Approximating polynomials with a Solidity code generator for Lockup Dynamic

[PaulRBerg]

Problem

Users want their streams to follow a polynomial curve rather than an exponential curve:

But Lockup Dynamic works with piecewise exponential functions, so it is not able to implement polynomials.

Solution

While polynomials cannot be directly implemented in Lockup Dynamic, they can be emulated. This is what we can do:

1. Implement the polynomial using any mathematical software (e.g. Wolfram) to generate the streamed amounts at the end of each moth over the desired period
2. For each month and amount, generate a segment
3. Each segment has an exponent of 1 (which enables linear streaming)

The overall result is that the implied curve approximates the desired polynomial fairly accurately

Implementation

Rust program that takes an array of objects as input, with each object containing the following data:

• Amount as an integer
• Date as a string (e.g. Sep 30, 2024)

Which then produces some (literal) Solidity code, e.g.

pragma solidity >=0.8.19;

import { LockupDynamic } from "@sablier/v2-core/src/types/DataTypes.sol";

function getSegments() pure returns (LockupDynamic.SegmentWithDelta[] memory) {
    LockupDynamic.SegmentWithDelta[] memory segments = new LockupDynamic.SegmentWithDelta[](1);
    segments[0] = LockupDynamic.SegmentWithDelta({ amount: 100e18, milestone: 1727643600, exponent: ud2x18(1) });
}

Examples

• https://github.com/sablier-labs/sablier-v2-exactly-protocol
• https://github.com/sablier-labs/sablier-v2-steakhut
• https://gist.github.com/PaulRBerg/9c04afad3c6ea8ba0205f18261922c96

The Python script above generates this plot: