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Whitepapers, Derivations and Useful Resources

Whitepapers

  • Curve DAO

    Whitepaper on the general structure and workingsof the Curve DAO.

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  • Stableswap

    Whitepaper on the Stableswap invariant.

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  • Cryptoswap

    Whitepaper on the Cryptoswap invariant.

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  • Curve Stablecoin & LLAMMA

    Whitepaper on the workings of the Curve Stablecoin and Linear-Liquidation Automated Market Maker Algorithm (LLAMMA).

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StableSwap Derivations

The StableSwap invariant is defined as:

\[ A n^n \sum x_i + D = A D n^n + \frac{D^{n+1}}{n^n \prod x_i} \]

Where: - \(A\) is the amplification coefficient
- \(n\) is the number of coins
- \(x_i\) is the amount of the \(i\)-th coin
- \(D\) is the invariant (i.e. the total virtual balance when all coins are equal)

Newton’s Method for Solving \(D\)

We derive Newton's iteration for solving \(D\) given \(\{x_i\}_{i=1}^n\) and \(A\).

Start with:

\[ f(D) = A n^n \sum x_i + D(1 - A n^n) - \frac{D^{n+1}}{n^n \prod x_i} = 0 \]

Or equivalently:

\[ f(D) = \frac{D^{n+1}}{n^n \prod x_i} - D(1 - A n^n) - A n^n \sum x_i \]

Taking the derivative:

\[ f'(D) = \frac{(n + 1) D^n}{n^n \prod x_i} + A n^n - 1 \]

Newton Iteration

Using Newton’s formula:

\[ D_{\text{new}} = D - \frac{f(D)}{f'(D)} = \frac{D f'(D) - f(D)}{f'(D)} \]

Substituting \(f(D)\) and \(f'(D)\):

\[ \begin{aligned} D_{\text{new}} &= \frac{D \left(\frac{(n+1) D^n}{n^n \prod x_i} + A n^n - 1\right) - \left(\frac{D^{n+1}}{n^n \prod x_i} - D(1 - A n^n) - A n^n \sum x_i \right)}{\frac{(n+1) D^n}{n^n \prod x_i} + A n^n - 1} \\ &= \frac{\frac{(n+1) D^{n+1}}{n^n \prod x_i} + A n^n D - D - \frac{D^{n+1}}{n^n \prod x_i} + D - A n^n D + A n^n \sum x_i}{\frac{(n+1) D^n}{n^n \prod x_i} + A n^n - 1} \\ &= \frac{\frac{n D^{n+1}}{n^n \prod x_i} + A n^n \sum x_i}{\frac{(n+1) D^n}{n^n \prod x_i} + A n^n - 1} \end{aligned} \]

This corresponds to the newton_D function in math.vy.

Newton’s Method for Solving \(x_j\)

Now derive Newton’s iteration for solving \(x_j\), given \(\{x_i\}_{i \neq j}\), \(A\), and \(D\).

Start with the invariant:

\[ A n^n \sum x_i + D = A D n^n + \frac{D^{n+1}}{n^n \prod x_i} \]

Isolate \(x_j\):

\[ A n^n \left(x_j + \sum_{i \neq j} x_i\right) + D = A D n^n + \frac{D^{n+1}}{n^n (x_j \prod_{i \neq j} x_i)} \]

Multiply both sides by \(x_j\):

\[ A n^n \left(x_j^2 + x_j \sum_{i \neq j} x_i \right) + D x_j = A D n^n x_j + \frac{D^{n+1}}{n^n \prod_{i \neq j} x_i} \]

Divide by \(A n^n\):

\[ x_j^2 + x_j \sum_{i \neq j} x_i + \frac{D}{A n^n} x_j = D x_j + \frac{D^{n+1}}{A n^{2n} \prod_{i \neq j} x_i} \]

Bring terms to one side:

\[ x_j^2 + x_j \left(\sum_{i \neq j} x_i + \frac{D}{A n^n} - D\right) - \frac{D^{n+1}}{A n^{2n} \prod_{i \neq j} x_i} = 0 \]

Define:

\[ f(x_j) = x_j^2 + x_j \left(\sum_{i \neq j} x_i + \frac{D}{A n^n} - D\right) - \frac{D^{n+1}}{A n^{2n} \prod_{i \neq j} x_i} \]

Taking the derivative:

\[ f'(x_j) = 2 x_j + \sum_{i \neq j} x_i + \frac{D}{A n^n} - D \]

Newton Iteration

\[ x_j^{\text{new}} = x_j - \frac{f(x_j)}{f'(x_j)} = \frac{x_j f'(x_j) - f(x_j)}{f'(x_j)} \]

Substitute in the expressions:

\[ \begin{aligned} x_j^{\text{new}} &= \frac{x_j \left(2 x_j + \sum_{i \neq j} x_i + \frac{D}{A n^n} - D\right) - \left(x_j^2 + x_j \left(\sum_{i \neq j} x_i + \frac{D}{A n^n} - D\right) - \frac{D^{n+1}}{A n^{2n} \prod_{i \neq j} x_i} \right)}{2 x_j + \sum_{i \neq j} x_i + \frac{D}{A n^n} - D} \\ &= \frac{x_j^2 + \frac{D^{n+1}}{A n^{2n} \prod_{i \neq j} x_i}}{2 x_j + \sum_{i \neq j} x_i + \frac{D}{A n^n} - D} \end{aligned} \]

This corresponds to the newton_x function in math.vy.

Here’s a cleaned-up and well-structured version of your Cryptoswap Derivations document. The content remains mathematically equivalent, but the presentation is clearer, more readable, and easier to follow:

Cryptoswap Derivations

Newton Step for newton_D() in Tricrypto and Twocrypto

This derivation explains the mathematical logic behind the newton_D() function used in Curve’s tricrypto and twocrypto pools.

Invariant Function Definitions

We start with the core function:

\[ F = K D^{n-1} S + P - K D^n - \left(\frac{D}{n}\right)^n \]

Where: - \( D \): the invariant - \( S = \sum x_i \) - \( P = \prod x_i \) - \( n \): number of tokens (typically 2 or 3) - \( \gamma \): price scale parameter - \( A \): amplification coefficient

Intermediate Definitions

\[ K = \frac{A K_0 \gamma^2}{(\gamma + 1 - K_0)^2}, \quad K_0 = \frac{P n^n}{D^n} \]
\[ g = \gamma + 1 - K_0, \quad \hat{A} = n^n A \]
\[ m_1 = \frac{D g^2}{\hat{A} \gamma^2}, \quad m_2 = \frac{2n K_0}{g} \]
\[ \text{neg_fprime} = S + S m_2 + \frac{m_1 n}{K_0} - m_2 D \]

Derivative of \( K \)

\[ K' = \left( \frac{A K_0 \gamma^2}{(\gamma + 1 - K_0)^2} \right)' \]
\[ = \left( \frac{A \gamma^2}{(\gamma + 1 - K_0)^2} + \frac{2 A K_0 \gamma^2}{(\gamma + 1 - K_0)^3} \right) \cdot \left( -n \frac{P n^n}{D^{n+1}} \right) \]
\[ = -n \left( \frac{A \gamma^2}{g^2} + \frac{2 A K_0 \gamma^2}{g^3} \right) \cdot \frac{K_0}{D} \]

Derivative of \( F \)

\[ F = K D^{n-1} S + P - K D^n - \left( \frac{D}{n} \right)^n \]

Taking the derivative:

\[ F' = (n-1) K D^{n-2} S - n K D^{n-1} - \frac{D^{n-1}}{n^{n-1}} + K' D^{n-1} (S - D) \]

Substitute \( K \) and \( K' \):

\[ F' = - \frac{A K_0 \gamma^2}{g^2} D^{n-2} S - \frac{2 n A K_0^2 \gamma^2}{g^3} D^{n-2} (S - D) - \frac{D^{n-1}}{n^{n-1}} \]

Derivation of \( \frac{F}{F'} \)

\[ \frac{F}{F'} = \frac { \frac{A K_0 \gamma^2}{g^2} D^{n-1} S + P - \frac{A K_0 \gamma^2}{g^2} D^n - \left( \frac{D}{n} \right)^n } { - \frac{A K_0 \gamma^2}{g^2} D^{n-2} S + \frac{2 n A K_0^2 \gamma^2}{g^3} D^{n-2}(D - S) - \left( \frac{D}{n} \right)^{n-1} } \]

Divide numerator and denominator by \( D^n / n^n \):

\[ = \frac { \frac{\hat{A} K_0 \gamma^2}{g^2} D^{-1} S + K_0 - \frac{\hat{A} K_0 \gamma^2}{g^2} - 1 } { - \frac{\hat{A} K_0 \gamma^2}{g^2} D^{-2} S + \frac{2n \hat{A} K_0^2 \gamma^2}{g^3} D^{-2} (D - S) - \frac{n}{D} } \]

Divide numerator and denominator by \( \frac{\hat{A} \gamma^2}{g^2 D} \):

\[ = \frac { K_0 S + m_1 (K_0 - 1) - K_0 D } { - \frac{K_0 S}{D} + \frac{m_2 K_0}{D} (D - S) - \frac{n m_1}{D} } \]

Multiply numerator and denominator by \( D \):

\[ = \frac { K_0 S D + m_1 (K_0 - 1) D - K_0 D^2 } { - K_0 S + m_2 K_0 (D - S) - n m_1 } \]

Divide numerator and denominator by \( K_0 \):

\[ = \frac { S D + m_1 \left(1 - \frac{1}{K_0}\right) D - D^2 } { - S + m_2 (D - S) - \frac{n m_1}{K_0} } \]

Distribute:

\[ = \frac { S D + m_1 \left(1 - \frac{1}{K_0} \right) D - D^2 } { - S - m_2 S + m_2 D - \frac{n m_1}{K_0} } \]

Substitute the denominator with \(-\text{neg_fprime}\):

\[ = \frac{S D + m_1 \left(1 - \frac{1}{K_0} \right) D - D^2}{-\text{neg_fprime}} \]

Newton Iteration Step

\[ D_{k+1} = D_k - \frac{F}{F'} = D_k + \frac{S D_k + m_1 \left(1 - \frac{1}{K_0} \right) D_k - D_k^2}{\text{neg_fprime}} \]
\[ = \frac{\text{neg_fprime} D_k + S D_k + m_1 \left(1 - \frac{1}{K_0} \right) D_k - D_k^2}{\text{neg_fprime}} \]

Final Form: Positive and Negative Contributions

Separate into two parts:

Positive Term \( D_+ \):

\[ D_+ = \frac{(\text{neg_fprime} + S) D_k}{\text{neg_fprime}} \]

Negative Term \( D_- \):

\[ D_- = \frac{D_k^2 - m_1 \left( \frac{K_0 - 1}{K_0} \right) D_k}{\text{neg_fprime}} \]

Final Newton Step:

\[ D_{k+1} = D_+ - D_- \]

Useful Resources

Stableswap

Cryptoswap

Curve Stablecoin (crvUSD)

Lending

Curve Integration