Abstract
Mathematical computations over a sizelimited fixedpoint infrastructure, bare a fundamental tradeoff between Accuracy and Valid Input Range.
Within any computation, the only type of operation which can reduce the accuracy of the output is division.
The typical solution is to rearrange the computation such that division takes place only as the very last step.
Subsequently, intermediate results of the computation become more likely to overflow and revert the entire transaction.
This means that rearranging the computation will effectively reduce the range of input which can be handled successfully.
For example, consider a / b + c / d
with a = 100, b = 51, c = 1, d = 25
, and a sizelimit of 8 bits (maximum value being 255).
The true value of this expression is 100 / 51 + 1 / 25 = 2.00078...
, so ideally on a fixedpoint infrastructure, we would want our implementation to yield 2
.
The straightforward implementation a / b + c / d
is very innacurate of course.
It yields the output 100 / 51 + 1 / 25 = 1 + 0 = 1
, which reflects an inaccuracy of more than 50% comparing with the true value.
But rearranging it such that division takes place as the very last step using (a * d + b * c) / (b * d)
reverts the transaction.
It yields the intermediate computation of a * d = 100 * 25
, which overflows the given sizelimit.
This package consists of the following libraries:
 NaturalNum  a set of functions over natural numbers
 RationalNum  a set of functions over rational numbers
NaturalNum handles the sizelimit internally, thus allowing to rearrange any computation and achieve maximum accuracy without reducing the size of valid input range.
In the example above, it yields (100 * 25 + 51 * 1) / (51 * 25) = 2
, which reflects an inaccuracy of less than 0.04% comparing with the true value.
RationalNum maintains the entire expression as a rational number, thus allowing to avoid the rearrangement and implement the computation in its "native form".
Using it will typically be less costeffective, hence it is advisable to use NaturalNum whenever performance is considered critical.
NaturalNum
This module implements arithmetic over natural numbers of any conceivable size.
The data structure used for representing a natural number is a uint256
dynamic array.
For performance considerations, we have refrained from wrapping it within a struct
.
Hopefully, future compilers will support the usage of type
over this nonprimitive type.
This module supports the following operations:
 Function
encode(uint256 val)
=>uint256[]
 Function
decode(uint256[] num)
=>uint256
 Function
eq(uint256[] x, uint256[] y)
=>bool
 Function
gt(uint256[] x, uint256[] y)
=>bool
 Function
lt(uint256[] x, uint256[] y)
=>bool
 Function
gte(uint256[] x, uint256[] y)
=>bool
 Function
lte(uint256[] x, uint256[] y)
=>bool
 Function
and(uint256[] x, uint256[] y)
=>uint256[]
 Function
or(uint256[] x, uint256[] y)
=>uint256[]
 Function
xor(uint256[] x, uint256[] y)
=>uint256[]
 Function
add(uint256[] x, uint256[] y)
=>uint256[]
 Function
sub(uint256[] x, uint256[] y)
=>uint256[]
 Function
mul(uint256[] x, uint256[] y)
=>uint256[]
 Function
div(uint256[] x, uint256[] y)
=>uint256[]
 Function
mod(uint256[] x, uint256[] y)
=>uint256[]
 Function
pow(uint256[] x, uint256 n)
=>uint256[]
 Function
shl(uint256[] x, uint256 n)
=>uint256[]
 Function
shr(uint256[] x, uint256 n)
=>uint256[]
 Function
bitLength(uint256[] x)
=>uint256
This module assumes that every uint256[]
input can ultimately be traced back to (i.e., created by) function encode
.
Since the length of dynamic arrays is bounded by 2**641
, the maximum representable value is (2**256)**(2**641)1
.
RationalNum
This module implements arithmetic over rational numbers of any conceivable size.
The data structure used for representing a rational number is:
struct Rnum {
bool s; // the represented value's negativity
uint256[] n; // the represented value's numerator
uint256[] d; // the represented value's denominator
}
This module supports the following operations:
 Function
encode(bool s, uint256 n, uint256 d)
=>Rnum
 Function
decode(Rnum num)
=>(bool, uint256, uint256)
 Function
eq(Rnum x, Rnum y)
=>bool
 Function
gt(Rnum x, Rnum y)
=>bool
 Function
lt(Rnum x, Rnum y)
=>bool
 Function
gte(Rnum x, Rnum y)
=>bool
 Function
lte(Rnum x, Rnum y)
=>bool
 Function
add(Rnum x, Rnum y)
=>Rnum
 Function
sub(Rnum x, Rnum y)
=>Rnum
 Function
mul(Rnum x, Rnum y)
=>Rnum
 Function
div(Rnum x, Rnum y)
=>Rnum
Testing
Prerequisites
node 16.13.0

yarn 1.22.10
ornpm 8.1.0
Installation

yarn install
ornpm install
Compilation

yarn build
ornpm run build
Execution

yarn test
ornpm run test
Verification

yarn verify
ornpm run verify