Crate malachite_nz

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This crate defines Naturals (non-negative integers) and Integers. Unlike primitive integers (u32, i32, and so on), these may be arbitrarily large. The name of this crate refers to the mathematical symbols for natural numbers and integers, $\N$ and $\Z$.

  • There are many functions defined on Naturals and Integers. These include

    • All the ones you’d expect, like addition, subtraction, multiplication, and integer division;
    • Implementations of DivRound, which provides division that rounds according to a specified RoundingMode;
    • Various mathematical functions, like implementations of FloorSqrt and Gcd;
    • Modular arithmetic functions, like implementations of ModAdd and ModPow, and of traits for arithmetic modulo a power of 2, like ModPowerOf2Add and ModPowerOf2Pow;
    • Various functions for logic and bit manipulation, like BitAnd and BitAccess.
  • The implementations of these functions use high-performance algorithms that work efficiently for large numbers. For example, multiplication uses the naive quadratic algorithm, or one of 13 variants of Toom-Cook multiplication, or Schönhage-Strassen (FFT) multiplication, depending on the input size.

  • Small numbers are also handled efficiently. Any Natural smaller than $2^{64}$ does not use any allocated memory, and working with such numbers is almost as fast as working with primitive integers. As a result, Malachite does not provide implementations for e.g. adding a Natural to a u64, since the u64 can be converted to a Natural very cheaply.

  • Malachite handles memory intelligently. Consider the problem of adding a 1000-bit Natural and a 500-bit Natural. If we only have references to the Naturals, then we must allocate new memory for the result, and this is what the &Natural + &Natural implementation does. However, if we can take the first (larger) Natural by value, then we do not need to allocate any memory (except in the unlikely case of a carry): we can reuse the memory of the first Natural to store the result, and this is what the Natural + &Natural implementation does. On the other hand, if we can only take the second (smaller) Natural by value, then we only have 500 bits of memory available, which is not enough to store the sum. In this case, the Vec containing the smaller Natural’s data can be extended to hold 1000 bits, in hopes that this will be more efficient than allocating 1000 bits in a completely new Vec. Finally, if both Naturals are taken by value, then the Natural + Natural implementation chooses to reuse the memory of the larger one.

    Now consider what happens when evaluating the expression &x + &y + &z, where each Natural has $n$ bits. Malachite must allocate about $n$ bits for the result, but what about the intermediate sum &x + &y? Does Malachite need to allocate another $n$ bits for that, for a total of $2n$ bits? No! Malachite first allocates $n$ bits for &x + &y, but then that partial sum is taken by value using the Natural + &Natural implementation described above; so those $n$ bits are reused for the final sum.


Large Naturals and Integers store their data as Vecs of some primitive type. The elements of these Vecs are called “limbs” in GMP terminology, since they’re large digits. By default, the type of a Limb is u64, but you can set it to u32 using the 32_bit_limbs feature.

Demos and benchmarks

This crate comes with a bin target that can be used for running demos and benchmarks.

  • Almost all of the public functions in this crate have an associated demo. Running a demo shows you a function’s behavior on a large number of inputs. For example, to demo the mod_pow function on Naturals, you can use the following command:
    cargo run --features bin_build --release -- -l 10000 -m exhaustive -d demo_natural_mod_pow
    This command uses the exhaustive mode, which generates every possible input, generally starting with the simplest input and progressing to more complex ones. Another mode is random. The -l flag specifies how many inputs should be generated.
  • You can use a similar command to run benchmarks. The following command benchmarks various GCD algorithms for u64s:
    cargo run --features bin_build --release -- -l 1000000 -m random -b \
        benchmark_natural_gcd_algorithms -o
    or GCD implementations of other libraries:
    cargo run --features bin_build --release -- -l 1000000 -m random -b \
        benchmark_natural_gcd_library_comparison -o
    This creates a file called You can use gnuplot to create an SVG from it like so:
    gnuplot -e "set terminal svg; l \"\"" > gcd-bench.svg

The list of available demos and benchmarks is not documented anywhere; you must find them by browsing through bin_util/demo_and_bench.


  • 32_bit_limbs: Sets the type of Limb to u32 instead of the default, u64.
  • test_build: A large proportion of the code in this crate is only used for testing. For a typical user, building this code would result in an unnecessarily long compilation time and an unnecessarily large binary. Some of it is also used for testing malachite-q, so it can’t just be confined to the tests directory. My solution is to only build this code when the test_build feature is enabled. If you want to run unit tests, you must enable test_build. However, doctests don’t require it, since they only test the public interface.
  • bin_build: This feature is used to build the code for demos and benchmarks, which also takes a long time to build. Enabling this feature also enables test_build.


  • Integer, a type representing integers with arbitrarily large absolute values.
  • Natural, a type representing arbitrarily large non-negative integers.