The Bignum extensions add the following features to the Javascript language while being 100% backward compatible:
n suffix is necessary as in the TC39 BigInt proposal1).
BigFloat) in base 2 using the IEEE 754 semantics.
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.
The extensions are independent from each other except the math
mode which relies on the bigint mode and the operator overloading.
If the operands of an operator have at least one object type, a custom
operator method is searched before doing the legacy Javascript
ToNumber conversion.
For unary operators, the custom function is looked up in the object and has the following name:
unary +Symbol.operatorPlus
unary -Symbol.operatorNeg
++Symbol.operatorInc
--Symbol.operatorDec
~Symbol.operatorNot
For binary operators:
Symbol.operatorOrder is looked up in both
constructors and converted to Int32. The operator is then
looked in the constructor with the larger Symbol.operatorOrder
value. A TypeError is raised if both constructors have the same
Symbol.operatorOrder value.
The operator is looked up with the following name:
+Symbol.operatorAdd
-Symbol.operatorSub
*Symbol.operatorMul
/Symbol.operatorDiv
%Symbol.operatorMod
% (math mode)Symbol.operatorMathMod
**Symbol.operatorPow
|Symbol.operatorOr
^Symbol.operatorXor
&Symbol.operatorAnd
<<Symbol.operatorShl
>>Symbol.operatorShr
<Symbol.operatorCmpLT
>Symbol.operatorCmpLT, operands swapped
<=Symbol.operatorCmpLE
>=Symbol.operatorCmpLE, operands swapped
==, !=Symbol.operatorCmpEQ
The return value of Symbol.operatorCmpLT, Symbol.operatorCmpLE and
Symbol.operatorCmpEQ is converted to Boolean.
Symbol constructorThe following global symbols are added for the operator overloading:
operatorOrderoperatorAddoperatorSuboperatorMuloperatorDivoperatorModoperatorPowoperatorShloperatorShroperatorAndoperatorOroperatorXoroperatorCmpLToperatorCmpLEoperatorCmpEQoperatorPlusoperatorNegoperatorNotoperatorIncoperatorDecThe bigint mode is enabled with the "use bigint" directive. It
propagates the same way as the strict mode. In bigint mode, all
integers are considered as bigint (arbitrarily large integer,
similar to the TC39 BigInt
proposal2)
instead of 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:
In standard mode, the semantics of each operation is modified so that
when it returns a 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:
| Internal type | Observable type (standard mode) | Observable type (bigint mode) |
|---|---|---|
| SmallInt | number | bigint |
| BigInt | bigint | bigint |
| Float | number | number |
There is no incompatibility with Javascript.
The following changes are visible:
typeof 1.0 === "number" and typeof 1 === "bigint". Another consequence is that 1.0 === 1 is false.
2**53 + 1 === 2**53. This is no longer true with the bignum extensions.
0x800000000 | 1 === 0x800000001 while it gives 1 in standard mode.
0x1f i.e. 1 << 32 === 4294967296 while it gives 1 in standard mode. However, the >>> operator (unsigned right shift) which is useless with bignums keeps its standard mode behavior4.
Object.is(0, -0) === true. Use -0.0 to create a minus zero floating point value.
ToPrimitive abstract operation is called with the "integer" preferred type when an integer is required (e.g. for bitwise binary or shift operations).
Number.prototype. InsteadObject.getPrototypeOf(1) === BigInt.prototype. The prototype of floats remains Number.prototype.
bigints.
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 + operator also accepts strings as input and behaves like
standard Javascript in this case.
The binary operator % 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 % in math mode returns the Euclidian
remainder of the division i.e. it is always positive.
The binary operator / returns a float.
The binary operator / in math mode returns a float if one of
the operands is float. Otherwise, BigInt[Symbol.operatorDiv] is
invoked.
The returned type of a ** b is Float if a or b
are Float. If a and b are integers:
BigInt[Symbol.operatorPow] is invoked.
The unary - and unary + return the same type as their
operand. They performs no floating point rounding when the result is a
float.
The unary operators ++ and -- 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.
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 << and <<, the shift can be positive or negative. So
a << b is defined as \lfloor a/2^{-b} \rfloor and
a >> b is defined as \lfloor a/2^{b} \rfloor.
The operator >>> is supported for backward compatibility and
behaves the same way as Javascript i.e. implicit conversion to Uint32.
If the result fits a SmallInt it is converted to a SmallInt. Otherwise it is a BigInt.
The relational operators <, <=, >, >=, ==, != work as expected with
integers and floating point numbers (e.g. 1.0 == 1 is true).
The strict equality operators === and !== have the usual Javascript
semantics. In particular, different types never equal, so 1.0
=== 1 is false.
Number literals in bigint mode have a slightly different behavior than in standard Javascript:
p letter assuming a base 2. The same convention is used by
C99. Example: 0x1p3 is the same as 8.0.
BigInt functionThe 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.
BigInt properties:
asIntN(bits, a)Set b=a \pmod{2^{bits}}. Return b if b < 2^{bits-1} otherwise b-2^{bits}.
asUintN(bits, a)Return a \pmod{2^{bits}}.
tdiv(a, b)Return trunc(a/b). b = 0 raises a RangeError
exception.
fdiv(a, b)Return \lfloor a/b \rfloor. b = 0 raises a RangeError
exception.
cdiv(a, b)Return \lceil a/b \rceil. b = 0 raises a RangeError
exception.
ediv(a, b)Return sgn(b) \lfloor a/{|b|} \rfloor (Euclidian
division). b = 0 raises a RangeError exception.
tdivrem(a, b)fdivrem(a, b)cdivrem(a, b)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.
sqrt(a)Return \lfloor \sqrt(a) \rfloor. A RangeError exception is raised if a < 0.
sqrtrem(a)Return an array of two elements. The first element is \lfloor \sqrt{a} \rfloor. The second element is a-\lfloor \sqrt{a} \rfloor^2. A RangeError exception is raised if a < 0.
floorLog2(a)Return -1 if a \leq 0 otherwise return \lfloor \log2(a) \rfloor.
ctz(a)Return the number of trailing zeros in the two’s complement binary representation of a. Return -1 if a=0.
BigInt.prototypeIt is a normal object.
Number constructorThe number constructor returns its argument rounded to a Float using the global floating point environment. In bigint mode, the Number constructor returns a Float. In standard mode, it returns a SmallInt if the value fits it, otherwise a Float.
Number.prototypeThe following properties are modified:
toString(radix)In bigint mode, integers are converted to the specified radix with infinite precision.
toPrecision(p)toFixed(p)toExponential(p)In bigint mode, integers are accepted and converted to string with infinite precision.
parseInt(string, radix)In bigint mode, an integer is returned and the conversion is done with infinite precision.
Math objectThe following properties are modified:
abs(x)Absolute value. Return an integer if x is an Integer. Otherwise
return a Float. No rounding is performed.
min(a, b)max(a, b)No rounding is performed. The returned type is the same one as the minimum (resp. maximum) value.
This extension adds the BigFloat primitive type. The
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 NaN, +/-Infinity, +0 and -0
are supported. The mantissa and exponent can have any bit length with
an implementation specific minimum and maximum.
Each floating point operation operates with infinite precision and
then rounds the result according to the specified floating point
environment (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
RNDN (“round to nearest with ties to even”)5. The status flags of the global environment cannot be
read6. The precision of the global environment is
BigFloatEnv.prec. The number of exponent bits of the global
environment is BigFloatEnv.expBits. If BigFloatEnv.expBits is
strictly smaller than the maximum allowed number of exponent bits
(BigFloatEnv.expBitsMax), then the global environment subnormal
flag is set to true. Otherwise it is set to false;
For example, prec = 53 and expBits = 11 give exactly
the same precision as the IEEE 754 64 bit floating point. The
default precision is prec = 113 and expBits = 15 (IEEE
754 128 bit floating point).
The global floating point environment can only be modified temporarily
when calling a function (see BigFloatEnv.setPrec). Hence a
function can change the global floating point environment for its
callees but not for its caller.
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.
typeof applied on a BigFloat returns 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. 0.0 ===
0.0l is false).
BigFloat literals are floating point numbers with a trailing l
suffix. BigFloat literals have an infinite precision. They are rounded
according to the global floating point environment when they are
evaluated.7
BigFloat functionThe 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.
BigFloat properties:
LN2PIGetter. 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.
MIN_VALUEMAX_VALUEEPSILONGetter. Return the minimum, maximum and epsilon BigFloat values
(same definition as the corresponding Number constants).
fpRound(a[, e])Round the floating point number a according to the floating
point environment e or the global environment if e is
undefined.
parseFloat(a[, radix[, e]])Parse the string a as a floating point number in radix
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 e
or the global environment if e is undefined.
isFinite(a)Return true if a is a finite bigfloat.
isNaN(a)Return true if a is a NaN bigfloat.
add(a, b[, e])sub(a, b[, e])mul(a, b[, e])div(a, b[, e])Perform the specified floating point operation and round the floating
point number a according to the floating point environment
e or the global environment if e is undefined. If
e is specified, the floating point status flags are updated.
floor(x)ceil(x)round(x)trunc(x)Round to an integer. No additional rounding is performed.
fmod(x, y[, e])remainder(x, y[, e])Floating point remainder. The quotient is truncated to zero (fmod) or
to the nearest integer with ties to even (remainder). e is an
optional floating point environment.
sqrt(x[, e])Square root. Return a rounded floating point number. e is an
optional floating point environment.
sin(x[, e])cos(x[, e])tan(x[, e])asin(x[, e])acos(x[, e])atan(x[, e])atan2(x, y[, e])exp(x[, e])log(x[, e])pow(x, y[, e])Transcendental operations. Return a rounded floating point
number. e is an optional floating point environment.
BigFloat.prototypeThe following properties are modified:
toString(radix)For floating point numbers:
toPrecision(p[, rnd_mode])toFixed(p[, rnd_mode])toExponential(p[, rnd_mode])Same semantics as the corresponding Number functions with
BigFloats. There is no limit on the accepted precision p. The
rounding mode can be optionally specified. It is set by default to
BigFloatEnv.RNDNA.
BigFloatEnv constructorThe BigFloatEnv([p, [,rndMode]] constructor cannot be invoked as a
function. The floating point environment contains:
BigFloatEnv.prototype.clearStatus() or with the various status flag setters.
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
p, the number of exponent bits is set to expBitsMax and the
subnormal flags is set to false. If rndMode is
undefined, the rounding mode is set to RNDN.
BigFloatEnv properties:
precGetter. Return the precision in bits of the global floating point
environment. The initial value is 53.
expBitsGetter. Return the exponent size in bits of the global floating point
environment assuming an IEEE 754 representation. If expBits <
expBitsMax, then subnormal numbers are supported. The initial value
is 11.
setPrec(f, p[, e])Set the precision of the global floating point environment to p
and the exponent size to e then call the function
f. Then the Float precision and exponent size are reset to
their precious value and the return value of f is returned (or
an exception is raised if f raised an exception). If e
is undefined it is set to BigFloatEnv.expBitsMax. p
must be >= 53 and e must be >= 11 so that the global precision
is at least equivalent to the IEEE 754 64 bit doubles.
precMinRead-only integer. Return the minimum allowed precision. Must be at least 2.
precMaxRead-only integer. Return the maximum allowed precision. Must be at least 53.
expBitsMinRead-only integer. Return the minimum allowed exponent size in bits. Must be at least 3.
expBitsMaxRead-only integer. Return the maximum allowed exponent size in bits. Must be at least 11.
RNDNRead-only integer. Round to nearest, with ties to even rounding mode.
RNDZRead-only integer. Round to zero rounding mode.
RNDDRead-only integer. Round to -Infinity rounding mode.
RNDURead-only integer. Round to +Infinity rounding mode.
RNDNARead-only integer. Round to nearest, with ties away from zero rounding mode.
RNDNURead-only integer. Round to nearest, with ties to +Infinity rounding mode.
RNDF8Read-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.
BigFloatEnv.prototype properties:
precGetter and setter (Integer). Return or set the precision in bits.
expBitsGetter and setter (Integer). Return or set the exponent size in bits assuming an IEEE 754 representation.
rndModeGetter and setter (Integer). Return or set the rounding mode.
subnormalGetter and setter (Boolean). subnormal flag. It is false when
expBits = expBitsMax.
clearStatus()Clear the status flags.
invalidOperationdivideByZerooverflowunderflowinexactGetter and setter (Boolean). Status flags.
Math objectThe following properties are modified:
abs(x)Absolute value. If x is a BigFloat, its absolute value is
returned as a BigFloat. No rounding is performed.
min(a, b)max(a, b)The returned type is the same one as the minimum (resp. maximum)
value, so BigFloat values are accepted. When a BigFloat
is returned, no rounding is performed.
A new math mode is enabled with the "use math"
directive. "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 BigFloat type by default.
It propagates the same way as the strict mode. In this mode:
^ operator is a similar to the power operator (**).
^ and **) grammar is modified so that -2^2 is allowed and yields -4.
^^ operator.
BigInt[Symbol.operatorDiv] in case both operands are integers.
BigInt[Symbol.operatorPow] in case both operands are integers and the exponent is strictly negative.
BigFloat by default (i.e. a l suffix is implied).
Symbol constructorThe following global symbol is added for the operator overloading:
operatorMathModNumber literals.
https://tc39.github.io/proposal-bigint/
https://tc39.github.io/proposal-bigint/
Could be extended to 53 bits without changing the principle.
The unsigned right right operator could be removed in bigint mode.
The rationale is that the rounding mode changes must always be explicit.
The rationale is to avoid side effects for the built-in operators.
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.
Could be removed in case a deterministic behavior for floating point operations is required.