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@afourpaper
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@titlepage
@afourpaper
@sp 7
@center @titlefont{Javascript Bignum Extensions}
@sp 3
@center Version 2020-01-11
@sp 3
@center Author: Fabrice Bellard
@end titlepage

@setfilename jsbignum.info
@settitle Javascript Bignum Extensions

@contents

@chapter Introduction

The Bignum extensions add the following features to the Javascript
language while being 100% backward compatible:

@itemize

@item Operator overloading with a dispatch logic inspired from the proposal available at @url{https://github.com/tc39/proposal-operator-overloading/}.

@item Arbitrarily large floating point numbers (@code{BigFloat}) in base 2 using the IEEE 754 semantics.

@item Arbitrarily large floating point numbers (@code{BigDecimal}) in base 10 based on the proposal available at
@url{https://github.com/littledan/proposal-bigdecimal}.

@item @code{math} mode: arbitrarily large integers and floating point numbers are available by default. The integer division and power can be overloaded for example to return a fraction. The modulo operator (@code{%}) is defined as the Euclidian
remainder. @code{^} is an alias to the power operator
(@code{**}). @code{^^} is used as the exclusive or operator.

@end itemize

The extensions are independent from each other except the @code{math}
mode which relies on BigFloat and operator overloading.

@chapter Operator overloading

Operator overloading is inspired from the proposal available at
@url{https://github.com/tc39/proposal-operator-overloading/}. It
implements the same dispatch logic but finds the operator sets by
looking at the @code{Symbol.operatorSet} property in the objects. The
changes were done in order to simplify the implementation.

More precisely, the following modifications were made:

@itemize

@item @code{with operators from} is not supported. Operator overloading is always enabled.

@item The dispatch is not based on a static @code{[[OperatorSet]]} field in all instances. Instead, a dynamic lookup of the @code{Symbol.operatorSet} property is done. This property is typically added in the prototype of each object.

@item @code{Operators.create(...dictionaries)} is used to create a new OperatorSet object. The @code{Operators} function is supported as an helper to be closer to the TC39 proposal.

@item @code{[]} cannot be overloaded.

@item In math mode, the BigInt division and power operators can be overloaded with @code{Operators.updateBigIntOperators(dictionary)}.

@end itemize

@chapter BigInt extensions

A few properties are added to the BigInt object:

@table @code

@item tdiv(a, b)
Return @math{trunc(a/b)}. @code{b = 0} raises a RangeError
exception.

@item fdiv(a, b)
Return @math{\lfloor a/b \rfloor}. @code{b = 0} raises a RangeError
exception.

@item cdiv(a, b)
Return @math{\lceil a/b \rceil}. @code{b = 0} raises a RangeError
exception.

@item ediv(a, b)
Return @math{sgn(b) \lfloor a/{|b|} \rfloor} (Euclidian
division). @code{b = 0} raises a RangeError exception.

@item tdivrem(a, b)
@item fdivrem(a, b)
@item cdivrem(a, b)
@item edivrem(a, b)
Return an array of two elements. The first element is the quotient,
the second is the remainder. The same rounding is done as the
corresponding division operation.

@item sqrt(a)
Return @math{\lfloor \sqrt(a) \rfloor}. A RangeError exception is
raised if @math{a < 0}.

@item sqrtrem(a)
Return an array of two elements. The first element is @math{\lfloor
\sqrt{a} \rfloor}. The second element is @math{a-\lfloor \sqrt{a}
\rfloor^2}. A RangeError exception is raised if @math{a < 0}.

@item floorLog2(a)
Return -1 if @math{a \leq 0} otherwise return @math{\lfloor \log2(a) \rfloor}.

@item ctz(a)
Return the number of trailing zeros in the two's complement binary representation of a. Return -1 if @math{a=0}.

@end table

@chapter BigFloat

@section Introduction

This extension adds the @code{BigFloat} primitive type. The
@code{BigFloat} type represents floating point numbers in base 2
with the IEEE 754 semantics. A floating
point number is represented as a sign, mantissa and exponent. The
special values @code{NaN}, @code{+/-Infinity}, @code{+0} and @code{-0}
are supported. The mantissa and exponent can have any bit length with
an implementation specific minimum and maximum.

@section Floating point rounding

Each floating point operation operates with infinite precision and
then rounds the result according to the specified floating point
environment (@code{BigFloatEnv} object). The status flags of the
environment are also set according to the result of the operation.

If no floating point environment is provided, the global floating
point environment is used.

The rounding mode of the global floating point environment is always
@code{RNDN} (``round to nearest with ties to even'')@footnote{The
rationale is that the rounding mode changes must always be
explicit.}. The status flags of the global environment cannot be
read@footnote{The rationale is to avoid side effects for the built-in
operators.}. The precision of the global environment is
@code{BigFloatEnv.prec}. The number of exponent bits of the global
environment is @code{BigFloatEnv.expBits}. The global environment
subnormal flag is set to @code{true}.

For example, @code{prec = 53} and @code{ expBits = 11} exactly give
the same precision as the IEEE 754 64 bit floating point format. The
default precision is @code{prec = 113} and @code{ expBits = 15} (IEEE
754 128 bit floating point format).

The global floating point environment can only be modified temporarily
when calling a function (see @code{BigFloatEnv.setPrec}). Hence a
function can change the global floating point environment for its
callees but not for its caller.

@section Operators

The builtin operators are extended so that a BigFloat is returned if
at least one operand is a BigFloat. The computations are always done
with infinite precision and rounded according to the global floating
point environment.

@code{typeof} applied on a @code{BigFloat} returns @code{bigfloat}.

BigFloat can be compared with all the other numeric types and the
result follows the expected mathematical relations.

However, since BigFloat and Number are different types they are never
equal when using the strict comparison operators (e.g. @code{0.0 ===
0.0l} is false).

@section BigFloat literals

BigFloat literals are floating point numbers with a trailing @code{l}
suffix. BigFloat literals have an infinite precision. They are rounded
according to the global floating point environment when they are
evaluated.@footnote{Base 10 floating point literals cannot usually be
exactly represented as base 2 floating point number. In order to
ensure that the literal is represented accurately with the current
precision, it must be evaluated at runtime.}

@section Builtin Object changes

@subsection @code{BigFloat} function

The @code{BigFloat} function cannot be invoked as a constructor. When
invoked as a function: the parameter is converted to a primitive
type. If the result is a numeric type, it is converted to BigFloat
without rounding. If the result is a string, it is converted to
BigFloat using the precision of the global floating point environment.

@code{BigFloat} properties:

@table @code

@item LN2
@item PI
Getter. Return the value of the corresponding mathematical constant
rounded to nearest, ties to even with the current global
precision. The constant values are cached for small precisions.

@item MIN_VALUE
@item MAX_VALUE
@item EPSILON
Getter. Return the minimum, maximum and epsilon @code{BigFloat} values
(same definition as the corresponding @code{Number} constants).

@item fpRound(a[, e])
Round the floating point number @code{a} according to the floating
point environment @code{e} or the global environment if @code{e} is
undefined.

@item parseFloat(a[, radix[, e]])
Parse the string @code{a} as a floating point number in radix
@code{radix}. The radix is 0 (default) or from 2 to 36. The radix 0
means radix 10 unless there is a hexadecimal or binary prefix. The
result is rounded according to the floating point environment @code{e}
or the global environment if @code{e} is undefined.

@item isFinite(a)
Return true if @code{a} is a finite bigfloat.

@item isNaN(a)
Return true if @code{a} is a NaN bigfloat.

@item add(a, b[, e])
@item sub(a, b[, e])
@item mul(a, b[, e])
@item div(a, b[, e])
Perform the specified floating point operation and round the floating
point number @code{a} according to the floating point environment
@code{e} or the global environment if @code{e} is undefined. If
@code{e} is specified, the floating point status flags are updated.

@item floor(x)
@item ceil(x)
@item round(x)
@item trunc(x)
Round to an integer. No additional rounding is performed.

@item abs(x)
Return the absolute value of x. No additional rounding is performed.

@item fmod(x, y[, e])
@item remainder(x, y[, e])
Floating point remainder. The quotient is truncated to zero (fmod) or
to the nearest integer with ties to even (remainder). @code{e} is an
optional floating point environment.

@item sqrt(x[, e])
Square root. Return a rounded floating point number. @code{e} is an
optional floating point environment.

@item sin(x[, e])
@item cos(x[, e])
@item tan(x[, e])
@item asin(x[, e])
@item acos(x[, e])
@item atan(x[, e])
@item atan2(x, y[, e])
@item exp(x[, e])
@item log(x[, e])
@item pow(x, y[, e])
Transcendental operations. Return a rounded floating point
number. @code{e} is an optional floating point environment.

@end table

@subsection @code{BigFloat.prototype}

The following properties are modified:

@table @code
@item valueOf()
Return the bigfloat primitive value corresponding to @code{this}.

@item toString(radix)

For floating point numbers:

@itemize
@item
If the radix is a power of two, the conversion is done with infinite
precision.
@item
Otherwise, the number is rounded to nearest with ties to even using
the global precision. It is then converted to string using the minimum
number of digits so that its conversion back to a floating point using
the global precision and round to nearest gives the same number. 

@end itemize

The exponent letter is @code{e} for base 10, @code{p} for bases 2, 8,
16 with a binary exponent and @code{@@} for the other bases.

@item toPrecision(p, rnd_mode = BigFloatEnv.RNDNA, radix = 10)
@item toFixed(p, rnd_mode = BigFloatEnv.RNDNA, radix = 10)
@item toExponential(p, rnd_mode = BigFloatEnv.RNDNA, radix = 10)
Same semantics as the corresponding @code{Number} functions with
BigFloats. There is no limit on the accepted precision @code{p}. The
rounding mode and radix can be optionally specified. The radix must be
between 2 and 36.

@end table

@subsection @code{BigFloatEnv} constructor

The @code{BigFloatEnv([p, [,rndMode]]} constructor cannot be invoked as a
function. The floating point environment contains:

@itemize
@item the mantissa precision in bits

@item the exponent size in bits assuming an IEEE 754 representation;

@item the subnormal flag (if true, subnormal floating point numbers can
be generated by the floating point operations).

@item the rounding mode

@item the floating point status. The status flags can only be set by the floating point operations. They can be reset with @code{BigFloatEnv.prototype.clearStatus()} or with the various status flag setters.

@end itemize

@code{new BigFloatEnv([p, [,rndMode]]} creates a new floating point
environment. The status flags are reset. If no parameter is given the
precision, exponent bits and subnormal flags are copied from the
global floating point environment. Otherwise, the precision is set to
@code{p}, the number of exponent bits is set to @code{expBitsMax} and the
subnormal flags is set to @code{false}. If @code{rndMode} is
@code{undefined}, the rounding mode is set to @code{RNDN}.

@code{BigFloatEnv} properties:

@table @code

@item prec
Getter. Return the precision in bits of the global floating point
environment. The initial value is @code{113}.

@item expBits
Getter. Return the exponent size in bits of the global floating point
environment assuming an IEEE 754 representation. The initial value is
@code{15}.

@item setPrec(f, p[, e])
Set the precision of the global floating point environment to @code{p}
and the exponent size to @code{e} then call the function
@code{f}. Then the Float precision and exponent size are reset to
their precious value and the return value of @code{f} is returned (or
an exception is raised if @code{f} raised an exception). If @code{e}
is @code{undefined} it is set to @code{BigFloatEnv.expBitsMax}.

@item precMin
Read-only integer. Return the minimum allowed precision. Must be at least 2.

@item precMax
Read-only integer. Return the maximum allowed precision. Must be at least 113.

@item expBitsMin
Read-only integer. Return the minimum allowed exponent size in
bits. Must be at least 3.

@item expBitsMax
Read-only integer. Return the maximum allowed exponent size in
bits. Must be at least 15.

@item RNDN
Read-only integer. Round to nearest, with ties to even rounding mode.

@item RNDZ
Read-only integer. Round to zero rounding mode.

@item RNDD
Read-only integer. Round to -Infinity rounding mode.

@item RNDU
Read-only integer. Round to +Infinity rounding mode.

@item RNDNA
Read-only integer. Round to nearest, with ties away from zero rounding mode.

@item RNDA
Read-only integer. Round away from zero rounding mode.

@item RNDF@footnote{Could be removed in case a deterministic behavior for floating point operations is required.}
Read-only integer. Faithful rounding mode. The result is
non-deterministically rounded to -Infinity or +Infinity. This rounding
mode usually gives a faster and deterministic running time for the
floating point operations.

@end table

@code{BigFloatEnv.prototype} properties:

@table @code

@item prec
Getter and setter (Integer). Return or set the precision in bits.

@item expBits
Getter and setter (Integer). Return or set the exponent size in bits
assuming an IEEE 754 representation.

@item rndMode
Getter and setter (Integer). Return or set the rounding mode.

@item subnormal
Getter and setter (Boolean). subnormal flag. It is false when
@code{expBits = expBitsMax}.

@item clearStatus()
Clear the status flags.

@item invalidOperation
@item divideByZero
@item overflow
@item underflow
@item inexact
Getter and setter (Boolean). Status flags.

@end table

@chapter BigDecimal

This extension adds the @code{BigDecimal} primitive type. The
@code{BigDecimal} type represents floating point numbers in base
10. It is inspired from the proposal available at
@url{https://github.com/littledan/proposal-bigdecimal}.

The @code{BigDecimal} floating point numbers are always normalized and
finite. There is no concept of @code{-0}, @code{Infinity} or
@code{NaN}. By default, all the computations are done with infinite
precision.

@section Operators

The following builtin operators support BigDecimal:

@table @code

@item +
@item -
@item *
Both operands must be BigDecimal. The result is computed with infinite
precision.
@item %
Both operands must be BigDecimal. The result is computed with infinite
precision. A range error is throws in case of division by zero.

@item /
Both operands must be BigDecimal. A range error is throws in case of
division by zero or if the result cannot be represented with infinite
precision (use @code{BigDecimal.div} to specify the rounding).

@item **
Both operands must be BigDecimal. The exponent must be a positive
integer. The result is computed with infinite precision.

@item ===
When one of the operand is a BigDecimal, return true if both operands
are a BigDecimal and if they are equal.

@item ==
@item !=
@item <=
@item >=
@item <
@item >

Numerical comparison. When one of the operand is not a BigDecimal, it is
converted to BigDecimal by using ToString(). Hence comparisons between
Number and BigDecimal do not use the exact mathematical value of the
Number value.

@end table

@section BigDecimal literals

BigDecimal literals are decimal floating point numbers with a trailing
@code{m} suffix.

@section Builtin Object changes

@subsection The @code{BigDecimal} function.

It returns @code{0m} if no parameter is provided. Otherwise the first
parameter is converted to a bigdecimal by using ToString(). Hence
Number values are not converted to their exact numerical value as
BigDecimal.

@subsection Properties of the @code{BigDecimal} object

@table @code

@item add(a, b[, e])
@item sub(a, b[, e])
@item mul(a, b[, e])
@item div(a, b[, e])
@item mod(a, b[, e])
@item sqrt(a, e)
@item round(a, e)
Perform the specified floating point operation and round the floating
point result according to the rounding object @code{e}. If the
rounding object is not present, the operation is executed with
infinite precision.

For @code{div}, a @code{RangeError} exception is thrown in case of
division by zero or if the result cannot be represented with infinite
precision if no rounding object is present.

For @code{sqrt}, a range error is thrown if @code{a} is less than
zero.

The rounding object must contain the following properties:
@code{roundingMode} is a string specifying the rounding mode
(@code{"floor"}, @code{"ceiling"}, @code{"down"}, @code{"up"},
@code{"half-even"}, @code{"half-up"}). Either
@code{maximumSignificantDigits} or @code{maximumFractionDigits} must
be present to specify respectively the number of significant digits
(must be >= 1) or the number of digits after the decimal point (must
be >= 0).

@end table

@subsection Properties of the @code{BigDecimal.prototype} object

@table @code
@item valueOf()
Return the bigdecimal primitive value corresponding to @code{this}.

@item toString()
Convert @code{this} to a string with infinite precision in base 10.

@item toPrecision(p, rnd_mode = "half-up")
@item toFixed(p, rnd_mode = "half-up")
@item toExponential(p, rnd_mode = "half-up")
Convert the BigDecimal @code{this} to string with the specified
precision @code{p}. There is no limit on the accepted precision
@code{p}. The rounding mode can be optionally
specified. @code{toPrecision} outputs either in decimal fixed notation
or in decimal exponential notation with a @code{p} digits of
precision.  @code{toExponential} outputs in decimal exponential
notation with @code{p} digits after the decimal point. @code{toFixed}
outputs in decimal notation with @code{p} digits after the decimal
point.

@end table

@chapter Math mode

A new @emph{math mode} is enabled with the @code{"use math"}
directive. It propagates the same way as the @emph{strict mode}. It is
designed so that arbitrarily large integers and floating point numbers
are available by default. In order to minimize the number of changes
in the Javascript semantics, integers are represented either as Number
or BigInt depending on their magnitude. Floating point numbers are
always represented as BigFloat.

The following changes are made to the Javascript semantics:

@itemize

@item Floating point literals (i.e. number with a decimal point or an exponent) are @code{BigFloat} by default (i.e. a @code{l} suffix is implied). Hence @code{typeof 1.0 === "bigfloat"}.

@item Integer literals (i.e. numbers without a decimal point or an exponent) with or without the @code{n} suffix are @code{BigInt} if their value cannot be represented as a safe integer. A safe integer is defined as a integer whose absolute value is smaller or equal to @code{2**53-1}. Hence @code{typeof 1 === "number "}, @code{typeof 1n === "number"} but @code{typeof 9007199254740992 === "bigint" }.

@item All the bigint builtin operators and functions are modified so that their result is returned as a Number if it is a safe integer. Otherwise the result stays a BigInt.

@item The builtin operators are modified so that they return an exact result (which can be a BigInt) if their operands are safe integers. Operands between Number and BigInt are accepted provided the Number operand is a safe integer. The integer power with a negative exponent returns a BigFloat as result. The integer division returns a BigFloat as result.

@item The @code{^} operator is an alias to the power operator (@code{**}).

@item The power operator (both @code{^} and @code{**}) grammar is modified so that @code{-2^2} is allowed and yields @code{-4}.

@item The logical xor operator is still available with the @code{^^} operator.

@item The modulo operator (@code{%}) returns the Euclidian remainder (always positive) instead of the truncated remainder.

@item The integer division operator can be overloaded with @code{Operators.updateBigIntOperators(dictionary)}.

@item The integer power operator with a non zero negative exponent can be overloaded with @code{Operators.updateBigIntOperators(dictionary)}.

@end itemize

@bye