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@iftex
@afourpaper
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@titlepage
@afourpaper
@sp 7
@center @titlefont{Javascript Bignum Extensions}
@sp 3
@center Version 2018-06-16
@sp 3
@center Author: Fabrice Bellard
@end titlepage

@setfilename spec.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 Overloading of the standard operators
to support new types such as complex numbers, fractions or matrixes.

@item Bigint mode where arbitrarily large integers are available by default (no @code{n} suffix is necessary as in the TC39 BigInt proposal@footnote{@url{https://tc39.github.io/proposal-bigint/}}).

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

@item Optional @code{math} mode which modifies the semantics of the division, modulo and power operator. The division and power operator return a fraction with integer operands and the modulo operator is defined as the Euclidian remainder.

@end itemize

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

@chapter Operator overloading

@section Introduction

If the operands of an operator have at least one object type, a custom
operator method is searched before doing the legacy Javascript
@code{ToNumber} conversion.

For unary operators, the custom function is looked up in the object
and has the following name:

@table @code
@item unary + 
@code{Symbol.operatorPlus}

@item unary -
@code{Symbol.operatorNeg}

@item ++
@code{Symbol.operatorInc}

@item --
@code{Symbol.operatorDec}

@item ~
@code{Symbol.operatorNot}

@end table

For binary operators:

@itemize

@item
If both operands have the same constructor function, then the operator
is looked up in the constructor.

@item
Otherwise, the property @code{Symbol.operatorOrder} is looked up in both
constructors and converted to @code{Int32}. The operator is then
looked in the constructor with the larger @code{Symbol.operatorOrder}
value. A @code{TypeError} is raised if both constructors have the same
@code{Symbol.operatorOrder} value.

@end itemize

The operator is looked up with the following name:

@table @code
@item + 
@code{Symbol.operatorAdd}

@item -
@code{Symbol.operatorSub}

@item *
@code{Symbol.operatorMul}

@item /
@code{Symbol.operatorDiv}

@item %
@code{Symbol.operatorMod}

@item % (math mode)
@code{Symbol.operatorMathMod}

@item **
@code{Symbol.operatorPow}

@item |
@code{Symbol.operatorOr}

@item ^
@code{Symbol.operatorXor}

@item &
@code{Symbol.operatorAnd}

@item <<
@code{Symbol.operatorShl}

@item >>
@code{Symbol.operatorShr}

@item <
@code{Symbol.operatorCmpLT}

@item >
@code{Symbol.operatorCmpLT}, operands swapped

@item <=
@code{Symbol.operatorCmpLE}

@item >=
@code{Symbol.operatorCmpLE}, operands swapped

@item ==, !=
@code{Symbol.operatorCmpEQ}

@end table

The return value of @code{Symbol.operatorCmpLT}, @code{Symbol.operatorCmpLE} and
@code{Symbol.operatorCmpEQ} is converted to @code{Boolean}.

@section Builtin Object changes

@subsection @code{Symbol} constructor

The following global symbols are added for the operator overloading:
@table @code
@item operatorOrder
@item operatorAdd
@item operatorSub
@item operatorMul
@item operatorDiv
@item operatorMod
@item operatorPow
@item operatorShl
@item operatorShr
@item operatorAnd
@item operatorOr
@item operatorXor
@item operatorCmpLT
@item operatorCmpLE
@item operatorCmpEQ
@item operatorPlus
@item operatorNeg
@item operatorNot
@item operatorInc
@item operatorDec
@end table


@chapter The BigInt Mode

@section Introduction

The bigint mode is enabled with the @code{"use bigint"} directive. It
propagates the same way as the strict mode. In bigint mode, all
integers are considered as @code{bigint} (arbitrarily large integer,
similar to the TC39 BigInt
proposal@footnote{@url{https://tc39.github.io/proposal-bigint/}})
instead of @code{number} (floating point number). In order to be able
to exchange data between standard and bigint modes, numbers are
internally represented as 3 different types:

@itemize

@item Small integer (SmallInt): 32 bit integer@footnote{Could be extended to 53 bits without changing the principle.}.

@item Big integer (BigInt): arbitrarily large integer.

@item Floating point number (Float).

@end itemize

In standard mode, the semantics of each operation is modified so that
when it returns a @code{number}, it is either of SmallInt or
Float. But the difference between SmallInt and Float is not observable
in standard mode.

In bigint mode, each operation behaves differently whether its
operands are integer or float. The difference between SmallInt and
BigInt is not observable (i.e. they are both integers).

The following table summarizes the observable types:

@multitable @columnfractions .3 .3 .3
@headitem Internal type @tab Observable type@* (standard mode) @tab Observable type@* (bigint mode)
@item SmallInt @tab number @tab bigint
@item BigInt @tab bigint @tab bigint
@item Float @tab number @tab number
@end multitable

@section Changes that introduce incompatibilities with Javascript

@subsection Standard mode

There is no incompatibility with Javascript.

@subsection Bigint mode

The following changes are visible:

@itemize

@item Integer and Float are different types. Constants are typed. For example: @code{typeof 1.0 === "number"} and @code{typeof 1 === "bigint"}. Another consequence is that @code{1.0 === 1} is false.

@item The range of integers is unlimited. In standard mode: @code{2**53 + 1 === 2**53}. This is no longer true with the bignum extensions.

@item Binary bitwise operators do not truncate to 32 bits i.e. @code{0x800000000 | 1 === 0x800000001} while it gives @code{1} in standard mode.

@item Bitwise shift operators do not truncate to 32 bits and do not mask the shift count with @code{0x1f} i.e. @code{1 << 32 === 4294967296} while it gives @code{1} in standard mode. However, the @code{>>>} operator (unsigned right shift) which is useless with bignums keeps its standard mode behavior@footnote{The unsigned right right operator could be removed in bigint mode.}.

@item Operators with integer operands never return the minus zero floating point value as result. Hence @code{Object.is(0, -0) === true}. Use @code{-0.0} to create a minus zero floating point value.

@item The @code{ToPrimitive} abstract operation is called with the @code{"integer"} preferred type when an integer is required (e.g. for bitwise binary or shift operations).

@item The prototype of integers is no longer @code{Number.prototype}. Instead@* @code{Object.getPrototypeOf(1) === BigInt.prototype}. The prototype of floats remains Number.prototype.

@item If the TC39 BigInt proposal is supported, there is no observable difference between integers and @code{bigint}s.

@end itemize

@section Operators

@subsection Arithmetic operators

The operands are converted to number values as in normal
Javascript. Then the general case is that an Integer is returned if
both operands are Integer. Otherwise, a float is returned.

The @code{+} operator also accepts strings as input and behaves like
standard Javascript in this case.

The binary operator @code{%} returns the truncated remainder of the
division. When the result is an Integer type, a dividend of zero yields a
RangeError exception.

The binary operator @code{%} in math mode returns the Euclidian
remainder of the division i.e. it is always positive. 

The binary operator @code{/} returns a float.

The binary operator @code{/} in math mode returns a float if one of
the operands is float. Otherwise, @code{BigInt[Symbol.operatorDiv]} is
invoked.

The returned type of @code{a ** b} is Float if @math{a} or @math{b}
are Float. If @math{a} and @math{b} are integers:
@itemize
@item @math{b < 0} returns a Float in bigint mode. In math mode, @code{BigInt[Symbol.operatorPow]} is invoked.

@item @math{b >= 0} returns an integer.
@end itemize

The unary @code{-} and unary @code{+} return the same type as their
operand. They performs no floating point rounding when the result is a
float.

The unary operators @code{++} and @code{--} return the same type as
their operand.

In standard mode:

If the operator returns an Integer and that the result fits a
SmallInt, it is converted to SmallInt. Otherwise, the Integer is
converted to a Float.

In bigint mode:

If the operator returns an Integer and that the result fits a
SmallInt, it is converted to SmallInt. Otherwise it is a BigInt.

@subsection Logical operators

In standard mode:

The operands have their standard behavior. If the result fits a
SmallInt it is converted to a SmallInt. Otherwise it is a Float.

In bigint mode:

The operands are converted to integer values. The floating point
values are converted to integer by rounding them to zero.

The logical operators are defined assuming the integers are
represented in two complement notation.

For @code{<<} and @code{<<}, the shift can be positive or negative. So
@code{a << b} is defined as @math{\lfloor a/2^{-b} \rfloor} and
@code{a >> b} is defined as @math{\lfloor a/2^{b} \rfloor}.

The operator @code{>>>} is supported for backward compatibility and
behaves the same way as Javascript i.e. implicit conversion to @code{Uint32}.

If the result fits a SmallInt it is converted to a SmallInt. Otherwise
it is a BigInt.

@subsection Relational operators

The relational operators <, <=, >, >=, ==, != work as expected with
integers and floating point numbers (e.g. @code{1.0 == 1} is true).

The strict equality operators === and !== have the usual Javascript
semantics. In particular, different types never equal, so @code{1.0
=== 1} is false.

@section Number literals

Number literals in bigint mode have a slightly different behavior than
in standard Javascript:

@enumerate

@item
A number literal without a decimal point or an exponent is considered
as an Integer. Otherwise it is a Float.

@item
Hexadecimal, octal or binary floating point literals are accepted with
a decimal point or an exponent. The exponent is specified with the
@code{p} letter assuming a base 2. The same convention is used by
C99. Example: @code{0x1p3} is the same as @code{8.0}.

@end enumerate

@section Builtin Object changes

@subsection @code{BigInt} function

The @code{BigInt} function cannot be invoked as a constructor. When
invoked as a function, it converts its first parameter to an
integer. When a floating point number is given as parameter, it is
truncated to an integer with infinite precision.

@code{BigInt} properties:

@table @code

@item asIntN(bits, a)
Set @math{b=a \pmod{2^{bits}}}. Return @math{b} if @math{b < 2^{bits-1}}
otherwise @math{b-2^{bits}}.

@item asUintN(bits, a)
Return @math{a \pmod{2^{bits}}}.

@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

@subsection @code{BigInt.prototype}

It is a normal object.

@subsection @code{Number} constructor

The number constructor returns its argument rounded to a Float using
the global floating point environement. In bigint mode, the Number
constructor returns a Float. In standard mode, it returns a SmallInt
if the value fits it, otherwise a Float.

@subsection @code{Number.prototype}

The following properties are modified:

@table @code
@item toString(radix)

In bigint mode, integers are converted to the specified radix with
infinite precision.

@item toPrecision(p)
@item toFixed(p)
@item toExponential(p)

In bigint mode, integers are accepted and converted to string with
infinite precision.

@item parseInt(string, radix)

In bigint mode, an integer is returned and the conversion is done with
infinite precision.

@end table

@subsection @code{Math} object

The following properties are modified:

@table @code
@item abs(x)
Absolute value. Return an integer if @code{x} is an Integer. Otherwise
return a Float. No rounding is performed.

@item min(a, b)
@item max(a, b)
No rounding is performed. The returned type is the same one as the
minimum (resp. maximum) value.

@end table

@chapter Arbitrarily large floating point numbers

@section Introduction

This extension adds the @code{BigFloat} primitive type. The
@code{BigFloat} type represents floating point numbers are 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}.  If @code{BigFloatEnv.expBits} is
strictly smaller than the maximum allowed number of exponent bits
(@code{BigFloatEnv.expBitsMax}), then the global environment subnormal
flag is set to @code{true}. Otherwise it is set to @code{false};

For example, @code{prec = 53} and @code{ expBits = 11} give exactly
the same precision as the IEEE 754 64 bit floating point type. It is
the default floating point precision.

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 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[, e])
@item ceil(x[, e])
@item round(x[, e])
@item trunc(x[, e])
Round to integer. A rounded @code{BigFloat} is returned. @code{e} is an
optional floating point environment.

@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 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

@item toPrecision(p[, rnd_mode])
@item toFixed(p[, rnd_mode])
@item toExponential(p[, rnd_mode])
Same semantics as the corresponding @code{Number} functions with
BigFloats. There is no limit on the accepted precision @code{p}. The
rounding mode can be optionally specified. It is set by default to
@code{BigFloatEnv.RNDNA}.

@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{53}.

@item expBits
Getter. Return the exponent size in bits of the global floating point
environment assuming an IEEE 754 representation. If @code{expBits <
expBitsMax}, then subnormal numbers are supported. The initial value
is @code{11}.

@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}. @code{p}
must be >= 53 and @code{e} must be >= 11 so that the global precision
is at least equivalent to the IEEE 754 64 bit doubles.

@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 53.

@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 11.

@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 RNDNU
Read-only integer. Round to nearest, with ties to +Infinity rounding mode.

@item RNDF@footnote{Could be removed in case a deterministic behvior for floating point operations is required.}
Read-only integer. Faithful rounding mode. The result is
non-deterministicly 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

@subsection @code{Math} object

The following properties are modified:

@table @code
@item abs(x)
Absolute value. If @code{x} is a BigFloat, its absolute value is
returned as a BigFloat. No rounding is performed.

@item min(a, b)
@item max(a, b)
The returned type is the same one as the minimum (resp. maximum)
value, so @code{BigFloat} values are accepted. When a @code{BigFloat}
is returned, no rounding is performed.

@end table

@chapter Math mode

@section Introduction

A new @emph{math mode} is enabled with the @code{"use math"}
directive. @code{"use bigint"} is implied in math mode. With this
mode, writing mathematical expressions is more intuitive, exact
results (e.g. fractions) can be computed for all operators and floating
point literals have the @code{BigFloat} type by default.

It propagates the same way as the @emph{strict mode}. In
this mode:

@itemize

@item The @code{^} operator is a similar 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 division operator invokes @code{BigInt[Symbol.operatorDiv]} in case both operands are integers.

@item The power operator invokes @code{BigInt[Symbol.operatorPow]} in case both operands are integers and the exponent is strictly negative.

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

@item Floating point literals are @code{BigFloat} by default (i.e. a @code{l} suffix is implied).

@end itemize

@section Builtin Object changes

@subsection @code{Symbol} constructor

The following global symbol is added for the operator overloading:
@table @code
@item operatorMathMod
@end table

@section Remaining issues

@enumerate

@item A new floating point literal suffix could be added for @code{Number} literals.

@end enumerate

@bye