//------------------------------------------------------------------------------ /* This file is part of rippled: https://github.com/ripple/rippled Copyright (c) 2023 Ripple Labs Inc. Permission to use, copy, modify, and/or distribute this software for any purpose with or without fee is hereby granted, provided that the above copyright notice and this permission notice appear in all copies. THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY SPECIAL , DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ //============================================================================== #include namespace ripple { STAmount ammLPTokens( STAmount const& asset1, STAmount const& asset2, Issue const& lptIssue) { auto const tokens = root2(asset1 * asset2); return toSTAmount(lptIssue, tokens); } /* * Equation 3: * t = T * [(b/B - (sqrt(f2**2 - b/(B*f1)) - f2)) / * (1 + sqrt(f2**2 - b/(B*f1)) - f2)] * where f1 = 1 - tfee, f2 = (1 - tfee/2)/f1 */ STAmount lpTokensIn( STAmount const& asset1Balance, STAmount const& asset1Deposit, STAmount const& lptAMMBalance, std::uint16_t tfee) { auto const f1 = feeMult(tfee); auto const f2 = feeMultHalf(tfee) / f1; Number const r = asset1Deposit / asset1Balance; auto const c = root2(f2 * f2 + r / f1) - f2; auto const t = lptAMMBalance * (r - c) / (1 + c); return toSTAmount(lptAMMBalance.issue(), t); } /* Equation 4 solves equation 3 for b: * Let f1 = 1 - tfee, f2 = (1 - tfee/2)/f1, t1 = t/T, t2 = 1 + t1, R = b/B * then * t1 = [R - sqrt(f2**2 + R/f1) + f2] / [1 + sqrt(f2**2 + R/f1] - f2] => * sqrt(f2**2 + R/f1)*(t1 + 1) = R + f2 + t1*f2 - t1 => * sqrt(f2**2 + R/f1)*t2 = R + t2*f2 - t1 => * sqrt(f2**2 + R/f1) = R/t2 + f2 - t1/t2, let d = f2 - t1/t2 => * sqrt(f2**2 + R/f1) = R/t2 + d => * f2**2 + R/f1 = (R/t2)**2 +2*d*R/t2 + d**2 => * (R/t2)**2 + R*(2*d/t2 - 1/f1) + d**2 - f2**2 = 0 */ STAmount ammAssetIn( STAmount const& asset1Balance, STAmount const& lptAMMBalance, STAmount const& lpTokens, std::uint16_t tfee) { auto const f1 = feeMult(tfee); auto const f2 = feeMultHalf(tfee) / f1; auto const t1 = lpTokens / lptAMMBalance; auto const t2 = 1 + t1; auto const d = f2 - t1 / t2; auto const a = 1 / (t2 * t2); auto const b = 2 * d / t2 - 1 / f1; auto const c = d * d - f2 * f2; return toSTAmount( asset1Balance.issue(), asset1Balance * solveQuadraticEq(a, b, c)); } /* Equation 7: * t = T * (c - sqrt(c**2 - 4*R))/2 * where R = b/B, c = R*fee + 2 - fee */ STAmount lpTokensOut( STAmount const& asset1Balance, STAmount const& asset1Withdraw, STAmount const& lptAMMBalance, std::uint16_t tfee) { Number const fr = asset1Withdraw / asset1Balance; auto const f1 = getFee(tfee); auto const c = fr * f1 + 2 - f1; auto const t = lptAMMBalance * (c - root2(c * c - 4 * fr)) / 2; return toSTAmount(lptAMMBalance.issue(), t); } /* Equation 8 solves equation 7 for b: * c - 2*t/T = sqrt(c**2 - 4*R) => * c**2 - 4*c*t/T + 4*t**2/T**2 = c**2 - 4*R => * -4*c*t/T + 4*t**2/T**2 = -4*R => * -c*t/T + t**2/T**2 = -R -=> * substitute c = R*f + 2 - f => * -(t/T)*(R*f + 2 - f) + (t/T)**2 = -R, let t1 = t/T => * -t1*R*f -2*t1 +t1*f +t1**2 = -R => * R = (t1**2 + t1*(f - 2)) / (t1*f - 1) */ STAmount withdrawByTokens( STAmount const& assetBalance, STAmount const& lptAMMBalance, STAmount const& lpTokens, std::uint16_t tfee) { auto const f = getFee(tfee); Number const t1 = lpTokens / lptAMMBalance; auto const b = assetBalance * (t1 * t1 - t1 * (2 - f)) / (t1 * f - 1); return toSTAmount(assetBalance.issue(), b); } Number square(Number const& n) { return n * n; } STAmount adjustLPTokens( STAmount const& lptAMMBalance, STAmount const& lpTokens, bool isDeposit) { // Force rounding downward to ensure adjusted tokens are less or equal // to requested tokens. saveNumberRoundMode rm(Number::setround(Number::rounding_mode::downward)); if (isDeposit) return (lptAMMBalance + lpTokens) - lptAMMBalance; return (lpTokens - lptAMMBalance) + lptAMMBalance; } std::tuple, STAmount> adjustAmountsByLPTokens( STAmount const& amountBalance, STAmount const& amount, std::optional const& amount2, STAmount const& lptAMMBalance, STAmount const& lpTokens, std::uint16_t tfee, bool isDeposit) { auto const lpTokensActual = adjustLPTokens(lptAMMBalance, lpTokens, isDeposit); if (lpTokensActual == beast::zero) { auto const amount2Opt = amount2 ? std::make_optional(STAmount{}) : std::nullopt; return std::make_tuple(STAmount{}, amount2Opt, lpTokensActual); } if (lpTokensActual < lpTokens) { // Equal trade if (amount2) { Number const fr = lpTokensActual / lpTokens; auto const amountActual = toSTAmount(amount.issue(), fr * amount); auto const amount2Actual = toSTAmount(amount2->issue(), fr * *amount2); return std::make_tuple( amountActual < amount ? amountActual : amount, amount2Actual < amount2 ? amount2Actual : amount2, lpTokensActual); } // Single trade auto const amountActual = [&]() { if (isDeposit) return ammAssetIn( amountBalance, lptAMMBalance, lpTokensActual, tfee); else return withdrawByTokens( amountBalance, lptAMMBalance, lpTokens, tfee); }(); return amountActual < amount ? std::make_tuple(amountActual, std::nullopt, lpTokensActual) : std::make_tuple(amount, std::nullopt, lpTokensActual); } assert(lpTokensActual == lpTokens); return {amount, amount2, lpTokensActual}; } Number solveQuadraticEq(Number const& a, Number const& b, Number const& c) { return (-b + root2(b * b - 4 * a * c)) / (2 * a); } // Minimize takerGets or takerPays std::optional solveQuadraticEqSmallest(Number const& a, Number const& b, Number const& c) { auto const d = b * b - 4 * a * c; if (d < 0) return std::nullopt; // use numerically stable citardauq formula for quadratic equation solution // https://people.csail.mit.edu/bkph/articles/Quadratics.pdf if (b > 0) return (2 * c) / (-b - root2(d)); else return (2 * c) / (-b + root2(d)); } } // namespace ripple