## y-cruncher Multi-Precision Library |
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## A Multi-threaded Multi-Precision Arithmetic Library## (Powered by y-cruncher) |

(Last updated: March 3, 2016)

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**YMP Library:**

**Large Number Objects:**

**Low-Level Programming:**

Twiddle Factor/Lookup Tables

Unlike most other bignum libraries, YMP large multiplication requires the use of precomputed twiddle factor tables. Since nearly all non-trivial functions reduce to large multiplication, pretty much everything requires these tables.

In YMP we use the term, "lookup tables" since twiddle factors aren't the only users of the precomputation framework.

**Why does YMP need them?**

Space-time tradeoff.

As of 2015, all practical large multiplication algorithms that run in quasi-linear time are based on Fast Fourier Transforms (FFTs). And FFT-based algorithms require twiddle factors. Since twiddle factors are generally expensive to compute, they need to be precomputed and cached to achieve an acceptable level of efficiency.

- Floating-point FFT twiddle factors require calls to sin/cos.
- Number-theoretic FFT twiddle factors require power-modulus.

There is one exception. The Schönhage–Strassen algorithm (SSA) is a number-theoretic FFT whose twiddle factors are powers of two. For this reason, they are trivial and do not need to be cached. This algorithm is precisely the one that used in mainstream libraries such as GMP. Which is why they don't need twiddle factor tables.

Unfortunately, the SSA algorithm is not (and never was) the fastest algorithm for large multiplication. Therefore, twiddle factors are a necessary evil to utilize the more powerful algorithms that are further down the space-time trade-off curve.

**Lookup Table Class:**

In y-cruncher/YMP, the * LookupTable* class is the object that holds these tables.

"cbitlen" stands for "Convolution Bit-Length". It is the size of the table.

A table with * cbitlen = x* will be able to handle any large multiplication of length

Anything that uses large multiplication requires this table. Using a table of insufficient size is undefined behavior. (Most of the time, it'll just throw an exception.)

Larger tables require more memory than smaller tables. But the table will max out at a few hundred megabytes. A "maxed out" table can handle any size up to the limit of YMP (10^{18} bits). But it will still only require a few hundred megabytes. This is by design. On the current generation of processors, using larger tables provides little benefit for increased memory usage and cache pollution.

**Table Management:**

The lookup table is an object that is shared across multiple threads. Therefore, YMP keeps a global instance for convenience. This global table starts with a size of zero and does not automatically resize. Therefore the user needs to manually resize it to be large enough to handle anything that will be encountered in the application.

As of the current version, the global table does not automatically resize because it is not possible to safely do so without sacrificing performance.

Resizing the global table is not thread-safe. Doing so while it is being used is undefined behavior. Therefore, it is strongly recommended to size the table at the start of the application before any computation begins. If you are not sure how large to make it, you can pass it -1 which will max out the table and consume up to a few hundred megabytes. (the exact size depends on various factors)

All functions in YMP that require a lookup table have overloads that allow the table to be passed in manually rather than defaulting to the global instance. This is useful for applications that wish to manage the table on their own. It is also useful for NUMA systems where it may be best to have multiple tables - one per NUMA node.

- Headers: ymp/LowLevel.h
- Sources: none

**Get Lookup Table:**

const LookupTable* get_global_table ();

**Description:**

Returns a pointer to the global lookup table.

**Get Lookup Table Size:**

uiL_t get_global_table_size ();

**Description:**

Returns the size of the global lookup table. The number returned is

, the largest convolution size measured in bits.cbitlen

**Ensure Lookup Table Size:**

void ensure_global_table<u32_t> (uiL_t cwordlen = -1);

void ensure_global_table<u64_t> (uiL_t cwordlen = -1);

**Description:**

Ensures that the global lookup table is of sufficient size.

The parameter specifies that the table should be large enough to accommodate any large multiply whose product is at most

words long.cwordlen

- If the table is already of sufficient size, this call is a no-op.
- If the table is not large enough, it is deleted and replaced with a new one of sufficient size.
If the parameter is -1 (or omitted), the table will be set large enough to handle any multiplication size that YMP supports. Depending on the target processor, the resulting "maxed out" table may consume several hundred megabytes. This may be the way to go since most machines nowadays have plenty of ram.

This function is not thread-safe with either itself or any other YMP function calls that use the global table. It is recommended to call this at the start of the application before any other YMP functions are used at all.

Use thse if you wish to manually manage tables.

**Make Lookup Table:**

dll_uptr<LookupTable> MakeLookupTable (uiL_t cbitlen = -1);

**Description:**

Construct the precomputed twiddle-factor/lookup table that is required by nearly all large multiply routines.

The parameter specifies that the table should be large enough to accommodate any large multiply whose product is at most

bits long.cbitlen

If the parameter is -1 (or omitted), the table will be set large enough to handle any multiplication size that YMP supports. Depending on the target processor, the resulting "maxed out" table may consume several hundred megabytes. This may be the way to go since most machines nowadays have plenty of ram.

This function returns a smart pointer for the sake of RAII safety. It uses a custom smart pointer because std::unique_ptr is not guaranteed to be compatible across the DLL boundary.