiplied by coin precisions.
    @return uint256 Fee charged.
    """
    return self._calc_token_fee(amounts, xp)

@view
@internal
def _calc_token_fee(amounts: uint256[N_COINS], xp: uint256[N_COINS]) -> uint256:
    # fee = sum(amounts_i - avg(amounts)) * fee' / sum(amounts)
    fee: uint256 = unsafe_div(
        unsafe_mul(self._fee(xp), N_COINS),
        unsafe_mul(4, unsafe_sub(N_COINS, 1))
    )

    S: uint256 = 0
    for _x in amounts:
        S += _x

    avg: uint256 = unsafe_div(S, N_COINS)
    Sdiff: uint256 = 0

    for _x in amounts:
        if _x > avg:
            Sdiff += unsafe_sub(_x, avg)
        else:
            Sdiff += unsafe_sub(avg, _x)

    return fee * Sdiff / S + NOISE_FEE
>>> soon  

remove_liquidity

TwoCrypto.remove_liquidity(_amount: uint256, min_amounts: uint256[N_COINS], receiver: address = msg.sender) -> uint256[N_COINS]:

Info

In case of any issues that result in a malfunctioning AMM state, users can safely withdraw liquidity using remove_liquidity. Withdrawal is based on balances proportional to the AMM balances, as this function does not perform complex math.

Function to remove liquidity from the pool and burn _amount of LP tokens. When removing liquidity with this function, no fees are charged as the coins are withdrawn in balanced proportions. This function also updates the xcp_oracle since liquidity was removed.

Returns: withdrawn balances (uint256[N_COINS]).

Emits: RemoveLiquidity

Input Type Description
_amount uint256 Amount of LP tokens to burn.
min_amounts uint256[N_COINS] Minimum amounts of tokens to withdraw.
receiver address Receiver of the coins; defaults to msg.sender.
Source code
event RemoveLiquidity:
    provider: indexed(address)
    token_amounts: uint256[N_COINS]
    token_supply: uint256

@external
@nonreentrant("lock")
def remove_liquidity(
    _amount: uint256,
    min_amounts: uint256[N_COINS],
    receiver: address = msg.sender,
) -> uint256[N_COINS]:
    """
    @notice This withdrawal method is very safe, does no complex math since
            tokens are withdrawn in balanced proportions. No fees are charged.
    @param _amount Amount of LP tokens to burn
    @param min_amounts Minimum amounts of tokens to withdraw
    @param receiver Address to send the withdrawn tokens to
    @return uint256[3] Amount of pool tokens received by the `receiver`
    """
    amount: uint256 = _amount
    balances: uint256[N_COINS] = self.balances
    withdraw_amounts: uint256[N_COINS] = empty(uint256[N_COINS])

    # -------------------------------------------------------- Burn LP tokens.

    total_supply: uint256 = self.totalSupply  # <------ Get totalSupply before
    self.burnFrom(msg.sender, _amount)  # ---- reducing it with self.burnFrom.

    # There are two cases for withdrawing tokens from the pool.
    #   Case 1. Withdrawal does not empty the pool.
    #           In this situation, D is adjusted proportional to the amount of
    #           LP tokens burnt. ERC20 tokens transferred is proportional
    #           to : (AMM balance * LP tokens in) / LP token total supply
    #   Case 2. Withdrawal empties the pool.
    #           In this situation, all tokens are withdrawn and the invariant
    #           is reset.

    if amount == total_supply:  # <----------------------------------- Case 2.

        for i in range(N_COINS):

            withdraw_amounts[i] = balances[i]

    else:  # <-------------------------------------------------------- Case 1.

        amount -= 1  # <---- To prevent rounding errors, favor LPs a tiny bit.

        for i in range(N_COINS):

            withdraw_amounts[i] = balances[i] * amount / total_supply
            assert withdraw_amounts[i] >= min_amounts[i]

    D: uint256 = self.D
    self.D = D - unsafe_div(D * amount, total_supply)  # <----------- Reduce D
    #      proportional to the amount of tokens leaving. Since withdrawals are
    #       balanced, this is a simple subtraction. If amount == total_supply,
    #                                                             D will be 0.

    # ---------------------------------- Transfers ---------------------------

    for i in range(N_COINS):
        # _transfer_out updates self.balances here. Update to state occurs
        # before external calls:
        self._transfer_out(i, withdraw_amounts[i], receiver)

    log RemoveLiquidity(msg.sender, withdraw_amounts, total_supply - _amount)

    # --------------------------- Upkeep xcp oracle --------------------------

    # Update xcp since liquidity was removed:
    xp: uint256[N_COINS] = self.xp(self.balances, self.cached_price_scale)
    last_xcp: uint256 = isqrt(xp[0] * xp[1])  # <----------- Cache it for now.

    last_timestamp: uint256[2] = self._unpack_2(self.last_timestamp)
    if last_timestamp[1] < block.timestamp:

        cached_xcp_oracle: uint256 = self.cached_xcp_oracle
        alpha: uint256 = MATH.wad_exp(
            -convert(
                unsafe_div(
                    unsafe_sub(block.timestamp, last_timestamp[1]) * 10**18,
                    self.xcp_ma_time  # <---------- xcp ma time has is longer.
                ),
                int256,
            )
        )

        self.cached_xcp_oracle = unsafe_div(
            last_xcp * (10**18 - alpha) + cached_xcp_oracle * alpha,
            10**18
        )
        last_timestamp[1] = block.timestamp

        # Pack and store timestamps:
        self.last_timestamp = self._pack_2(last_timestamp[0], last_timestamp[1])

    # Store last xcp
    self.last_xcp = last_xcp

    return withdraw_amounts
@external
@pure
def wad_exp(x: int256) -> int256:
    """
    @dev Calculates the natural exponential function of a signed integer with
        a precision of 1e18.
    @notice Note that this function consumes about 810 gas units. The implementation
            is inspired by Remco Bloemen's implementation under the MIT license here:
            https://xn--2-umb.com/22/exp-ln.
    @param x The 32-byte variable.
    @return int256 The 32-byte calculation result.
    """
    value: int256 = x

    # If the result is `< 0.5`, we return zero. This happens when we have the following:
    # "x <= floor(log(0.5e18) * 1e18) ~ -42e18".
    if (x <= -42_139_678_854_452_767_551):
        return empty(int256)

    # When the result is "> (2 ** 255 - 1) / 1e18" we cannot represent it as a signed integer.
    # This happens when "x >= floor(log((2 ** 255 - 1) / 1e18) * 1e18) ~ 135".
    assert x < 135_305_999_368_893_231_589, "Math: wad_exp overflow"

    # `x` is now in the range "(-42, 136) * 1e18". Convert to "(-42, 136) * 2 ** 96" for higher
    # intermediate precision and a binary base. This base conversion is a multiplication with
    # "1e18 / 2 ** 96 = 5 ** 18 / 2 ** 78".
    value = unsafe_div(x << 78, 5 ** 18)

    # Reduce the range of `x` to "(-½ ln 2, ½ ln 2) * 2 ** 96" by factoring out powers of two
    # so that "exp(x) = exp(x') * 2 ** k", where `k` is a signer integer. Solving this gives
    # "k = round(x / log(2))" and "x' = x - k * log(2)". Thus, `k` is in the range "[-61, 195]".
    k: int256 = unsafe_add(unsafe_div(value << 96, 54_916_777_467_707_473_351_141_471_128), 2 ** 95) >> 96
    value = unsafe_sub(value, unsafe_mul(k, 54_916_777_467_707_473_351_141_471_128))

    # Evaluate using a "(6, 7)"-term rational approximation. Since `p` is monic,
    # we will multiply by a scaling factor later.
    y: int256 = unsafe_add(unsafe_mul(unsafe_add(value, 1_346_386_616_545_796_478_920_950_773_328), value) >> 96, 57_155_421_227_552_351_082_224_309_758_442)
    p: int256 = unsafe_add(unsafe_mul(unsafe_add(unsafe_mul(unsafe_sub(unsafe_add(y, value), 94_201_549_194_550_492_254_356_042_504_812), y) >> 96,\
                        28_719_021_644_029_726_153_956_944_680_412_240), value), 4_385_272_521_454_847_904_659_076_985_693_276 << 96)

    # We leave `p` in the "2 ** 192" base so that we do not have to scale it up
    # again for the division.
    q: int256 = unsafe_add(unsafe_mul(unsafe_sub(value, 2_855_989_394_907_223_263_936_484_059_900), value) >> 96, 50_020_603_652_535_783_019_961_831_881_945)
    q = unsafe_sub(unsafe_mul(q, value) >> 96, 533_845_033_583_426_703_283_633_433_725_380)
    q = unsafe_add(unsafe_mul(q, value) >> 96, 3_604_857_256_930_695_427_073_651_918_091_429)
    q = unsafe_sub(unsafe_mul(q, value) >> 96, 14_423_608_567_350_463_180_887_372_962_807_573)
    q = unsafe_add(unsafe_mul(q, value) >> 96, 26_449_188_498_355_588_339_934_803_723_976_023)

    # The polynomial `q` has no zeros in the range because all its roots are complex.
    # No scaling is required, as `p` is already "2 ** 96" too large. Also,
    # `r` is in the range "(0.09, 0.25) * 2**96" after the division.
    r: int256 = unsafe_div(p, q)

    # To finalise the calculation, we have to multiply `r` by:
    #   - the scale factor "s = ~6.031367120",
    #   - the factor "2 ** k" from the range reduction, and
    #   - the factor "1e18 / 2 ** 96" for the base conversion.
    # We do this all at once, with an intermediate result in "2**213" base,
    # so that the final right shift always gives a positive value.

    # Note that to circumvent Vyper's safecast feature for the potentially
    # negative parameter value `r`, we first convert `r` to `bytes32` and
    # subsequently to `uint256`. Remember that the EVM default behaviour is
    # to use two's complement representation to handle signed integers.
    return convert(unsafe_mul(convert(convert(r, bytes32), uint256), 3_822_833_074_963_236_453_042_738_258_902_158_003_155_416_615_667) >>\
        convert(unsafe_sub(195, k), uint256), int256)
>>> soon  

remove_liquidity_one_coin

TwoCrypto.remove_liquidity_one_coin(token_amount: uint256, i: uint256, min_amount: uint256, receiver: address = msg.sender) -> uint256:

Function to burn token_amount LP tokens and withdraw liquidity in a single token i.

Returns: amount of coins withdrawn (uint256).

Emits: RemoveLiquidityOne

Input Type Description
token_amount uint256 Amount of LP tokens to burn.
i uint256 Index of the token to withdraw.
min_amount uint256 Minimum amount of token to withdraw.
receiver address Receiver of the coins; defaults to msg.sender.
Source code
event RemoveLiquidityOne:
    provider: indexed(address)
    token_amount: uint256
    coin_index: uint256
    coin_amount: uint256
    approx_fee: uint256
    packed_price_scale: uint256

@external
@nonreentrant("lock")
def remove_liquidity_one_coin(
    token_amount: uint256,
    i: uint256,
    min_amount: uint256,
    receiver: address = msg.sender
) -> uint256:
    """
    @notice Withdraw liquidity in a single token.
            Involves fees (lower than swap fees).
    @dev This operation also involves an admin fee claim.
    @param token_amount Amount of LP tokens to burn
    @param i Index of the token to withdraw
    @param min_amount Minimum amount of token to withdraw.
    @param receiver Address to send the withdrawn tokens to
    @return Amount of tokens at index i received by the `receiver`
    """

    self._claim_admin_fees()  # <--------- Auto-claim admin fees occasionally.

    A_gamma: uint256[2] = self._A_gamma()

    dy: uint256 = 0
    D: uint256 = 0
    p: uint256 = 0
    xp: uint256[N_COINS] = empty(uint256[N_COINS])
    approx_fee: uint256 = 0

    # ------------------------------------------------------------------------

    dy, D, xp, approx_fee = self._calc_withdraw_one_coin(
        A_gamma,
        token_amount,
        i,
        (self.future_A_gamma_time > block.timestamp),  # <------- During ramps
    )  #                                                  we need to update D.

    assert dy >= min_amount, "Slippage"

    # ---------------------------- State Updates -----------------------------

    # Burn user's tokens:
    self.burnFrom(msg.sender, token_amount)

    packed_price_scale: uint256 = self.tweak_price(A_gamma, xp, D, 0)
    #        Safe to use D from _calc_withdraw_one_coin here ---^

    # ------------------------- Transfers ------------------------------------

    # _transfer_out updates self.balances here. Update to state occurs before
    # external calls:
    self._transfer_out(i, dy, receiver)

    log RemoveLiquidityOne(
        msg.sender, token_amount, i, dy, approx_fee, packed_price_scale
    )

    return dy

@internal
@view
def _calc_withdraw_one_coin(
    A_gamma: uint256[2],
    token_amount: uint256,
    i: uint256,
    update_D: bool,
) -> (uint256, uint256, uint256[N_COINS], uint256):

    token_supply: uint256 = self.totalSupply
    assert token_amount <= token_supply  # dev: token amount more than supply
    assert i < N_COINS  # dev: coin out of range

    xx: uint256[N_COINS] = self.balances
    D0: uint256 = 0

    # -------------------------- Calculate D0 and xp -------------------------

    price_scale_i: uint256 = self.cached_price_scale * PRECISIONS[1]
    xp: uint256[N_COINS] = [
        xx[0] * PRECISIONS[0],
        unsafe_div(xx[1] * price_scale_i, PRECISION)
    ]
    if i == 0:
        price_scale_i = PRECISION * PRECISIONS[0]

    if update_D:  # <-------------- D is updated if pool is undergoing a ramp.
        D0 = MATH.newton_D(A_gamma[0], A_gamma[1], xp, 0)
    else:
        D0 = self.D

    D: uint256 = D0

    # -------------------------------- Fee Calc ------------------------------

    # Charge fees on D. Roughly calculate xp[i] after withdrawal and use that
    # to calculate fee. Precision is not paramount here: we just want a
    # behavior where the higher the imbalance caused the more fee the AMM
    # charges.

    # xp is adjusted assuming xp[0] ~= xp[1] ~= x[2], which is usually not the
    #  case. We charge self._fee(xp), where xp is an imprecise adjustment post
    #  withdrawal in one coin. If the withdraw is too large: charge max fee by
    #   default. This is because the fee calculation will otherwise underflow.

    xp_imprecise: uint256[N_COINS] = xp
    xp_correction: uint256 = xp[i] * N_COINS * token_amount / token_supply
    fee: uint256 = self._unpack_3(self.packed_fee_params)[1]  # <- self.out_fee.

    if xp_correction < xp_imprecise[i]:
        xp_imprecise[i] -= xp_correction
        fee = self._fee(xp_imprecise)

    dD: uint256 = unsafe_div(token_amount * D, token_supply)
    D_fee: uint256 = fee * dD / (2 * 10**10) + 1  # <------- Actual fee on D.

    # --------- Calculate `approx_fee` (assuming balanced state) in ith token.
    # -------------------------------- We only need this for fee in the event.
    approx_fee: uint256 = N_COINS * D_fee * xx[i] / D  # <------------------<---------- TODO: Check math.

    # ------------------------------------------------------------------------
    D -= (dD - D_fee)  # <----------------------------------- Charge fee on D.
    # --------------------------------- Calculate `y_out`` with `(D - D_fee)`.
    y: uint256 = MATH.get_y(A_gamma[0], A_gamma[1], xp, D, i)[0]
    dy: uint256 = (xp[i] - y) * PRECISION / price_scale_i
    xp[i] = y

    return dy, D, xp, approx_fee

@view
@internal
def _A_gamma() -> uint256[2]:
    t1: uint256 = self.future_A_gamma_time

    A_gamma_1: uint256 = self.future_A_gamma
    gamma1: uint256 = A_gamma_1 & 2**128 - 1
    A1: uint256 = A_gamma_1 >> 128

    if block.timestamp < t1:

        # --------------- Handle ramping up and down of A --------------------

        A_gamma_0: uint256 = self.initial_A_gamma
        t0: uint256 = self.initial_A_gamma_time

        t1 -= t0
        t0 = block.timestamp - t0
        t2: uint256 = t1 - t0

        A1 = ((A_gamma_0 >> 128) * t2 + A1 * t0) / t1
        gamma1 = ((A_gamma_0 & 2**128 - 1) * t2 + gamma1 * t0) / t1

    return [A1, gamma1]
@external
@view
def newton_D(ANN: uint256, gamma: uint256, x_unsorted: uint256[N_COINS], K0_prev: uint256 = 0) -> uint256:
    """
    Finding the invariant using Newton method.
    ANN is higher by the factor A_MULTIPLIER
    ANN is already A * N**N
    """

    # Safety checks
    assert ANN > MIN_A - 1 and ANN < MAX_A + 1  # dev: unsafe values A
    assert gamma > MIN_GAMMA - 1 and gamma < MAX_GAMMA + 1  # dev: unsafe values gamma

    # Initial value of invariant D is that for constant-product invariant
    x: uint256[N_COINS] = x_unsorted
    if x[0] < x[1]:
        x = [x_unsorted[1], x_unsorted[0]]

    assert x[0] > 10**9 - 1 and x[0] < 10**15 * 10**18 + 1  # dev: unsafe values x[0]
    assert unsafe_div(x[1] * 10**18, x[0]) > 10**14 - 1  # dev: unsafe values x[i] (input)

    S: uint256 = unsafe_add(x[0], x[1])  # can unsafe add here because we checked x[0] bounds

    D: uint256 = 0
    if K0_prev == 0:
        D = N_COINS * isqrt(unsafe_mul(x[0], x[1]))
    else:
        # D = isqrt(x[0] * x[1] * 4 / K0_prev * 10**18)
        D = isqrt(unsafe_mul(unsafe_div(unsafe_mul(unsafe_mul(4, x[0]), x[1]), K0_prev), 10**18))
        if S < D:
            D = S

    __g1k0: uint256 = gamma + 10**18
    diff: uint256 = 0

    for i in range(255):
        D_prev: uint256 = D
        assert D > 0
        # Unsafe division by D and D_prev is now safe

        # K0: uint256 = 10**18
        # for _x in x:
        #     K0 = K0 * _x * N_COINS / D
        # collapsed for 2 coins
        K0: uint256 = unsafe_div(unsafe_div((10**18 * N_COINS**2) * x[0], D) * x[1], D)

        _g1k0: uint256 = __g1k0
        if _g1k0 > K0:
            _g1k0 = unsafe_add(unsafe_sub(_g1k0, K0), 1)  # > 0
        else:
            _g1k0 = unsafe_add(unsafe_sub(K0, _g1k0), 1)  # > 0

        # D / (A * N**N) * _g1k0**2 / gamma**2
        mul1: uint256 = unsafe_div(unsafe_div(unsafe_div(10**18 * D, gamma) * _g1k0, gamma) * _g1k0 * A_MULTIPLIER, ANN)

        # 2*N*K0 / _g1k0
        mul2: uint256 = unsafe_div(((2 * 10**18) * N_COINS) * K0, _g1k0)

        # calculate neg_fprime. here K0 > 0 is being validated (safediv).
        neg_fprime: uint256 = (S + unsafe_div(S * mul2, 10**18)) + mul1 * N_COINS / K0 - unsafe_div(mul2 * D, 10**18)

        # D -= f / fprime; neg_fprime safediv being validated
        D_plus: uint256 = D * (neg_fprime + S) / neg_fprime
        D_minus: uint256 = unsafe_div(D * D,  neg_fprime)
        if 10**18 > K0:
            D_minus += unsafe_div(unsafe_div(D * unsafe_div(mul1, neg_fprime), 10**18) * unsafe_sub(10**18, K0), K0)
        else:
            D_minus -= unsafe_div(unsafe_div(D * unsafe_div(mul1, neg_fprime), 10**18) * unsafe_sub(K0, 10**18), K0)

        if D_plus > D_minus:
            D = unsafe_sub(D_plus, D_minus)
        else:
            D = unsafe_div(unsafe_sub(D_minus, D_plus), 2)

        if D > D_prev:
            diff = unsafe_sub(D, D_prev)
        else:
            diff = unsafe_sub(D_prev, D)

        if diff * 10**14 < max(10**16, D):  # Could reduce precision for gas efficiency here

            for _x in x:
                frac: uint256 = _x * 10**18 / D
                assert (frac >= 10**16 - 1) and (frac < 10**20 + 1)  # dev: unsafe values x[i]
            return D

    raise "Did not converge"

@external
@pure
def get_y(
    _ANN: uint256,
    _gamma: uint256,
    _x: uint256[N_COINS],
    _D: uint256,
    i: uint256
) -> uint256[2]:

    # Safety checks
    assert _ANN > MIN_A - 1 and _ANN < MAX_A + 1  # dev: unsafe values A
    assert _gamma > MIN_GAMMA - 1 and _gamma < MAX_GAMMA + 1  # dev: unsafe values gamma
    assert _D > 10**17 - 1 and _D < 10**15 * 10**18 + 1 # dev: unsafe values D

    ANN: int256 = convert(_ANN, int256)
    gamma: int256 = convert(_gamma, int256)
    D: int256 = convert(_D, int256)
    x_j: int256 = convert(_x[1 - i], int256)
    gamma2: int256 = unsafe_mul(gamma, gamma)

    # savediv by x_j done here:
    y: int256 = D**2 / (x_j * N_COINS**2)

    # K0_i: int256 = (10**18 * N_COINS) * x_j / D
    K0_i: int256 = unsafe_div(10**18 * N_COINS * x_j, D)
    assert (K0_i > 10**16 * N_COINS - 1) and (K0_i < 10**20 * N_COINS + 1)  # dev: unsafe values x[i]

    ann_gamma2: int256 = ANN * gamma2

    # a = 10**36 / N_COINS**2
    a: int256 = 10**32

    # b = ANN*D*gamma2/4/10000/x_j/10**4 - 10**32*3 - 2*gamma*10**14
    b: int256 = (
        D*ann_gamma2/400000000/x_j
        - convert(unsafe_mul(10**32, 3), int256)
        - unsafe_mul(unsafe_mul(2, gamma), 10**14)
    )

    # c = 10**32*3 + 4*gamma*10**14 + gamma2/10**4 + 4*ANN*gamma2*x_j/D/10000/4/10**4 - 4*ANN*gamma2/10000/4/10**4
    c: int256 = (
        unsafe_mul(10**32, convert(3, int256))
        + unsafe_mul(unsafe_mul(4, gamma), 10**14)
        + unsafe_div(gamma2, 10**4)
        + unsafe_div(unsafe_div(unsafe_mul(4, ann_gamma2), 400000000) * x_j, D)
        - unsafe_div(unsafe_mul(4, ann_gamma2), 400000000)
    )

    # d = -(10**18+gamma)**2 / 10**4
    d: int256 = -unsafe_div(unsafe_add(10**18, gamma) ** 2, 10**4)

    # delta0: int256 = 3*a*c/b - b
    delta0: int256 = 3 * a * c / b - b  # safediv by b

    # delta1: int256 = 9*a*c/b - 2*b - 27*a**2/b*d/b
    delta1: int256 = 3 * delta0 + b - 27*a**2/b*d/b

    divider: int256 = 1
    threshold: int256 = min(min(abs(delta0), abs(delta1)), a)
    if threshold > 10**48:
        divider = 10**30
    elif threshold > 10**46:
        divider = 10**28
    elif threshold > 10**44:
        divider = 10**26
    elif threshold > 10**42:
        divider = 10**24
    elif threshold > 10**40:
        divider = 10**22
    elif threshold > 10**38:
        divider = 10**20
    elif threshold > 10**36:
        divider = 10**18
    elif threshold > 10**34:
        divider = 10**16
    elif threshold > 10**32:
        divider = 10**14
    elif threshold > 10**30:
        divider = 10**12
    elif threshold > 10**28:
        divider = 10**10
    elif threshold > 10**26:
        divider = 10**8
    elif threshold > 10**24:
        divider = 10**6
    elif threshold > 10**20:
        divider = 10**2

    a = unsafe_div(a, divider)
    b = unsafe_div(b, divider)
    c = unsafe_div(c, divider)
    d = unsafe_div(d, divider)

    # delta0 = 3*a*c/b - b: here we can do more unsafe ops now:
    delta0 = unsafe_div(unsafe_mul(unsafe_mul(3, a), c), b) - b

    # delta1 = 9*a*c/b - 2*b - 27*a**2/b*d/b
    delta1 = 3 * delta0 + b - unsafe_div(unsafe_mul(unsafe_div(unsafe_mul(27, a**2), b), d), b)

    # sqrt_arg: int256 = delta1**2 + 4*delta0**2/b*delta0
    sqrt_arg: int256 = delta1**2 + unsafe_mul(unsafe_div(4*delta0**2, b), delta0)
    sqrt_val: int256 = 0
    if sqrt_arg > 0:
        sqrt_val = convert(isqrt(convert(sqrt_arg, uint256)), int256)
    else:
        return [
            self._newton_y(_ANN, _gamma, _x, _D, i),
            0
        ]

    b_cbrt: int256 = 0
    if b > 0:
        b_cbrt = convert(self._cbrt(convert(b, uint256)), int256)
    else:
        b_cbrt = -convert(self._cbrt(convert(-b, uint256)), int256)

    second_cbrt: int256 = 0
    if delta1 > 0:
        # second_cbrt = convert(self._cbrt(convert((delta1 + sqrt_val), uint256) / 2), int256)
        second_cbrt = convert(self._cbrt(convert(unsafe_add(delta1, sqrt_val), uint256) / 2), int256)
    else:
        # second_cbrt = -convert(self._cbrt(convert(unsafe_sub(sqrt_val, delta1), uint256) / 2), int256)
        second_cbrt = -convert(self._cbrt(unsafe_div(convert(unsafe_sub(sqrt_val, delta1), uint256), 2)), int256)

    # C1: int256 = b_cbrt**2/10**18*second_cbrt/10**18
    C1: int256 = unsafe_div(unsafe_mul(unsafe_div(b_cbrt**2, 10**18), second_cbrt), 10**18)

    # root: int256 = (10**18*C1 - 10**18*b - 10**18*b*delta0/C1)/(3*a), keep 2 safe ops here.
    root: int256 = (unsafe_mul(10**18, C1) - unsafe_mul(10**18, b) - unsafe_mul(10**18, b)/C1*delta0)/unsafe_mul(3, a)

    # y_out: uint256[2] =  [
    #     convert(D**2/x_j*root/4/10**18, uint256),   # <--- y
    #     convert(root, uint256)  # <----------------------- K0Prev
    # ]
    y_out: uint256[2] = [convert(unsafe_div(unsafe_div(unsafe_mul(unsafe_div(D**2, x_j), root), 4), 10**18), uint256), convert(root, uint256)]

    frac: uint256 = unsafe_div(y_out[0] * 10**18, _D)
    assert (frac >= 10**16 - 1) and (frac < 10**20 + 1)  # dev: unsafe value for y

    return y_out
>>> soon  

calc_token_amount

TwoCrypto.calc_token_amount(amounts: uint256[N_COINS], deposit: bool) -> uint256:

Function to calculate the LP tokens to be minted or burned for depositing or removing amounts of coins. This method takes fees into consideration.

Returns: amount of LP tokens deposited or withdrawn (uint256).

Input Type Description
amounts uint256[N_COINS] Amounts of tokens being deposited or withdrawn.
deposit bool true for deposit, false for withdrawal.
Source code
interface Factory:
    def views_implementation() -> address: view

@external
@view
def calc_token_amount(amounts: uint256[N_COINS], deposit: bool) -> uint256:
    """
    @notice Calculate LP tokens minted or to be burned for depositing or
            removing `amounts` of coins
    @dev Includes fee.
    @param amounts Amounts of tokens being deposited or withdrawn
    @param deposit True if it is a deposit action, False if withdrawn.
    @return uint256 Amount of LP tokens deposited or withdrawn.
    """
    view_contract: address = factory.views_implementation()
    return Views(view_contract).calc_token_amount(amounts, deposit, self)

@external
@view
def calc_token_fee(
    amounts: uint256[N_COINS], xp: uint256[N_COINS]
) -> uint256:
    """
    @notice Returns the fee charged on the given amounts for add_liquidity.
    @param amounts The amounts of coins being added to the pool.
    @param xp The current balances of the pool multiplied by coin precisions.
    @return uint256 Fee charged.
    """
    return self._calc_token_fee(amounts, xp)


@view
@internal
def _calc_token_fee(amounts: uint256[N_COINS], xp: uint256[N_COINS]) -> uint256:
    # fee = sum(amounts_i - avg(amounts)) * fee' / sum(amounts)
    fee: uint256 = unsafe_div(
        unsafe_mul(self._fee(xp), N_COINS),
        unsafe_mul(4, unsafe_sub(N_COINS, 1))
    )

    S: uint256 = 0
    for _x in amounts:
        S += _x

    avg: uint256 = unsafe_div(S, N_COINS)
    Sdiff: uint256 = 0

    for _x in amounts:
        if _x > avg:
            Sdiff += unsafe_sub(_x, avg)
        else:
            Sdiff += unsafe_sub(avg, _x)

    return fee * Sdiff / S + NOISE_FEE                
@view
@external
def calc_token_amount(
    amounts: uint256[N_COINS], deposit: bool, swap: address
) -> uint256:

    d_token: uint256 = 0
    amountsp: uint256[N_COINS] = empty(uint256[N_COINS])
    xp: uint256[N_COINS] = empty(uint256[N_COINS])

    d_token, amountsp, xp = self._calc_dtoken_nofee(amounts, deposit, swap)
    d_token -= (
        Curve(swap).calc_token_fee(amountsp, xp) * d_token / 10**10 + 1
    )

    return d_token

@internal
@view
def _calc_dtoken_nofee(
    amounts: uint256[N_COINS], deposit: bool, swap: address
) -> (uint256, uint256[N_COINS], uint256[N_COINS]):

    math: Math = Curve(swap).MATH()

    xp: uint256[N_COINS] = empty(uint256[N_COINS])
    precisions: uint256[N_COINS] = empty(uint256[N_COINS])
    price_scale: uint256 = 0
    D0: uint256 = 0
    token_supply: uint256 = 0
    A: uint256 = 0
    gamma: uint256 = 0

    xp, D0, token_supply, price_scale, A, gamma, precisions = self._prep_calc(swap)

    amountsp: uint256[N_COINS] = amounts
    if deposit:
        for k in range(N_COINS):
            xp[k] += amounts[k]
    else:
        for k in range(N_COINS):
            xp[k] -= amounts[k]

    xp = [
        xp[0] * precisions[0],
        xp[1] * price_scale * precisions[1] / PRECISION
    ]
    amountsp = [
        amountsp[0]* precisions[0],
        amountsp[1] * price_scale * precisions[1] / PRECISION
    ]

    D: uint256 = math.newton_D(A, gamma, xp, 0)
    d_token: uint256 = token_supply * D / D0

    if deposit:
        d_token -= token_supply
    else:
        d_token = token_supply - d_token

    return d_token, amountsp, xp
In  [1]:  Pool.calc_token_amount([1000000000000000000, 0], True)
Out [1]:  37590681591977081154

In  [1]:  Pool.calc_token_amount([0, 1000000000000000000], True)
Out [1]:  6622263874240447

In  [1]:  Pool.calc_token_amount([1000000000000000000, 0], False)
Out [1]:  37707043389433059543

calc_withdraw_one_coin

TwoCrypto.calc_withdraw_one_coin(token_amount: uint256, i: uint256) -> uint256:

Function to calculate the amount of output token i when burning token_amount of LP tokens. This method takes fees into consideration.

Returns: amount of tokens to receive (uint256).

Input Type Description
token_amount uint256 Amount of LP tokens burned.
i uint256 Index of the coin to withdraw.
Source code
@view
@external
def calc_withdraw_one_coin(token_amount: uint256, i: uint256) -> uint256:
    """
    @notice Calculates output tokens with fee
    @param token_amount LP Token amount to burn
    @param i token in which liquidity is withdrawn
    @return uint256 Amount of ith tokens received for burning token_amount LP tokens.
    """

    return self._calc_withdraw_one_coin(
        self._A_gamma(),
        token_amount,
        i,
        (self.future_A_gamma_time > block.timestamp)
    )[0]

@internal
@view
def _calc_withdraw_one_coin(
    A_gamma: uint256[2],
    token_amount: uint256,
    i: uint256,
    update_D: bool,
) -> (uint256, uint256, uint256[N_COINS], uint256):

    token_supply: uint256 = self.totalSupply
    assert token_amount <= token_supply  # dev: token amount more than supply
    assert i < N_COINS  # dev: coin out of range

    xx: uint256[N_COINS] = self.balances
    D0: uint256 = 0

    # -------------------------- Calculate D0 and xp -------------------------

    price_scale_i: uint256 = self.cached_price_scale * PRECISIONS[1]
    xp: uint256[N_COINS] = [
        xx[0] * PRECISIONS[0],
        unsafe_div(xx[1] * price_scale_i, PRECISION)
    ]
    if i == 0:
        price_scale_i = PRECISION * PRECISIONS[0]

    if update_D:  # <-------------- D is updated if pool is undergoing a ramp.
        D0 = MATH.newton_D(A_gamma[0], A_gamma[1], xp, 0)
    else:
        D0 = self.D

    D: uint256 = D0

    # -------------------------------- Fee Calc ------------------------------

    # Charge fees on D. Roughly calculate xp[i] after withdrawal and use that
    # to calculate fee. Precision is not paramount here: we just want a
    # behavior where the higher the imbalance caused the more fee the AMM
    # charges.

    # xp is adjusted assuming xp[0] ~= xp[1] ~= x[2], which is usually not the
    #  case. We charge self._fee(xp), where xp is an imprecise adjustment post
    #  withdrawal in one coin. If the withdraw is too large: charge max fee by
    #   default. This is because the fee calculation will otherwise underflow.

    xp_imprecise: uint256[N_COINS] = xp
    xp_correction: uint256 = xp[i] * N_COINS * token_amount / token_supply
    fee: uint256 = self._unpack_3(self.packed_fee_params)[1]  # <- self.out_fee.

    if xp_correction < xp_imprecise[i]:
        xp_imprecise[i] -= xp_correction
        fee = self._fee(xp_imprecise)

    dD: uint256 = unsafe_div(token_amount * D, token_supply)
    D_fee: uint256 = fee * dD / (2 * 10**10) + 1  # <------- Actual fee on D.

    # --------- Calculate `approx_fee` (assuming balanced state) in ith token.
    # -------------------------------- We only need this for fee in the event.
    approx_fee: uint256 = N_COINS * D_fee * xx[i] / D  # <------------------<---------- TODO: Check math.

    # ------------------------------------------------------------------------
    D -= (dD - D_fee)  # <----------------------------------- Charge fee on D.
    # --------------------------------- Calculate `y_out`` with `(D - D_fee)`.
    y: uint256 = MATH.get_y(A_gamma[0], A_gamma[1], xp, D, i)[0]
    dy: uint256 = (xp[i] - y) * PRECISION / price_scale_i
    xp[i] = y

    return dy, D, xp, approx_fee
@external
@view
def newton_D(ANN: uint256, gamma: uint256, x_unsorted: uint256[N_COINS], K0_prev: uint256 = 0) -> uint256:
    """
    Finding the invariant using Newton method.
    ANN is higher by the factor A_MULTIPLIER
    ANN is already A * N**N
    """

    # Safety checks
    assert ANN > MIN_A - 1 and ANN < MAX_A + 1  # dev: unsafe values A
    assert gamma > MIN_GAMMA - 1 and gamma < MAX_GAMMA + 1  # dev: unsafe values gamma

    # Initial value of invariant D is that for constant-product invariant
    x: uint256[N_COINS] = x_unsorted
    if x[0] < x[1]:
        x = [x_unsorted[1], x_unsorted[0]]

    assert x[0] > 10**9 - 1 and x[0] < 10**15 * 10**18 + 1  # dev: unsafe values x[0]
    assert unsafe_div(x[1] * 10**18, x[0]) > 10**14 - 1  # dev: unsafe values x[i] (input)

    S: uint256 = unsafe_add(x[0], x[1])  # can unsafe add here because we checked x[0] bounds

    D: uint256 = 0
    if K0_prev == 0:
        D = N_COINS * isqrt(unsafe_mul(x[0], x[1]))
    else:
        # D = isqrt(x[0] * x[1] * 4 / K0_prev * 10**18)
        D = isqrt(unsafe_mul(unsafe_div(unsafe_mul(unsafe_mul(4, x[0]), x[1]), K0_prev), 10**18))
        if S < D:
            D = S

    __g1k0: uint256 = gamma + 10**18
    diff: uint256 = 0

    for i in range(255):
        D_prev: uint256 = D
        assert D > 0
        # Unsafe division by D and D_prev is now safe

        # K0: uint256 = 10**18
        # for _x in x:
        #     K0 = K0 * _x * N_COINS / D
        # collapsed for 2 coins
        K0: uint256 = unsafe_div(unsafe_div((10**18 * N_COINS**2) * x[0], D) * x[1], D)

        _g1k0: uint256 = __g1k0
        if _g1k0 > K0:
            _g1k0 = unsafe_add(unsafe_sub(_g1k0, K0), 1)  # > 0
        else:
            _g1k0 = unsafe_add(unsafe_sub(K0, _g1k0), 1)  # > 0

        # D / (A * N**N) * _g1k0**2 / gamma**2
        mul1: uint256 = unsafe_div(unsafe_div(unsafe_div(10**18 * D, gamma) * _g1k0, gamma) * _g1k0 * A_MULTIPLIER, ANN)

        # 2*N*K0 / _g1k0
        mul2: uint256 = unsafe_div(((2 * 10**18) * N_COINS) * K0, _g1k0)

        # calculate neg_fprime. here K0 > 0 is being validated (safediv).
        neg_fprime: uint256 = (S + unsafe_div(S * mul2, 10**18)) + mul1 * N_COINS / K0 - unsafe_div(mul2 * D, 10**18)

        # D -= f / fprime; neg_fprime safediv being validated
        D_plus: uint256 = D * (neg_fprime + S) / neg_fprime
        D_minus: uint256 = unsafe_div(D * D,  neg_fprime)
        if 10**18 > K0:
            D_minus += unsafe_div(unsafe_div(D * unsafe_div(mul1, neg_fprime), 10**18) * unsafe_sub(10**18, K0), K0)
        else:
            D_minus -= unsafe_div(unsafe_div(D * unsafe_div(mul1, neg_fprime), 10**18) * unsafe_sub(K0, 10**18), K0)

        if D_plus > D_minus:
            D = unsafe_sub(D_plus, D_minus)
        else:
            D = unsafe_div(unsafe_sub(D_minus, D_plus), 2)

        if D > D_prev:
            diff = unsafe_sub(D, D_prev)
        else:
            diff = unsafe_sub(D_prev, D)

        if diff * 10**14 < max(10**16, D):  # Could reduce precision for gas efficiency here

            for _x in x:
                frac: uint256 = _x * 10**18 / D
                assert (frac >= 10**16 - 1) and (frac < 10**20 + 1)  # dev: unsafe values x[i]
            return D

    raise "Did not converge"

@external
@pure
def get_y(
    _ANN: uint256,
    _gamma: uint256,
    _x: uint256[N_COINS],
    _D: uint256,
    i: uint256
) -> uint256[2]:

    # Safety checks
    assert _ANN > MIN_A - 1 and _ANN < MAX_A + 1  # dev: unsafe values A
    assert _gamma > MIN_GAMMA - 1 and _gamma < MAX_GAMMA + 1  # dev: unsafe values gamma
    assert _D > 10**17 - 1 and _D < 10**15 * 10**18 + 1 # dev: unsafe values D

    ANN: int256 = convert(_ANN, int256)
    gamma: int256 = convert(_gamma, int256)
    D: int256 = convert(_D, int256)
    x_j: int256 = convert(_x[1 - i], int256)
    gamma2: int256 = unsafe_mul(gamma, gamma)

    # savediv by x_j done here:
    y: int256 = D**2 / (x_j * N_COINS**2)

    # K0_i: int256 = (10**18 * N_COINS) * x_j / D
    K0_i: int256 = unsafe_div(10**18 * N_COINS * x_j, D)
    assert (K0_i > 10**16 * N_COINS - 1) and (K0_i < 10**20 * N_COINS + 1)  # dev: unsafe values x[i]

    ann_gamma2: int256 = ANN * gamma2

    # a = 10**36 / N_COINS**2
    a: int256 = 10**32

    # b = ANN*D*gamma2/4/10000/x_j/10**4 - 10**32*3 - 2*gamma*10**14
    b: int256 = (
        D*ann_gamma2/400000000/x_j
        - convert(unsafe_mul(10**32, 3), int256)
        - unsafe_mul(unsafe_mul(2, gamma), 10**14)
    )

    # c = 10**32*3 + 4*gamma*10**14 + gamma2/10**4 + 4*ANN*gamma2*x_j/D/10000/4/10**4 - 4*ANN*gamma2/10000/4/10**4
    c: int256 = (
        unsafe_mul(10**32, convert(3, int256))
        + unsafe_mul(unsafe_mul(4, gamma), 10**14)
        + unsafe_div(gamma2, 10**4)
        + unsafe_div(unsafe_div(unsafe_mul(4, ann_gamma2), 400000000) * x_j, D)
        - unsafe_div(unsafe_mul(4, ann_gamma2), 400000000)
    )

    # d = -(10**18+gamma)**2 / 10**4
    d: int256 = -unsafe_div(unsafe_add(10**18, gamma) ** 2, 10**4)

    # delta0: int256 = 3*a*c/b - b
    delta0: int256 = 3 * a * c / b - b  # safediv by b

    # delta1: int256 = 9*a*c/b - 2*b - 27*a**2/b*d/b
    delta1: int256 = 3 * delta0 + b - 27*a**2/b*d/b

    divider: int256 = 1
    threshold: int256 = min(min(abs(delta0), abs(delta1)), a)
    if threshold > 10**48:
        divider = 10**30
    elif threshold > 10**46:
        divider = 10**28
    elif threshold > 10**44:
        divider = 10**26
    elif threshold > 10**42:
        divider = 10**24
    elif threshold > 10**40:
        divider = 10**22
    elif threshold > 10**38:
        divider = 10**20
    elif threshold > 10**36:
        divider = 10**18
    elif threshold > 10**34:
        divider = 10**16
    elif threshold > 10**32:
        divider = 10**14
    elif threshold > 10**30:
        divider = 10**12
    elif threshold > 10**28:
        divider = 10**10
    elif threshold > 10**26:
        divider = 10**8
    elif threshold > 10**24:
        divider = 10**6
    elif threshold > 10**20:
        divider = 10**2

    a = unsafe_div(a, divider)
    b = unsafe_div(b, divider)
    c = unsafe_div(c, divider)
    d = unsafe_div(d, divider)

    # delta0 = 3*a*c/b - b: here we can do more unsafe ops now:
    delta0 = unsafe_div(unsafe_mul(unsafe_mul(3, a), c), b) - b

    # delta1 = 9*a*c/b - 2*b - 27*a**2/b*d/b
    delta1 = 3 * delta0 + b - unsafe_div(unsafe_mul(unsafe_div(unsafe_mul(27, a**2), b), d), b)

    # sqrt_arg: int256 = delta1**2 + 4*delta0**2/b*delta0
    sqrt_arg: int256 = delta1**2 + unsafe_mul(unsafe_div(4*delta0**2, b), delta0)
    sqrt_val: int256 = 0
    if sqrt_arg > 0:
        sqrt_val = convert(isqrt(convert(sqrt_arg, uint256)), int256)
    else:
        return [
            self._newton_y(_ANN, _gamma, _x, _D, i),
            0
        ]

    b_cbrt: int256 = 0
    if b > 0:
        b_cbrt = convert(self._cbrt(convert(b, uint256)), int256)
    else:
        b_cbrt = -convert(self._cbrt(convert(-b, uint256)), int256)

    second_cbrt: int256 = 0
    if delta1 > 0:
        # second_cbrt = convert(self._cbrt(convert((delta1 + sqrt_val), uint256) / 2), int256)
        second_cbrt = convert(self._cbrt(convert(unsafe_add(delta1, sqrt_val), uint256) / 2), int256)
    else:
        # second_cbrt = -convert(self._cbrt(convert(unsafe_sub(sqrt_val, delta1), uint256) / 2), int256)
        second_cbrt = -convert(self._cbrt(unsafe_div(convert(unsafe_sub(sqrt_val, delta1), uint256), 2)), int256)

    # C1: int256 = b_cbrt**2/10**18*second_cbrt/10**18
    C1: int256 = unsafe_div(unsafe_mul(unsafe_div(b_cbrt**2, 10**18), second_cbrt), 10**18)

    # root: int256 = (10**18*C1 - 10**18*b - 10**18*b*delta0/C1)/(3*a), keep 2 safe ops here.
    root: int256 = (unsafe_mul(10**18, C1) - unsafe_mul(10**18, b) - unsafe_mul(10**18, b)/C1*delta0)/unsafe_mul(3, a)

    # y_out: uint256[2] =  [
    #     convert(D**2/x_j*root/4/10**18, uint256),   # <--- y
    #     convert(root, uint256)  # <----------------------- K0Prev
    # ]
    y_out: uint256[2] = [convert(unsafe_div(unsafe_div(unsafe_mul(unsafe_div(D**2, x_j), root), 4), 10**18), uint256), convert(root, uint256)]

    frac: uint256 = unsafe_div(y_out[0] * 10**18, _D)
    assert (frac >= 10**16 - 1) and (frac < 10**20 + 1)  # dev: unsafe value for y

    return y_out
In  [1]:  Pool.calc_withdraw_one_coin(1000000000000000000, 0)
Out [1]:  26501860406190437

In  [2]:  Pool.calc_withdraw_one_coin(1000000000000000000, 1)
Out [2]:  150537307454780254829

Fees and Pool Profits

The cryptoswap algorithm uses different fees, such as fee, mid_fee, out_fee, or fee_gamma to determine the fees charged, more on that here. All Fee values are denominated in 1e10 and can be changed by the admin.

Additionally, just as for other curve pools, there is an ADMIN_FEE, which is hardcoded to 50%. All twocrypto-ng pools share a universal fee_receiver, which is determined within the Factory contract. Unlike for most other Curve pools, there is no external method to claim the admin fees. They are claimed when removing liquidity single sided.

xcp_profit, xcp_profit_a, and last_xcp are used for tracking pool profits, which is necessary for the pool's rebalancing mechanism. These values are denominated in 1e18.

fee

TwoCrypto.fee() -> uint256:

Getter for the fee charged by the pool at the current state.

Returns: fee in bps (uint256).

Source code
@external
@view
def fee() -> uint256:
    """
    @notice Returns the fee charged by the pool at current state.
    @dev Not to be confused with the fee charged at liquidity action, since
        there the fee is calculated on `xp` AFTER liquidity is added or
        removed.
    @return uint256 fee bps.
    """
    return self._fee(self.xp(self.balances, self.cached_price_scale))

@internal
@view
def _fee(xp: uint256[N_COINS]) -> uint256:

    fee_params: uint256[3] = self._unpack_3(self.packed_fee_params)
    f: uint256 = xp[0] + xp[1]
    f = fee_params[2] * 10**18 / (
        fee_params[2] + 10**18 -
        (10**18 * N_COINS**N_COINS) * xp[0] / f * xp[1] / f
    )

    return unsafe_div(
        fee_params[0] * f + fee_params[1] * (10**18 - f),
        10**18
    )
In  [1]:  Pool.fee()
Out [1]:  30622026

mid_fee

TwoCrypto.mid_fee() -> uint256:

Getter for the mid_fee. This fee is the minimum fee and is charged when the pool is completely balanced.

Returns: mid fee (uint256).

Source code
packed_fee_params: public(uint256)  # <---- Packs mid_fee, out_fee, fee_gamma.

@view
@external
def mid_fee() -> uint256:
    """
    @notice Returns the current mid fee
    @return uint256 mid_fee value.
    """
    return self._unpack_3(self.packed_fee_params)[0]
In  [1]:  Pool.mid_fee()
Out [1]:  26000000

out_fee

TwoCrypto.out_fee() -> uint256:

Getter for the out_fee. This fee is the maximum fee and is charged when the pool is completely imbalanced.

Returns: out fee (uint256).

Source code
packed_fee_params: public(uint256)  # <---- Packs mid_fee, out_fee, fee_gamma.

@view
@external
def out_fee() -> uint256:
    """
    @notice Returns the current out fee
    @return uint256 out_fee value.
    """
    return self._unpack_3(self.packed_fee_params)[1]
In  [1]:  Pool.out_fee()
Out [1]:  45000000

fee_gamma

TwoCrypto.fee_gamma() -> uint256:

Getter for the current fee_gamma. This parameter modifies the rate at which fees rise as imbalance intensifies. Smaller values result in rapid fee hikes with growing imbalances, while larger values lead to more gradual increments in fees as imbalance expands.

Returns: fee gamma (uint256).

Source code
packed_fee_params: public(uint256)  # <---- Packs mid_fee, out_fee, fee_gamma.

@view
@external
def fee_gamma() -> uint256:
    """
    @notice Returns the current fee gamma
    @return uint256 fee_gamma value.
    """
    return self._unpack_3(self.packed_fee_params)[2]
In  [1]:  Pool.fee_gamma()
Out [1]:  230000000000000

packed_fee_params

TwoCrypto.packed_fee_params() -> uint256: view

Getter for the packed fee parameters.

Returns: packed fee params (uint256).

Source code
# Fee params that determine dynamic fees:
packed_fee_params: public(uint256)  # <---- Packs mid_fee, out_fee, fee_gamma.

@external
def __init__(
    _name: String[64],
    _symbol: String[32],
    _coins: address[N_COINS],
    _math: address,
    _salt: bytes32,
    packed_precisions: uint256,
    packed_gamma_A: uint256,
    packed_fee_params: uint256,
    packed_rebalancing_params: uint256,
    initial_price: uint256,
):
    ...
    self.packed_fee_params = packed_fee_params  # <-------------- Contains Fee
    #                                  params: mid_fee, out_fee and fee_gamma.
    ...
In  [1]:  Pool.packed_fee_params()
Out [1]:  8847341539944400050877843276543133320576000000

ADMIN_FEE

TwoCrypto.packed_fee_params() -> uint256: view

Getter for the admin fee of the pool. This value is hardcoded to 50% (5000000000) of the earned fees and can not be changed.

Returns: admin fee (uint256).

Source code
ADMIN_FEE: public(constant(uint256)) = 5 * 10**9  # <----- 50% of earned fees.
In  [1]:  Pool.ADMIN_FEE()
Out [1]:  5000000000

fee_receiver

TwoCrypto.fee_receiver() -> address:

Getter for the fee receiver of the admin fees. This address is set within the TwoCrypto-NG Factory. Every pool created through the Factory has the same fee receiver.

Returns: fee receiver (address).

Source code
interface Factory:
    def fee_receiver() -> address: view

@external
@view
def fee_receiver() -> address:
    """
    @notice Returns the address of the admin fee receiver.
    @return address Fee receiver.
    """
    return factory.fee_receiver()
In  [1]:  Pool.fee_receiver()
Out [1]:  '0xeCb456EA5365865EbAb8a2661B0c503410e9B347'

xcp_profit

TwoCrypto.xcp_profit() -> uint256: view

Getter for the current pool profits.

Returns: current profits (uint256).

Source code
xcp_profit: public(uint256)
In  [1]:  Pool.xcp_profit()
Out [1]:  1000280532115852216

xcp_profit_a

TwoCrypto.xcp_profit_a() -> uint256: view

Getter for the full profit at the last claim of admin fees.

Returns: profit at last claim (uint256).

Source code
xcp_profit_a: public(uint256)  # <--- Full profit at last claim of admin fees.

@external
def __init__(
    _name: String[64],
    _symbol: String[32],
    _coins: address[N_COINS],
    _math: address,
    _salt: bytes32,
    packed_precisions: uint256,
    packed_gamma_A: uint256,
    packed_fee_params: uint256,
    packed_rebalancing_params: uint256,
    initial_price: uint256,
):
    ...
    self.xcp_profit_a = 10**18
    ...
In  [1]:  Pool.xcp_profit_a()
Out [1]:  1000000000000000000

last_xcp

TwoCrypto.last_xcp() -> uint256: view

Getter for the last xcp action. This variable is updated by calling tweak_price or remove_liquidity.

Returns: timestamp of last xcp action (uint256).

Source code
last_xcp: public(uint256)
In  [1]:  Pool.last_xcp()
Out [1]:  4177413767556756716238

Price Scaling

Curve v2 pools automatically adjust liquidity to optimize depth close to the prevailing market rates, reducing slippage. More here. Price scaling parameter can be adjusted by the admin.

price_scale

TwoCrypto.price_scale() -> uint256:

Getter for the price scale of the coin at index 1 with regard to the coin at index 0. The price scale determines the price band around which liquidity is concentrated.

Returns: price scale (uint256).

Source code
cached_price_scale: uint256  # <------------------------ Internal price scale.

@external
@view
@nonreentrant("lock")
def price_scale() -> uint256:
    """
    @notice Returns the price scale of the coin at index `k` w.r.t the coin
            at index 0.
    @dev Price scale determines the price band around which liquidity is
        concentrated.
    @return uint256 Price scale of coin.
    """
    return self.cached_price_scale
In  [1]:  Pool.price_scale()
Out [1]:  176501696719232

allowed_extra_profit

TwoCrypto.allowed_extra_profit() -> uint256:

Getter for the allowed extra profit value.

Returns: allowed extra profit (uint256).

Source code
packed_rebalancing_params: public(uint256)  # <---------- Contains rebalancing
#               parameters allowed_extra_profit, adjustment_step, and ma_time.

@view
@external
def allowed_extra_profit() -> uint256:
    """
    @notice Returns the current allowed extra profit
    @return uint256 allowed_extra_profit value.
    """
    return self._unpack_3(self.packed_rebalancing_params)[0]
In  [1]:  Pool.allowed_extra_profit()
Out [1]:  2000000000000

adjustment_step

TwoCrypto.allowed_extra_profit() -> uint256:

Getter for the adjustment step value.

Returns: adjustment step (uint256).

Source code
packed_rebalancing_params: public(uint256)  # <---------- Contains rebalancing
#               parameters allowed_extra_profit, adjustment_step, and ma_time.

@view
@external
def adjustment_step() -> uint256:
    """
    @notice Returns the current adjustment step
    @return uint256 adjustment_step value.
    """
    return self._unpack_3(self.packed_rebalancing_params)[1]
In  [1]:  Pool.adjustment_step()
Out [1]:  146000000000000

packed_rebalancing_params

TwoCrypto.packed_rebalancing_params() -> uint256: view

Getter for the packed rebalancing parameters, consisting of allowed_extra_profit, adjustment_step, and ma_time.

Returns: packed rebalancing parameters (uint256).

Source code
packed_rebalancing_params: public(uint256)  # <---------- Contains rebalancing
#               parameters allowed_extra_profit, adjustment_step, and ma_time.
In  [1]:  Pool.packed_rebalancing_params()
Out [1]:  680564733841876929619973849625130958848000000000600

Bonding Curve Parameters

A bonding curve is used to determine asset prices according to the pool's supply of each asset, more here.

Bonding curve parameters A and gamma values are upgradable by the the pools admin.

A

TwoCrypto.A() -> uint256:

Getter for the current pool amplification parameter.

Returns: A (uint256).

Source code
@view
@external
def A() -> uint256:
    """
    @notice Returns the current pool amplification parameter.
    @return uint256 A param.
    """
    return self._A_gamma()[0]

@view
@internal
def _A_gamma() -> uint256[2]:
    t1: uint256 = self.future_A_gamma_time

    A_gamma_1: uint256 = self.future_A_gamma
    gamma1: uint256 = A_gamma_1 & 2**128 - 1
    A1: uint256 = A_gamma_1 >> 128

    if block.timestamp < t1:

        # --------------- Handle ramping up and down of A --------------------

        A_gamma_0: uint256 = self.initial_A_gamma
        t0: uint256 = self.initial_A_gamma_time

        t1 -= t0
        t0 = block.timestamp - t0
        t2: uint256 = t1 - t0

        A1 = ((A_gamma_0 >> 128) * t2 + A1 * t0) / t1
        gamma1 = ((A_gamma_0 & 2**128 - 1) * t2 + gamma1 * t0) / t1

    return [A1, gamma1]
In  [1]:  Pool.A()
Out [1]:  400000

gamma

TwoCrypto.gamma() -> uint256:

Getter for the current pool gamma parameter.

Returns: gamma (uint256).

Source code
@view
@external
def gamma() -> uint256:
    """
    @notice Returns the current pool gamma parameter.
    @return uint256 gamma param.
    """
    return self._A_gamma()[1]

@view
@internal
def _A_gamma() -> uint256[2]:
    t1: uint256 = self.future_A_gamma_time

    A_gamma_1: uint256 = self.future_A_gamma
    gamma1: uint256 = A_gamma_1 & 2**128 - 1
    A1: uint256 = A_gamma_1 >> 128

    if block.timestamp < t1:

        # --------------- Handle ramping up and down of A --------------------

        A_gamma_0: uint256 = self.initial_A_gamma
        t0: uint256 = self.initial_A_gamma_time

        t1 -= t0
        t0 = block.timestamp - t0
        t2: uint256 = t1 - t0

        A1 = ((A_gamma_0 >> 128) * t2 + A1 * t0) / t1
        gamma1 = ((A_gamma_0 & 2**128 - 1) * t2 + gamma1 * t0) / t1

    return [A1, gamma1]
In  [1]:  Pool.gamma()
Out [1]:  145000000000000

Oracle Methods

All pools have their own built in exponential moving average price oracle.

Prices and oracles are adjusted by when calling the internal tweak_price method, which happens at add_liquidity, remove_liquidity_one_coin and _exchange.

It is not called when removing liquidity one sided with remove_liquidity as this function does not alter prices.

tweak_price(A_gamma: uint256[2], _xp: uint256[N_COINS], new_D: uint256, K0_prev: uint256 = 0) -> uint256:
@internal
def tweak_price(
    <