max long comment cleanup

crates/hyperdrive-math/src/long/fees.rs

@@ -12,7 +12,6 @@ impl State {
     /// c(x) = \phi_{c} \cdot \left( \tfrac{1}{p} - 1 \right) \cdot x
     /// $$
     pub fn open_long_curve_fees(&self, base_amount: FixedPoint) -> FixedPoint {
-        // curve fee = ((1 / p) - 1) * phi_curve * dz
         self.curve_fee() * ((fixed!(1e18) / self.get_spot_price()) - fixed!(1e18)) * base_amount
     }
 
@@ -36,7 +35,7 @@ impl State {
         bond_amount: FixedPoint,
         normalized_time_remaining: FixedPoint,
     ) -> FixedPoint {
-        // ((1 - p) * phi_curve * d_y * t) / c
+        // curve_fee = ((1 - p) * phi_curve * d_y * t) / c
         self.curve_fee()
             * (fixed!(1e18) - self.get_spot_price())
             * bond_amount.mul_div_down(normalized_time_remaining, self.vault_share_price())
@@ -49,7 +48,7 @@ impl State {
         bond_amount: FixedPoint,
         normalized_time_remaining: FixedPoint,
     ) -> FixedPoint {
-        // flat fee = (d_y * (1 - t) * phi_flat) / c
+        // flat_fee = (d_y * (1 - t) * phi_flat) / c
         bond_amount.mul_div_down(
             fixed!(1e18) - normalized_time_remaining,
             self.vault_share_price(),

crates/hyperdrive-math/src/long/max.rs

@@ -24,6 +24,10 @@ impl State {
     }
 
     /// Gets the pool's solvency.
+    ///
+    /// $$
+    /// s = z - \tfrac{exposure}{c} - z_min
+    /// $$
     pub fn get_solvency(&self) -> FixedPoint {
         self.share_reserves()
             - self.long_exposure() / self.vault_share_price()
@@ -107,7 +111,7 @@ impl State {
             // numbers.
             //
             // Proceed to the next step of Newton's method. Once we have a
-            // candidate solution, we check to see if the pool is solvent if
+            // candidate solution, we check to see if the pool is solvent after
             // a long is opened with the candidate amount. If the pool isn't
             // solvent, then we're done.
             let maybe_derivative = self.solvency_after_long_derivative(max_base_amount);
@@ -157,7 +161,9 @@ impl State {
         //
         // y_t = (mu * z_t) * ((1 + curveFee * (1 / p_0 - 1) * (1 - flatFee)) / (1 - flatFee)) ** (1 / t_s)
         //
-        // We can use this formula to solve our YieldSpace invariant for z_t:
+        // Our equation for price is the inverse of that used by YieldSpace, which must be considered when
+        // deriving the invariant from the price equation.
+        // With this in mind, we can use this formula to solve our YieldSpace invariant for z_t:
         //
         // k = (c / mu) * (mu * z_t) ** (1 - t_s) +
         //     (
@@ -256,7 +262,7 @@ impl State {
     /// exposure after opening a long with $x$ base as:
     ///
     /// \begin{aligned}
-    /// z(x) &= z_0 + \tfrac{x - g(x)}{c} - z_{min} \\
+    /// z(x) &= z_0 + \tfrac{x - g(x)}{c} \\
     /// e(x) &= e_0 + min(exposure_{c}, 0) + 2 \cdot y(x) - x + g(x) \\
     ///      &= e_0 + min(exposure_{c}, 0) + 2 \cdot p_r^{-1} \cdot x -
     ///             2 \cdot c(x) - x + g(x)
@@ -268,7 +274,7 @@ impl State {
     /// to calculate the approximate ending solvency of:
     ///
     /// $$
-    /// s(x) \approx z(x) - \tfrac{e(x)}{c} - z_{min}
+    /// s(x) \approx z(x) - \tfrac{e(x) - min(exposure_{c}, 0)}{c} - z_{min}
     /// $$
     ///
     /// If we let the initial solvency be given by $s_0$, we can solve for