y-cruncher - Algorithms and Internals

By Alexander J. Yee

 

(Last updated: August 22, 2018)

 

 

Back To:

 

Expanded Articles:

 

General:

 

Large Number Arithmetic:

 

Implementation (as of v0.7.6):

 

General Information:

 

Libraries and Dependencies:

y-cruncher has no other non-system dependencies. No Boost. No GMP. Pretty much everything that isn't provided by C++ is built from ground up.

Furthermore, the Cilk and TBB dependencies can be trivially removed without affecting the core functionality of the program.

 

 

Compilers:

 

Other Internal Requirements:

 

Code Organization:

 

y-cruncher's root source tree is (roughly) broken up into the following subdirectories. They are listed in order of build dependency.

Module Files Lines of Code Open Sourced? Description
Public Libs 136 13,244 Yes

The public portion of the support libraries.

Private Libs 367 43,399 No The private portion of the support libraries.
Dynamic Linking 3 213 No Nothing here yet.
Launcher 10 852 Yes

The CPU dispatcher that picks the optimal binary to run.

It's the module that builds the y-cruncher(.exe) binary.

Digit Viewer 91 10,186 Yes

The bundled Digit Viewer.

Digit Viewer 2 84 11,043 Not Yet Work in-progress rewrite of the Digit Viewer.
BBPv2 32 4,385 No

The bundled BBP digit extraction app for Pi.

Modules

(expanded below)

1,978 316,150 No

Low-level arbitrary-precision arithmetic: Addition, subtraction, multiplication, radix conversion, and checksum hashing.

Objects 83 14,708 Partial

Large number objects. (BigInt, BigFloat, etc...)

Functions 28 4,348 No

Non-trivial math: Division, square root, and string conversions.

YMP Library 14 2,228 Headers Only

A public interface to the internal large number library.

Number Factory 31 3,414 Yes

Research infrastructure and test app for the YMP library.

y-cruncher 392 57,158 No

y-cruncher itself. This has most of the console UI and the implementations for all the constants.

Experimental 102 14,476 No

Sandboxes for experimental code.

Misc. 9 1,274 No

Settings, versioning, and development sandbox.

Total: 3,360 497,078  

Software bloat anyone?

 

Modules
Sub-Module Files Lines of Code Description
Cache 6 348

Dead Code - unfinished experiment with caching.

Intrinsics 11 851

Double-word multiply, bit-reversal, length conversions, etc...

Profiles 14 890

Processor-specific tuning settings.

Linear Ops 9 973

Parallel and out-of-core memset(), memcpy(), and scanning.

Checksum Hashing 13 1,310 Modular Redundancy Checks. Ram-only + parallel + out-of-core.
Random 9 707

Pseudorandom number generators. Ram-only + parallel + out-of-core.

Carryout 13 2,675 Kogge-Stone Parallel Carryout. Ram-only + parallel + out-of-core.
Addition 17 1,753 Large integer addition and subtraction. Ram-only.
Word Multiply 15 1,822

Single-word multiplication. Ram-only + parallel + out-of-core.

Parameters 11 864

Parameters structures.

Large Multiply 1,834 299,614

Large integer multiplication. Ram-only + parallel + out-of-core.

CBRv2 19 3,230 Radix Conversion via Scaled Remainder Tree. Ram-only + parallel + out-of-core.
Testers 1 264

Deprecated test scripts.

Misc. 6 849

 

Total: 1,978 316,150

 

 

 

Notes:

 

 

 

Limits:

 

Like most other programs, there are theoretical limits to y-cruncher. Most of these limits are enforced by the program.

  32-bit 64-bit Comments

Ram Usage

~1.8 GiB ~ 1 EiB (1018 bytes)

Limited by memory address space.

Disk Usage

~ 1 EiB

Limited by 64-bit address space.

Task Decomposition

65,536

Arbitrary limit.

RAID - Level 1

8 paths

 

RAID - Level 2

64 x Level 1 RAID groups

Limited by the # of bits in largest integer.

Will likely be increased in the future.

Largest Multiplication

(2.02 * 1018) x (2.02 * 1018) bits
(6.7 * 1017) x (6.7 * 1017)
decimal digits

Small Primes Number-Theoretic Transform:

  • 5 x 63-bit primes
  • Transform Length: 7 * 252

Convolution Length

4.03 * 1018 bits
1.34 * 1018 decimal digits

Computation Size

(for all constants)

1015 decimal digits

Limited by double-precision floating-point.*

BBP Hexadecimal Offset

246 - 1

Implementation-specific limitation.

*y-cruncher uses double-precision floating-point for things such as:

The result of these calculations are generally rounded to integers and must be accurate to +/- 1 for the program to operate correctly. The problem is that double-precision floating-point only has 53 bits of precision which will run out at around 9 * 1015. Since there is round-off error, the limit will certainly be lower. The exact limit is unknown and will vary with the different constants. Therefore y-cruncher arbitrarily caps it to 1015 decimal digits. Colloquially, I call this the "float-indexing limit".

 

There are currently no plans to raise this limit since it is already well beyond the capability of current hardware (as of 2015).

 

It is worth mentioning that the float-indexing limit is the only thing left that prevents y-cruncher from going all the way up to the 64-bit limit. Without it, it should be possible to reach 6.7 * 1017 decimal digits (the limit of the Small Primes NTT).

 

Getting rid of the float-indexing limit will require a floating-point type with at least a 64-bit mantissa. A viable option is to use 80-bit extended-precision via the x87 FPU although some compilers don't support it. But since "float indexing" isn't exactly a performance bottleneck, any sort of software emulation will probably work as well.