Approximation of the cumulative distribution function (cdf) for arbitrary 18 decimal fixed point parameters x, μ, σ and a target absolute error of 1e-8.

The complementary error function (erfc) is approximated rationally using a variant from the algorithm presented in this paper. The rational polynomial is composed of a polynomial of degree 11 as numerator, and a polynomial of degree 4 as denominator.

The coefficients of the rational polynomial are scaled by a factor of 2^96 and then codified as part of the erfc function. This scaling allows for the use of cheaper sar (shift arithmetic right) operations instead of more expensive sdiv (signed division) operations (former costs 3 gas, while latter 5). Additionally, this function always returns 0 when the input is more than 4.0523 as the maximum absolute error would not surpass the target error.

The erfc and cdf functions are differentially fuzzed against Python and Javascript implementations to ensure that they provide an absolute error of less than 1e-8. These functions are also benchmarked against solstat, gud-cdf, and solidity-cdf. It's important to note that the benchmark does not verify the precision of these external libraries, such as solidity-cdf, which has an error of more than 1e-8.

To run the gas benchmark: forge t --gas-report

Run the following for the dependencies used in the (differential) tests:

1. python -m venv venv && source venv/bin/activate
2. pip install mpmath

1. (cd differential && yarn)

To run the tests: forge t

This repository is inspired by or directly modified from many sources, primarily:

• High-Precision Bootstrapping of RNS-CKKS Homomorphic Encryption Using Optimal Minimax Polynomial Approximation and Inverse Sine Function: paper for algorithm variant
• gud-cdf: codification script
• solidity-cdf: inspiration for erfc function inside cdf
• gaussian: javascript differential scripts