TITLE

Big Numbers

VERSION

$Id: pdd14_bignum.pod,v 1.3 2002/04/02 07:08:43 josh Exp $

CURRENT

Maintainer: Alex Gough (alex@earth.li)
Class: Internals
PDD Number: 14
Version: $Id: pdd14_bignum.pod,v 1.3 2002/04/02 07:08:43 josh Exp $
Status: Informational
Last Modified: $Id: pdd14_bignum.pod,v 1.3 2002/04/02 07:08:43 josh Exp $
PDD Format: 1
Language: English

HISTORY

version 1.1

The first version is more a description of the state of the art than a plan for the future. The author believes this is akin to learning how to crawl before trying to run. Leaving the ground is considered Right Out with attempts being left as an exercise for the interested reader.

CHANGES

Version 1.0

None. First version

Version 1.1

Special values added, exception handling updated to suit EDA, more rounding types added, TODO added. PDD number bagged.

ABSTRACT

This document describes the big number library, the functionality it provides and some internal details of interest to people making use of the library. Some of the areas in which the big number library meet with the rest of Parrot are also discussed.

DESCRIPTION

The big number library provides Parrot with both a collection of (nearly) infinite precision numeric types and an implementation of an extended decimal arithmetic (EDA).

Why decimal arithmetic?

There are benefits in using the big number library to provide both values of effectively unlimited precision and a defined arithmetic, complete with rounding and exceptional conditions, for values which are otherwise easily represented using standard low-level types. Both require the same range of operations but differ in the environment under which those operations occur. The effort required to produce a library which implements a decimal arithmetic is not much greater than that needed to provide a base-2 big number library. It is true that some trade-off in both space and speed is made but given the nature of dynamic languages, this should not present too great a burden.

Numeric types provided

The bignumber library provides the following data types to Parrot:

Big integers (BigInt)

Whole numbers with no limits on their size.

Big floats (BigNum)

Numbers with decimal fractional parts, again with no limit on size.

Big floats with fixed fractional parts

Numbers with a fixed maximum number of digits in their fractional part, again with no limit on size.

The library implements these different forms of number using the same internal representation, and differentiates between them only when performing rounding operations. A number has the following abstract form:

[ sign, string of digits, exponent ]

If sign is zero, the number is positive, if equal to one, the number is negative. The number has the value

sign, string of digits * 10 ** exponent

A big integer must always have a non-negative exponent, a big float may have any exponent and a float with a fixed fractional part will have an exponent greater than a given (negative) number. These limits are not attached to a numeric value, but instead are enforced by giving any operation involving the numbers a context.

In general, Parrot functions will not need to care about what the bignum objects are or do, they should merely be used as arguments to big number functions, the objects will be managed by Parrot's garbage collection in a similar manner to strings.

Special Values

Additionally the library provides special values, which represent the result of otherwise undefined operations (division by zero, for instance). +Infinity, -Infinity and both quiet and signalling Not a Number (NaN) are available. In general, the result of an operation with at least one argument which is NaN will be NaN, if the argument is a signalling NaN, an exception will also be raised. See the EDA for full details.

Context

All operations occur within a defined context. This tells the operations how they should be treating their arguments, what sort of rounding to perform and what to do if information is lost through rounding.

The context provides the environment in which an operation occurs, in particular the following options are available:

precision

A positive precision indicates that big floats are to be used, these cannot have more than precision digits in their coefficient before or after any operation. Arguments to operations with more than precision digits will be truncated and rounded appropriately and results of operations will not have more than precision digits in their coefficients, with any extra digits accumulated during the calculation of the operation being truncated and rounded as required.

A precision of zero indicates that integer operations are to be performed. Arguments to operations are rounded so that they have no fractional part, and the result of all operations will be rounded to be integers.

A negative value of precision indicates that a fixed number of fractional digits are to be provided, with arguments and results being truncated after those digits.

With non-positive values of precision the total number of digits in the coefficient is limited only by available memory.

rounding

The rounding part of the context defines the rounding algorithm which is to be applied when truncating digits from a number's coefficient. The available rounding forms are outlined below.

traps and flags

The traps part of the context defines how exceptions are raised by the library. Seven distinct classes of error can occur, if the corresponding trap is set (enabled) then an exception is raised by the library (and dispatched to Parrot) otherwise execution continues with the exception class being recorded in flags. For more details see the extended decimal arithmetic standard.

It is therefore the current context which determines which numeric type is being considered during a particular operation, this makes it easy to upgrade from one numeric form to another, and also allows for considerable code-reuse within the library.

Exception Classes

The following exception classes are defined:

Lost Digits

Non-zero digits have been removed from an argument to a function during rounding before the operation.

Division By Zero

Division by zero was attempted.

Inexact

Because arguments were rounded, or because the result of an operation has lost significant digits, the result is inexact.

Invalid Operation

An invalid operation was attempted, for instance a sNaN was provided as an argument to a function. This also covers recoverable errors like 0/0, which signals Invalid Operation and can return NaN.

Overflow

The exponent of a number has overflowed.

Rounded

An argument has been rounded.

Underflow

The exponent of a number has underflowed.

Rounding

The rounding part of the context defines the rounding algorithm to be used, the following are provided (examples assume a precision of 5):

Round down

Any unwanted digits are simply truncated from the coefficient. This rounds towards zero.

[0, 1234567, 10] => [0, 12345, 12]

Round half up

The first lost digit is examined, if this is in the range 0-4, the coefficient is truncated directly, if in the range 5-9, one is added to the final digit of the coefficient. If this leads to a coefficient with more than precision digits, the number is rounded again, removing the trailing zero. This is essentially rounding to nearest.

[0, 1234567, 10] => [0, 12346, 12]
[0, 1234549, 10] => [0, 12345, 12]
[0, 9999950, 10] => [0, 10000, 13]

Round half even

The first lost digit is examined, if it lies in the range 0-4, the coefficient is truncated directly, if in the range 6-9, the coefficient is rounded up. If the first lost digit is equal to 5, and the remaining lost digits in the coefficient are non-zero, the number is also rounded up. If the lost digits are equal to exactly half, the number is rounded up if the least significant retained digit is odd, and rounded down if it is even.

Round Floor

If the digits to be discarded are non zero, and the number is negative, the coefficient is rounded up, otherwise it remains the same.

This is rounding towards -Infinity.

Round Ceiling

If the digits to be discarded are non zero, and the number is positive, the coefficient is rounded up, otherwise it remains the same.

This is rounding towards +Infinity.

Operations

The following operations are provided by the library, they function exactly as those described in the Standard Decimal Arithmetic (SDA) (see references below) with some extension to cope with integer and fixed fractional part numbers. Only the deviations are outlined here.

In all cases, the sequence of rounding and promotion to zero outlined by the SDA are followed, even where the context implies integer operations.

Addition, Subtraction

Multiplication

Division

Under integer conditions, division is halted once the first fractional digit is calculated, with the result then being rounded to an integer and returned. Under fixed-fraction conditions, one more digit than needed is calculated, with the coefficient then being rounded and returned.

If a floating point value is required, or if inexact division by a very small number is attempted, it may be wise to follow big float arithmetic to limit the number of digits returned. It is safe to chose a precision at least as big the largest number of digits of either argument to the division function.

Integer division, Remainder

For both integer and fixed-fraction numbers, the result returned by the remainder function will be an integer or fixed-fraction number. The result of integer division will be an integer.

Rounding

Plus / Minus

Comparison

Comparison returns a big number which is equal to 1, 0 or -1 if the first argument is larger, equal to or smaller than the second. An alternative form which returns an INTVAL is also provided.

Rescale

Power

Square Root

Conversion to and from strings

A one to one conversion between the abstract representation above and a string is provided by the library, and acts as defined by the standard decimal arithmetic. Other conversation operations may also be implemented, and these may not provide one to one mapping.

A pedantic error checking conversion is available within the library, but only works with native strings. Versions which work with Parrot strings will also be provided, although in a separate file to the rest of the library (they will share a common private header file).

IMPLEMENTATION

Functions are provided with implement the arithmetic, conversion, creation and destruction of big numbers by dealing with otherwise opaque big number objects.

Big number representation

A big number is represented by the following structure, this is capable of being allocated, tracked and destroyed by the Parrot garbage collection system.

typedef struct {
   BN_NIB* buffer; /* string of nibbles */
   UINTVAL nibs;   /* nibs allocated, in sizeof(BN_NIB) */
   UINTVAL flags;  /* private flags store: 001 Inf,  010 qNAN, 110 sNAN */
   INTVAL digits;  /* digits used */
   int sign;       /* sign of number, 0=> positive or zero, 1 => negative */
   INTVAL expn;    /* exponent of number */
} parrot_bignum_t;

Within the library, individual decimal digits can be accessed using macros, outside the library, access must be made via exported functions. BN_NIB is likely to be a UINTVAL, but this is not essential.

Special values are represented by setting digits to zero and setting appropriate private flags, using internal macros. Infinity has one flag field, NaN another flag field and sNaN a third. In general the flags should not be examined directly, even within the module.

Context

typedef struct {
   INTVAL precision;     /* number of digs to retain */
   BN_ROUNDING rounding; /* rounding type to perform */
   BOOLVAL extended;     /* do we use extended or base semantics? */
   unsigned char flags;       /* records possible errors */
   unsigned char traps;       /* throw errors or not? */
} parrot_bignum_context;

BN_ROUNDING is an enumeration of the possible rounding types as described above. traps is a bitmask of exception traps, 0 implies that a trap is disabled and 1 implies it is enabled. flags is a bitmask which records exceptional conditions and has the same fields at flags.

It is expected that language level types implement big floats using a global floating point context which is tagged onto an interpreter structure (and can thus be modified by called the right opcodes). That big integers and fixed-fraction number are provided by creating a context with an appropriate precision whenever a call into the library is made.

Exceptional Conditions

When an exceptional condition is raised by the module, control is passed to BN_nonfatal(), this examines the error which has occurred and the current context to determine which class of error has occurred. If the corresponding trap handler is not enabled, the context's flags are updated and control is returned to the bignumber library, otherwise the exception becomes fatal. How this mechanism interacts with Parrot's own is yet to be decided.

The possible exceptions are detailed in the extended decimal arithmetic.

Tests

The Standard Decimal Arithmetic provides a collection of tests for both its base and extended behaviour. Initially it is hoped that this library can pass all base tests, with extended tests to be included at a later date as it is extended to cope with values such as +Inf.

TODO

Fill in the remaining functions from the EDA, verify that the test suite still passes, integrate the library into the rest of Parrot, provide PMC types and suitable opcodes. Conversion to and from Parrot strings, conversion to and from floating point types, sprintf output of bignumbers.

ATTACHMENTS

FOOTNOTES

REFERENCES

IBM's Standard Decimal Arithmetic, with tests http://www2.hursley.ibm.com/decimal/

Perl's Math::Big* modules.

cpanm Scheme

perl -MCPAN -e shell
install Scheme