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Trapezoidal (n=2) #91

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merged 7 commits into from
Nov 16, 2023
Merged

Trapezoidal (n=2) #91

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9dac5b9
feat: trapezoidal riemann (n=2)
MathisGD Nov 16, 2023
c1633da
docs
MathisGD Nov 16, 2023
f75a09e
chore: fmt
MathisGD Nov 16, 2023
0531b5c
Merge branch 'feat/riemann-avg' into feat/trapezoidal
MathisGD Nov 16, 2023
e46719d
docs: more concise comments for trapeze N=2
QGarchery Nov 16, 2023
7667991
fix: missing bracket
QGarchery Nov 16, 2023
97794c6
Merge pull request #95 from morpho-org/docs/trapezoidal
MathisGD Nov 16, 2023
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38 changes: 17 additions & 21 deletions src/SpeedJumpIrm.sol
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Original file line number Diff line number Diff line change
Expand Up @@ -10,10 +10,6 @@ import {MarketParamsLib} from "../lib/morpho-blue/src/libraries/MarketParamsLib.
import {Id, MarketParams, Market} from "../lib/morpho-blue/src/interfaces/IMorpho.sol";
import {MathLib as MorphoMathLib} from "../lib/morpho-blue/src/libraries/MathLib.sol";

/// @dev Number of steps used in the Riemann sum.
/// @dev 4 steps allows to have a relative error below 30% for 15 days at err=1 or err=-1.
int256 constant N_STEPS = 4;

/// @title AdaptiveCurveIrm
/// @author Morpho Labs
/// @custom:contact security@morpho.org
Expand Down Expand Up @@ -147,24 +143,24 @@ contract AdaptiveCurveIrm is IIrm {
} else {
// Formula of the average rate that should be returned to Morpho Blue:
// avg = 1/T ∫_0^T curve(startRateAtTarget * exp(speed * x), err) dx
// The integral is approximated with a Riemann sum (steps of length T/N). To underestimate the rate, a
// left Riemann (a=0, b=N-1) is done when the rate goes up (err>0) and a right Riemann (a=1, b=N) is
// done when the rate goes down (err<0).
// avg ~= 1/T Σ_i=a^b curve(startRateAtTarget * exp(speed * T/N * i), err) * T / N
// ~= Σ_i=a^b curve(startRateAtTarget * exp(linearAdaptation/N * i), err) / N
// curve is linear in startRateAtTarget, so:
// ~= curve(Σ_i=a^b startRateAtTarget * exp(linearAdaptation/N * i), err) / N
// ~= curve(Σ_i=a^b startRateAtTarget * exp(linearAdaptation/N * i) / N, err)
int256 sumRateAtTarget;
int256 step = linearAdaptation / N_STEPS;
// Compute the terms 1 to N_STEPS - 1.
for (int256 k = 1; k < N_STEPS; k++) {
sumRateAtTarget += _newRateAtTarget(startRateAtTarget, step * k);
}
// The integral is approximated with the trapezoidal rule:
// avg ~= 1/T Σ_i=1^N (f((i-1) * T/N) + f(i * T/N)) / 2 * T/N
// Where f(x) = curve(startRateAtTarget * exp(speed * x), err).
// avg ~= 1/N * ( (f(0)+f(T))/2 + Σ_i=1^(N-1) f(i*T/N) )
// avg ~= 1/N * ( (f(startRateAtTarget)+f(endRateAtTarget))/2 + Σ_i=1^(N-1) curve(startRateAtTarget *
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// exp(speed * i*T/N), err))
// As curve is linear in rateAtTarget:
// avg ~= 1/N * curve((startRateAtTarget+endRateAtTarget)/2 + Σ_i=1^(N-1) startRateAtTarget * exp(speed
// * i*T/N), err)
// avg ~= curve((startRateAtTarget+endRateAtTarget)/(2*N) + Σ_i=1^(N-1) startRateAtTarget * exp(speed *
// i*T/N) / N, err)
// With N=2:
// avg ~= curve((startRateAtTarget+endRateAtTarget)/4 + startRateAtTarget * exp(speed * T/2) / 2, err)
// avg ~= curve((startRateAtTarget + endRateAtTarget + 2 * startRateAtTarget * exp(speed * T/2)) / 4,
// err)
endRateAtTarget = _newRateAtTarget(startRateAtTarget, linearAdaptation);
// Add the term 0 for a left Riemann or the term N_STEPS for a right Riemann.
sumRateAtTarget += err < 0 ? endRateAtTarget : startRateAtTarget;
avgRateAtTarget = sumRateAtTarget / N_STEPS;
int256 midRateAtTarget = _newRateAtTarget(startRateAtTarget, linearAdaptation / 2);
avgRateAtTarget = (startRateAtTarget + 2 * midRateAtTarget + endRateAtTarget) / 4;
}
}

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