y-cruncher - Algorithms and Internals

By Alexander J. Yee

 

(Last updated: March 13, 2017)

 

 

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Expanded Articles:

 

General:

 

Large Number Arithmetic:

 

Implementation (as of v0.7.2):

 

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 Plus dependency is not a hard dependency. It 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 68 6,391 Yes

The public portion of the support library: Console I/O, File I/O, time, smart pointers, serialization, strings, and basic environment info.

Arch-Specific Libs 83 9,014 No ISA-specific libraries for SSE, AVX, and AVX512.
Basic Libs 78 8,639 No Simple and (mostly) self-contained libraries and interfaces: Command line options, template containers, file readers, configuration files, checkpoint-restart, thread pool, and parallelism interface.
Private Libs 157 18,708 No

Formerly held all the internal libs. It has since been split up leaving behind only the ugly OS-specific libraries: Hardware detection, memory allocators, thread library, parallel frameworks, and multi-layer raid-file.

Launcher 9 671 No

The CPU dispatcher that picks the optimal binary to run.

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

Digit Viewer 87 9,876 Yes

The bundled Digit Viewer.

BBPv2 32 4,347 No

The bundled BBP digit extraction app for Pi.

Modules 1,602 268,954 No

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

Objects 62 10,993 Partial

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

Functions 25 3,425 No

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

YMP Library 14 2,138 Headers Only

A public interface to the internal large number library.

Number Factory 31 3,408 Yes

Research infrastructure and test app for the YMP library.

y-cruncher 290 41,439 No

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

Experimental 83 10,856 No

Sandboxes for experimental code.

Misc. 8 3,639 No

Settings, versioning, and development sandbox.

Total: 2,629 402,498  

Software bloat anyone?

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.