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Tricrypto-NG Oracles

Tricrypto-NG pools have the following oracle:

  • price_oracle

    An exponential moving-average (EMA) price oracle with a periodicity determined by ma_time. It returns the price relative to the coin at index 0 in the pool.

Example: Price Oracle for TriCRV

The TriCRV pool consists of crvUSD <> wETH <> CRV.

Because crvUSD is coin[0], the prices of wETH and CRV are returned with regard to crvUSD.

>>> price_oracle(0) = 3670949576287168254655
3670.94957629       # price of wETH w.r.t crvUSD

>>> price_oracle(1) = 724988309167051066
0.72498830916       # price of CRV w.r.t crvUSD

In order to get the reverse EMA (e.g. price of crvUSD with regard to wETH):

1036price_oracle(0)=2.7240908e+14

The formula to calculate the exponential moving-average essentially comes down to:

Source Code for Calculating the EMA 
@internal
def tweak_price(
    A_gamma: uint256[2],
    _xp: uint256[N_COINS],
    new_D: uint256,
    K0_prev: uint256 = 0,
) -> uint256:
    """
    @notice Tweaks price_oracle, last_price and conditionally adjusts
            price_scale. This is called whenever there is an unbalanced
            liquidity operation: _exchange, add_liquidity, or
            remove_liquidity_one_coin.
    @dev Contains main liquidity rebalancing logic, by tweaking `price_scale`.
    @param A_gamma Array of A and gamma parameters.
    @param _xp Array of current balances.
    @param new_D New D value.
    @param K0_prev Initial guess for `newton_D`.
    """

    # ---------------------------- Read storage ------------------------------

    rebalancing_params: uint256[3] = self._unpack(
        self.packed_rebalancing_params
    )  # <---------- Contains: allowed_extra_profit, adjustment_step, ma_time.
    price_oracle: uint256[N_COINS - 1] = self._unpack_prices(
        self.price_oracle_packed
    )
    last_prices: uint256[N_COINS - 1] = self._unpack_prices(
        self.last_prices_packed
    )
    packed_price_scale: uint256 = self.price_scale_packed
    price_scale: uint256[N_COINS - 1] = self._unpack_prices(
        packed_price_scale
    )

    total_supply: uint256 = self.totalSupply
    old_xcp_profit: uint256 = self.xcp_profit
    old_virtual_price: uint256 = self.virtual_price
    last_prices_timestamp: uint256 = self.last_prices_timestamp

    # ----------------------- Update MA if needed ----------------------------

    if last_prices_timestamp < block.timestamp:

        #   The moving average price oracle is calculated using the last_price
        #      of the trade at the previous block, and the price oracle logged
        #              before that trade. This can happen only once per block.

        # ------------------ Calculate moving average params -----------------

        alpha: uint256 = MATH.wad_exp(
            -convert(
                unsafe_div(
                    (block.timestamp - last_prices_timestamp) * 10**18,
                    rebalancing_params[2]  # <----------------------- ma_time.
                ),
                int256,
            )
        )

        for k in range(N_COINS - 1):

            # ----------------- We cap state price that goes into the EMA with
            #                                                 2 x price_scale.
            price_oracle[k] = unsafe_div(
                min(last_prices[k], 2 * price_scale[k]) * (10**18 - alpha) +
                price_oracle[k] * alpha,  # ^-------- Cap spot price into EMA.
                10**18
            )

        self.price_oracle_packed = self._pack_prices(price_oracle)
        self.last_prices_timestamp = block.timestamp  # <---- Store timestamp.

    #                  price_oracle is used further on to calculate its vector
    #            distance from price_scale. This distance is used to calculate
    #                  the amount of adjustment to be done to the price_scale.

    # ------------------ If new_D is set to 0, calculate it ------------------

    D_unadjusted: uint256 = new_D
    if new_D == 0:  #  <--------------------------- _exchange sets new_D to 0.
        D_unadjusted = MATH.newton_D(A_gamma[0], A_gamma[1], _xp, K0_prev)

    # ----------------------- Calculate last_prices --------------------------

    last_prices = MATH.get_p(_xp, D_unadjusted, A_gamma)
    for k in range(N_COINS - 1):
        last_prices[k] = unsafe_div(last_prices[k] * price_scale[k], 10**18)
    self.last_prices_packed = self._pack_prices(last_prices)

    # ---------- Update profit numbers without price adjustment first --------

    xp: uint256[N_COINS] = empty(uint256[N_COINS])
    xp[0] = unsafe_div(D_unadjusted, N_COINS)
    for k in range(N_COINS - 1):
        xp[k + 1] = D_unadjusted * 10**18 / (N_COINS * price_scale[k])

    # ------------------------- Update xcp_profit ----------------------------

    xcp_profit: uint256 = 10**18
    virtual_price: uint256 = 10**18

    if old_virtual_price > 0:

        xcp: uint256 = MATH.geometric_mean(xp)
        virtual_price = 10**18 * xcp / total_supply

        xcp_profit = unsafe_div(
            old_xcp_profit * virtual_price,
            old_virtual_price
        )  # <---------------- Safu to do unsafe_div as old_virtual_price > 0.

        #       If A and gamma are not undergoing ramps (t < block.timestamp),
        #         ensure new virtual_price is not less than old virtual_price,
        #                                        else the pool suffers a loss.
        if self.future_A_gamma_time < block.timestamp:
            assert virtual_price > old_virtual_price, "Loss"

    self.xcp_profit = xcp_profit

    # ------------ Rebalance liquidity if there's enough profits to adjust it:
    if virtual_price * 2 - 10**18 > xcp_profit + 2 * rebalancing_params[0]:
        #                          allowed_extra_profit --------^

        # ------------------- Get adjustment step ----------------------------

        #                Calculate the vector distance between price_scale and
        #                                                        price_oracle.
        norm: uint256 = 0
        ratio: uint256 = 0
        for k in range(N_COINS - 1):

            ratio = unsafe_div(price_oracle[k] * 10**18, price_scale[k])
            # unsafe_div because we did safediv before ----^

            if ratio > 10**18:
                ratio = unsafe_sub(ratio, 10**18)
            else:
                ratio = unsafe_sub(10**18, ratio)
            norm = unsafe_add(norm, ratio**2)

        norm = isqrt(norm)  # <-------------------- isqrt is not in base 1e18.
        adjustment_step: uint256 = max(
            rebalancing_params[1], unsafe_div(norm, 5)
        )  #           ^------------------------------------- adjustment_step.

        if norm > adjustment_step:  # <---------- We only adjust prices if the
            #          vector distance between price_oracle and price_scale is
            #             large enough. This check ensures that no rebalancing
            #           occurs if the distance is low i.e. the pool prices are
            #                                     pegged to the oracle prices.

            # ------------------------------------- Calculate new price scale.

            p_new: uint256[N_COINS - 1] = empty(uint256[N_COINS - 1])
            for k in range(N_COINS - 1):
                p_new[k] = unsafe_div(
                    price_scale[k] * unsafe_sub(norm, adjustment_step)
                    + adjustment_step * price_oracle[k],
                    norm
                )  # <- norm is non-zero and gt adjustment_step; unsafe = safe

            # ---------------- Update stale xp (using price_scale) with p_new.
            xp = _xp
            for k in range(N_COINS - 1):
                xp[k + 1] = unsafe_div(_xp[k + 1] * p_new[k], price_scale[k])
                # unsafe_div because we did safediv before ----^

            # ------------------------------------------ Update D with new xp.
            D: uint256 = MATH.newton_D(A_gamma[0], A_gamma[1], xp, 0)

            for k in range(N_COINS):
                frac: uint256 = xp[k] * 10**18 / D  # <----- Check validity of
                assert (frac > 10**16 - 1) and (frac < 10**20 + 1)  #   p_new.

            xp[0] = D / N_COINS
            for k in range(N_COINS - 1):
                xp[k + 1] = D * 10**18 / (N_COINS * p_new[k])  # <---- Convert
                #                                           xp to real prices.

            # ---------- Calculate new virtual_price using new xp and D. Reuse
            #              `old_virtual_price` (but it has new virtual_price).
            old_virtual_price = unsafe_div(
                10**18 * MATH.geometric_mean(xp), total_supply
            )  # <----- unsafe_div because we did safediv before (if vp>1e18)

            # ---------------------------- Proceed if we've got enough profit.
            if (
                old_virtual_price > 10**18 and
                2 * old_virtual_price - 10**18 > xcp_profit
            ):

                packed_price_scale = self._pack_prices(p_new)

                self.D = D
                self.virtual_price = old_virtual_price
                self.price_scale_packed = packed_price_scale

                return packed_price_scale

    # --------- price_scale was not adjusted. Update the profit counter and D.
    self.D = D_unadjusted
    self.virtual_price = virtual_price

    return packed_price_scale
α=epower
power=(block.timestamp−last_prices_timestamp)×1018ma_time
EMA=min(last_prices,2×price_scale)×(1018−α)+price_oracle×α1018

Note: The state price that goes into the EMA is capped with 2 x price_scale to prevent manipulation.

Variable Description
block.timestamp Timestamp of the block. Since all transactions within a block share the same timestamp, EMA oracles can only be updated once per block.
last_prices_timestamp Timestamp when the EMA oracle was last updated.
ma_time Time window for the moving-average oracle.
last_prices Last stored spot price of the coin to calculate the price oracle for.
price_scale Price scale value of the coin to calculate the price oracle for.
price_oracle Price oracle value of the coin to calculate the price oracle for.
alpha Weighting multiplier that adjusts the impact of the latest spot value versus the previous EMA in the new EMA calculation.

The AMM implementation uses several private variables to pack and store key values, which are used for calculating the EMA oracle.

Info

Some storage variables pack multiple values into a single entry to save on gas costs. These values are unpacked when needed for use.

Source Code 
PRICE_SIZE: constant(uint128) = 256 / (N_COINS - 1)
PRICE_MASK: constant(uint256) = 2**PRICE_SIZE - 1

last_prices_packed: uint256

@internal
@view
def _pack_prices(prices_to_pack: uint256[N_COINS-1]) -> uint256:
    """
    @notice Packs N_COINS-1 prices into a uint256.
    @param prices_to_pack The prices to pack
    @return uint256 An integer that packs prices
    """
    packed_prices: uint256 = 0
    p: uint256 = 0
    for k in range(N_COINS - 1):
        packed_prices = packed_prices << PRICE_SIZE
        p = prices_to_pack[N_COINS - 2 - k]
        assert p < PRICE_MASK
        packed_prices = p | packed_prices
    return packed_prices

PRICE_SIZE: constant(uint128) = 256 / (N_COINS - 1)
PRICE_MASK: constant(uint256) = 2**PRICE_SIZE - 1

last_prices_packed: uint256

@internal
@view
def _unpack_prices(_packed_prices: uint256) -> uint256[2]:
    """
    @notice Unpacks N_COINS-1 prices from a uint256.
    @param _packed_prices The packed prices
    @return uint256[2] Unpacked prices
    """
    unpacked_prices: uint256[N_COINS-1] = empty(uint256[N_COINS-1])
    packed_prices: uint256 = _packed_prices
    for k in range(N_COINS - 1):
        unpacked_prices[k] = packed_prices & PRICE_MASK
        packed_prices = packed_prices >> PRICE_SIZE

    return unpacked_prices

  • last_prices_packed stores the latest prices of all coins, packaging them into a single variable. When using the prices, they must be unpacked using the _unpack_prices method.

  • price_oracle_packed stores the moving average values of the coins. This variable includes two values: the first represents the moving average for coin(1) relative to coin(0), and the second represents the moving average for coin(2) relative to coin(0). The price_oracle method unpacks these values and return the price oracle of the coin at index k.

  • price_scale_packed functions similarly by packing the price scales of coin 1 and 2 with respect to coin 0. The price_scale method unpacks these values and returns the price scale of the coin at index k.

  • last_prices_timestamp marks the timestamp when the price_oracle for a coin was last updated.

Oracle and Price Methods¶

price_oracle¶

CurveTricryptoOptimizedWETH.price_oracle(k: uint256) -> uint256:

Revert

This function reverts if i >= 2.

Function to calculate the exponential moving average (EMA) price for the coin at index k with regard to the coin at index 0. The oracle is an exponential moving average, with a periodicity determined by ma_time. The aggregated prices are cached state prices (dy/dx) calculated AFTER the latest trade.

The moving average price oracle is calculated using the last_price of the trade at the previous block, and the price oracle logged before that trade. This can happen only once per block.

Returns: EMA price of coin k (uint256).

Input Type Description
k uint256 Index of the coin.
Source code 
price_scale_packed: uint256  # <------------------------ Internal price scale.
price_oracle_packed: uint256  # <------- Price target given by moving average.

last_prices_packed: uint256
last_prices_timestamp: public(uint256)

@external
@view
@nonreentrant("lock")
def price_oracle(k: uint256) -> uint256:
    """
    @notice Returns the oracle price of the coin at index `k` w.r.t the coin
            at index 0.
    @dev The oracle is an exponential moving average, with a periodicity
        determined by `self.ma_time`. The aggregated prices are cached state
        prices (dy/dx) calculated AFTER the latest trade.
    @param k The index of the coin.
    @return uint256 Price oracle value of kth coin.
    """
    price_oracle: uint256 = self._unpack_prices(self.price_oracle_packed)[k]
    price_scale: uint256 = self._unpack_prices(self.price_scale_packed)[k]
    last_prices_timestamp: uint256 = self.last_prices_timestamp

    if last_prices_timestamp < block.timestamp:  # <------------ Update moving
        #                                                   average if needed.

        last_prices: uint256 = self._unpack_prices(self.last_prices_packed)[k]
        ma_time: uint256 = self._unpack(self.packed_rebalancing_params)[2]
        alpha: uint256 = MATH.wad_exp(
            -convert(
                (block.timestamp - last_prices_timestamp) * 10**18 / ma_time,
                int256,
            )
        )

        # ---- We cap state price that goes into the EMA with 2 x price_scale.
        return (
            min(last_prices, 2 * price_scale) * (10**18 - alpha) +
            price_oracle * alpha
        ) / 10**18

    return price_oracle

>>> CurveTricryptoOptimizedWETH.price_oracle(0)
66466761042718407573921

>>> CurveTricryptoOptimizedWETH.price_oracle(1)
3243401255685792725933

price_scale¶

CurveTricryptoOptimizedWETH.price_scale(k: uint256) -> uint256:

Revert

This function reverts if i >= 2.

Getter for the price scale of the coin at index k.

Returns: price scale of the coin k (uint256).

Input Type Description
k uint256 Index of the coin.
Source code 
price_scale_packed: uint256  # <------------------------ Internal price scale.

@external
@view
def price_scale(k: uint256) -> 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.
    @param k The index of the coin.
    @return uint256 Price scale of coin.
    """
    return self._unpack_prices(self.price_scale_packed)[k]

>>> CurveTricryptoOptimizedWETH.price_scale(0)
64955165867890305070839

>>> CurveTricryptoOptimizedWETH.price_scale(1)
3133935659389092150237

last_prices¶

CurveTricryptoOptimizedWETH.last_prices(k: uint256) -> uint256:

Revert

This function reverts if i >= 2.

Getter method for the last stored price for coin at index value k, stored in last_prices_packed.

Returns: last stored spot price of coin k (uint256).

Input Type Description
k uint256 Index of the coin.
Source code 
last_prices_packed: uint256

@external
@view
def last_prices(k: uint256) -> uint256:
    """
    @notice Returns last price of the coin at index `k` w.r.t the coin
            at index 0.
    @dev last_prices returns the quote by the AMM for an infinitesimally small swap
        after the last trade. It is not equivalent to the last traded price, and
        is computed by taking the partial differential of `x` w.r.t `y`. The
        derivative is calculated in `get_p` and then multiplied with price_scale
        to give last_prices.
    @param k The index of the coin.
    @return uint256 Last logged price of coin.
    """
    return self._unpack_prices(self.last_prices_packed)[k]

@internal
@view
def _unpack_prices(_packed_prices: uint256) -> uint256[2]:
    """
    @notice Unpacks N_COINS-1 prices from a uint256.
    @param _packed_prices The packed prices
    @return uint256[2] Unpacked prices
    """
    unpacked_prices: uint256[N_COINS-1] = empty(uint256[N_COINS-1])
    packed_prices: uint256 = _packed_prices
    for k in range(N_COINS - 1):
        unpacked_prices[k] = packed_prices & PRICE_MASK
        packed_prices = packed_prices >> PRICE_SIZE

    return unpacked_prices

>>> CurveTricryptoOptimizedWETH.last_prices(0)
66512510695325991643669

>>> CurveTricryptoOptimizedWETH.last_prices(1)
3249719806881710136102

last_prices_timestamp¶

CurveTricryptoOptimizedWETH.last_prices_timestamp() -> uint256: view

Getter for the timestamp when the EMA price oracle was updated the last time.

Returns: timestamp (uint256).

Source code 
last_prices_timestamp: public(uint256)

>>> CurveTricryptoOptimizedWETH.last_prices_timestamp()
1713167903

ma_time¶

CurveTricryptoOptimizedWETH.ma_time() -> uint256: view

Getter for the exponential moving average time for the price oracle. This value can be adjusted via commit_new_parameters(), as detailed in the admin controls section.

Returns: periodicity of the EMA (uint256)

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

@view
@external
def ma_time() -> uint256:
    """
    @notice Returns the current moving average time in seconds
    @dev To get time in seconds, the parameter is multipled by ln(2)
        One can expect off-by-one errors here.
    @return uint256 ma_time value.
    """
    return self._unpack(self.packed_rebalancing_params)[2] * 694 / 1000

>>> CurveTricryptoOptimizedWETH.ma_time()
600

lp_price¶

CurveTricryptoOptimizedWETH.lp_price() -> uint256: view

Getter for the price of the LP token, calculated as follows:

lp token price=3×virtual_price×(price_oracle[0]×price_oracle[1])10243

Returns: LP token price (uint256).

Source code 
price_oracle_packed: uint256  # <------- Price target given by moving average.

@external
@view
@nonreentrant("lock")
def lp_price() -> uint256:
    """
    @notice Calculates the current price of the LP token w.r.t coin at the
            0th index
    @return uint256 LP price.
    """

    price_oracle: uint256[N_COINS-1] = self._unpack_prices(
        self.price_oracle_packed
    )
    return (
        3 * self.virtual_price * MATH.cbrt(price_oracle[0] * price_oracle[1])
    ) / 10**24

>>> CurveTricryptoOptimizedWETH.lp_price()
1809325066767569054740

get_virtual_price¶

CurveTricryptoOptimizedWETH.get_virtual_price() -> uint256:

Function to calculate the current virtual price of the pool 's LP token.

Returns: virtual price (uint256).

Source code 
totalSupply: public(uint256)

@external
@view
@nonreentrant("lock")
def get_virtual_price() -> uint256:
    """
    @notice Calculates the current virtual price of the pool LP token.
    @dev Not to be confused with `self.virtual_price` which is a cached
        virtual price.
    @return uint256 Virtual Price.
    """
    return 10**18 * self.get_xcp(self.D) / self.totalSupply

>>> CurveTricryptoOptimizedWETH.get_virtual_price()
1005849271542625678

virtual_price¶

CurveTricryptoOptimizedWETH.virtual_price() -> uint256: view

Getter for the cached virtual price.

Returns: cached virtual price (uint256).

Source code 
virtual_price: public(uint256)  # <------ Cached (fast to read) virtual price.
#                          The cached `virtual_price` is also used internally.

>>> CurveTricryptoOptimizedWETH.virtual_price()
1005849271542625678

Updating Oracles and Other Storage Variables¶

The EMA oracle and other storage variables are updated each time the internal tweak_price function is called. This function tweaksprice_oracle and last_price, and conditionally adjusts price_scale.

The tweak_price function is called whenever there is an unbalanced liquidity operation, including:

  • _exchange
  • add_liquidity
  • remove_liquidity_one_coin
Source Code 
@internal
def tweak_price(
    A_gamma: uint256[2],
    _xp: uint256[N_COINS],
    new_D: uint256,
    K0_prev: uint256 = 0,
) -> uint256:
    """
    @notice Tweaks price_oracle, last_price and conditionally adjusts
            price_scale. This is called whenever there is an unbalanced
            liquidity operation: _exchange, add_liquidity, or
            remove_liquidity_one_coin.
    @dev Contains main liquidity rebalancing logic, by tweaking `price_scale`.
    @param A_gamma Array of A and gamma parameters.
    @param _xp Array of current balances.
    @param new_D New D value.
    @param K0_prev Initial guess for `newton_D`.
    """

    # ---------------------------- Read storage ------------------------------

    rebalancing_params: uint256[3] = self._unpack(
        self.packed_rebalancing_params
    )  # <---------- Contains: allowed_extra_profit, adjustment_step, ma_time.
    price_oracle: uint256[N_COINS - 1] = self._unpack_prices(
        self.price_oracle_packed
    )
    last_prices: uint256[N_COINS - 1] = self._unpack_prices(
        self.last_prices_packed
    )
    packed_price_scale: uint256 = self.price_scale_packed
    price_scale: uint256[N_COINS - 1] = self._unpack_prices(
        packed_price_scale
    )

    total_supply: uint256 = self.totalSupply
    old_xcp_profit: uint256 = self.xcp_profit
    old_virtual_price: uint256 = self.virtual_price
    last_prices_timestamp: uint256 = self.last_prices_timestamp

    # ----------------------- Update MA if needed ----------------------------

    if last_prices_timestamp < block.timestamp:

        #   The moving average price oracle is calculated using the last_price
        #      of the trade at the previous block, and the price oracle logged
        #              before that trade. This can happen only once per block.

        # ------------------ Calculate moving average params -----------------

        alpha: uint256 = MATH.wad_exp(
            -convert(
                unsafe_div(
                    (block.timestamp - last_prices_timestamp) * 10**18,
                    rebalancing_params[2]  # <----------------------- ma_time.
                ),
                int256,
            )
        )

        for k in range(N_COINS - 1):

            # ----------------- We cap state price that goes into the EMA with
            #                                                 2 x price_scale.
            price_oracle[k] = unsafe_div(
                min(last_prices[k], 2 * price_scale[k]) * (10**18 - alpha) +
                price_oracle[k] * alpha,  # ^-------- Cap spot price into EMA.
                10**18
            )

        self.price_oracle_packed = self._pack_prices(price_oracle)
        self.last_prices_timestamp = block.timestamp  # <---- Store timestamp.

    #                  price_oracle is used further on to calculate its vector
    #            distance from price_scale. This distance is used to calculate
    #                  the amount of adjustment to be done to the price_scale.

    # ------------------ If new_D is set to 0, calculate it ------------------

    D_unadjusted: uint256 = new_D
    if new_D == 0:  #  <--------------------------- _exchange sets new_D to 0.
        D_unadjusted = MATH.newton_D(A_gamma[0], A_gamma[1], _xp, K0_prev)

    # ----------------------- Calculate last_prices --------------------------

    last_prices = MATH.get_p(_xp, D_unadjusted, A_gamma)
    for k in range(N_COINS - 1):
        last_prices[k] = unsafe_div(last_prices[k] * price_scale[k], 10**18)
    self.last_prices_packed = self._pack_prices(last_prices)

    # ---------- Update profit numbers without price adjustment first --------

    xp: uint256[N_COINS] = empty(uint256[N_COINS])
    xp[0] = unsafe_div(D_unadjusted, N_COINS)
    for k in range(N_COINS - 1):
        xp[k + 1] = D_unadjusted * 10**18 / (N_COINS * price_scale[k])

    # ------------------------- Update xcp_profit ----------------------------

    xcp_profit: uint256 = 10**18
    virtual_price: uint256 = 10**18

    if old_virtual_price > 0:

        xcp: uint256 = MATH.geometric_mean(xp)
        virtual_price = 10**18 * xcp / total_supply

        xcp_profit = unsafe_div(
            old_xcp_profit * virtual_price,
            old_virtual_price
        )  # <---------------- Safu to do unsafe_div as old_virtual_price > 0.

        #       If A and gamma are not undergoing ramps (t < block.timestamp),
        #         ensure new virtual_price is not less than old virtual_price,
        #                                        else the pool suffers a loss.
        if self.future_A_gamma_time < block.timestamp:
            assert virtual_price > old_virtual_price, "Loss"

    self.xcp_profit = xcp_profit

    # ------------ Rebalance liquidity if there's enough profits to adjust it:
    if virtual_price * 2 - 10**18 > xcp_profit + 2 * rebalancing_params[0]:
        #                          allowed_extra_profit --------^

        # ------------------- Get adjustment step ----------------------------

        #                Calculate the vector distance between price_scale and
        #                                                        price_oracle.
        norm: uint256 = 0
        ratio: uint256 = 0
        for k in range(N_COINS - 1):

            ratio = unsafe_div(price_oracle[k] * 10**18, price_scale[k])
            # unsafe_div because we did safediv before ----^

            if ratio > 10**18:
                ratio = unsafe_sub(ratio, 10**18)
            else:
                ratio = unsafe_sub(10**18, ratio)
            norm = unsafe_add(norm, ratio**2)

        norm = isqrt(norm)  # <-------------------- isqrt is not in base 1e18.
        adjustment_step: uint256 = max(
            rebalancing_params[1], unsafe_div(norm, 5)
        )  #           ^------------------------------------- adjustment_step.

        if norm > adjustment_step:  # <---------- We only adjust prices if the
            #          vector distance between price_oracle and price_scale is
            #             large enough. This check ensures that no rebalancing
            #           occurs if the distance is low i.e. the pool prices are
            #                                     pegged to the oracle prices.

            # ------------------------------------- Calculate new price scale.

            p_new: uint256[N_COINS - 1] = empty(uint256[N_COINS - 1])
            for k in range(N_COINS - 1):
                p_new[k] = unsafe_div(
                    price_scale[k] * unsafe_sub(norm, adjustment_step)
                    + adjustment_step * price_oracle[k],
                    norm
                )  # <- norm is non-zero and gt adjustment_step; unsafe = safe

            # ---------------- Update stale xp (using price_scale) with p_new.
            xp = _xp
            for k in range(N_COINS - 1):
                xp[k + 1] = unsafe_div(_xp[k + 1] * p_new[k], price_scale[k])
                # unsafe_div because we did safediv before ----^

            # ------------------------------------------ Update D with new xp.
            D: uint256 = MATH.newton_D(A_gamma[0], A_gamma[1], xp, 0)

            for k in range(N_COINS):
                frac: uint256 = xp[k] * 10**18 / D  # <----- Check validity of
                assert (frac > 10**16 - 1) and (frac < 10**20 + 1)  #   p_new.

            xp[0] = D / N_COINS
            for k in range(N_COINS - 1):
                xp[k + 1] = D * 10**18 / (N_COINS * p_new[k])  # <---- Convert
                #                                           xp to real prices.

            # ---------- Calculate new virtual_price using new xp and D. Reuse
            #              `old_virtual_price` (but it has new virtual_price).
            old_virtual_price = unsafe_div(
                10**18 * MATH.geometric_mean(xp), total_supply
            )  # <----- unsafe_div because we did safediv before (if vp>1e18)

            # ---------------------------- Proceed if we've got enough profit.
            if (
                old_virtual_price > 10**18 and
                2 * old_virtual_price - 10**18 > xcp_profit
            ):

                packed_price_scale = self._pack_prices(p_new)

                self.D = D
                self.virtual_price = old_virtual_price
                self.price_scale_packed = packed_price_scale

                return packed_price_scale

    # --------- price_scale was not adjusted. Update the profit counter and D.
    self.D = D_unadjusted
    self.virtual_price = virtual_price

    return packed_price_scale