If I understand correctly, JavaScript numbers are always stored as double precision floating point numbers, following the international IEEE 754 standard. Which mean it uses 52 bits for fraction significand. But in the picture above, it seems like 0.57 in binary uses 54 bits.
Another thing is (if I understand correctly) 0.55 in binary is also an repeating number. But why 0.55 + 1 = 1.55 (no loss) and 0.57 + 1 = 1.5699999999999998
Which mean it uses 52 bits for fraction significand. But in the picture above, it seems like 0.57 in binary uses 54 bits.
JavaScript’s Number type, which is essentially IEEE 754 basic 64-bit binary floating-point, has 53-bit significands. 52 bits are encoded in the “trailing significand” field. The leading bit is encoded via the exponent field (an exponent field of 1-2046 means the leading bit is one, an exponent field of 0 means the leading bit is zero, and an exponent field of 2047 is used for infinity or NaN).
The value you see for .57 has 53 significant bits. The leading “0.” is produced by the toString operation; it is not part of the encoding of the number.
But why 0.55 + 1 = 1.55 (no loss) and 0.57 + 1 = 1.5699999999999998.
When JavaScript is formatting some Number x for display with its default rules, those rules say to produce the shortest decimal numeral (in its significant digits, not counting decorations like a leading “0.”) that, when converted back to the Number format, produces x. Purposes of this rule include (a) always ensuring the display uniquely identifies which exact Number value was the source value and (b) not using more digits than necessary to accomplish (a).
Thus, if you start with a decimal numeral such as .57 and convert it to a Number, you get some value x that is a result of the conversion having to round to a number representable in the Number format. Then, when x is formatted for display, you get the original number, because the rule that says to produce the shortest number that converts back to x naturally produces the number you started with.
(But that x does not exactly represent 0.57. The nearest double to 0.57 is slightly below it; see the decimal and binary64 representations of it on an IEEE double calculator).
On the other hand, when you perform some operation such as .57 + 1, you are doing some arithmetic that produces a number y that did not start as a simple decimal numeral. So, when formatting such a number for display, the rule may require more digits be used for it. In other words. when you add .57 and 1, the result in the Number format is not the same number as you get from 1.57. So, to format the result of .57 + 1, JavaScript has to use more digits to distinguish that number from the number you get from 1.57—they are different and must be displayed differently.
If 0.57 was exactly representable as a double, the pre-rounding result of the sum would be exactly 1.57, so 1 + 0.57 would round to the same double as 1.57.
But that's not the case, it's actually 1 + nearest_double(0.57) =
1.569999999999999951150186916493 (pre-rounding, not a double) which rounds down to
1.56999999999999984012788445398. These decimal representations of numbers have many more digits than we need to distinguish 1ulp (unit in the last place) of the significand, or even the 0.5 ulp max rounding error.
1.57 rounds to ~1.57000000000000006217248937901, so that's not an option for printing the result of 1 + 0.57. The decimal string needs to distinguish the number from adjacent binary64 values.
It just so happens that the rounding that occurs in .55 + 1 yields the same number one gets from converting 1.55 to Number, so displaying the result of .55 + 1 produces “1.55”.
toString(2) prints string up to last non-zero digit.
1.57 has different bit representation than 1 + 0.57 ( but it's not impossible to get result 1.57), but 1 + 0.55 in binary equals 1.55 as you can see in snippet below:
console.log(1.57)
console.log(1.57.toString(2))
console.log((1+.57).toString(2))
console.log("1.32 + 0.25 = ",1.32 + .25)
console.log((1.32 + .25).toString(2))
console.log(1.55)
console.log(1.55.toString(2))
console.log((1+.55).toString(2))
Remember that computer performs operations on binary numbers, 1.57 or 1.55 is just a human-readable output
Number.prototype.toString roughly implements the following section of the ES262 spec:
7.1.12.1 NumberToString(m)
let n, k, and s be integers such that k ≥ 1, 10 ** k-1 ≤ s < 10 ** k,
the Number value for s × 10 ** n-k is m,
and k is as small as possible.
Therefore toString just estimates the value, it does not return the exact bytes stored.
What you see in the console is not an exact representation either.
Related
See this code:
var jsonString = '{"id":714341252076979033,"type":"FUZZY"}';
var jsonParsed = JSON.parse(jsonString);
console.log(jsonString, jsonParsed);
When I see my console in Firefox 3.5, the value of jsonParsed is the number rounded:
Object id=714341252076979100 type=FUZZY
Tried different values, the same outcome (number rounded).
I also don't get its rounding rules. 714341252076979136 is rounded to 714341252076979200, whereas 714341252076979135 is rounded to 714341252076979100.
Why is this happening?
You're overflowing the capacity of JavaScript's number type, see §8.5 of the spec for details. Those IDs will need to be strings.
IEEE-754 double-precision floating point (the kind of number JavaScript uses) can't precisely represent all numbers (of course). Famously, 0.1 + 0.2 == 0.3 is false. That can affect whole numbers just like it affects fractional numbers; it starts once you get above 9,007,199,254,740,991 (Number.MAX_SAFE_INTEGER).
Beyond Number.MAX_SAFE_INTEGER + 1 (9007199254740992), the IEEE-754 floating-point format can no longer represent every consecutive integer. 9007199254740991 + 1 is 9007199254740992, but 9007199254740992 + 1 is also 9007199254740992 because 9007199254740993 cannot be represented in the format. The next that can be is 9007199254740994. Then 9007199254740995 can't be, but 9007199254740996 can.
The reason is we've run out of bits, so we no longer have a 1s bit; the lowest-order bit now represents multiples of 2. Eventually, if we keep going, we lose that bit and only work in multiples of 4. And so on.
Your values are well above that threshold, and so they get rounded to the nearest representable value.
As of ES2020, you can use BigInt for integers that are arbitrarily large, but there is no JSON representation for them. You could use strings and a reviver function:
const jsonString = '{"id":"714341252076979033","type":"FUZZY"}';
// Note it's a string −−−−^−−−−−−−−−−−−−−−−−−^
const obj = JSON.parse(jsonString, (key, value) => {
if (key === "id" && typeof value === "string" && value.match(/^\d+$/)) {
return BigInt(value);
}
return value;
});
console.log(obj);
(Look in the real console, the snippets console doesn't understand BigInt.)
If you're curious about the bits, here's what happens: An IEEE-754 binary double-precision floating-point number has a sign bit, 11 bits of exponent (which defines the overall scale of the number, as a power of 2 [because this is a binary format]), and 52 bits of significand (but the format is so clever it gets 53 bits of precision out of those 52 bits). How the exponent is used is complicated (described here), but in very vague terms, if we add one to the exponent, the value of the significand is doubled, since the exponent is used for powers of 2 (again, caveat there, it's not direct, there's cleverness in there).
So let's look at the value 9007199254740991 (aka, Number.MAX_SAFE_INTEGER):
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110011 1111111111111111111111111111111111111111111111111111
= 9007199254740991 (Number.MAX_SAFE_INTEGER)
That exponent value, 10000110011, means that every time we add one to the significand, the number represented goes up by 1 (the whole number 1, we lost the ability to represent fractional numbers much earlier).
But now that significand is full. To go past that number, we have to increase the exponent, which means that if we add one to the significand, the value of the number represented goes up by 2, not 1 (because the exponent is applied to 2, the base of this binary floating point number):
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110100 0000000000000000000000000000000000000000000000000000
= 9007199254740992 (Number.MAX_SAFE_INTEGER + 1)
Well, that's okay, because 9007199254740991 + 1 is 9007199254740992 anyway. But! We can't represent 9007199254740993. We've run out of bits. If we add just 1 to the significand, it adds 2 to the value:
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110100 0000000000000000000000000000000000000000000000000001
= 9007199254740994 (Number.MAX_SAFE_INTEGER + 3)
The format just cannot represent odd numbers anymore as we increase the value, the exponent is too big.
Eventually, we run out of significand bits again and have to increase the exponent, so we end up only being able to represent multiples of 4. Then multiples of 8. Then multiples of 16. And so on.
What you're seeing here is actually the effect of two roundings. Numbers in ECMAScript are internally represented double-precision floating-point. When id is set to 714341252076979033 (0x9e9d9958274c359 in hex), it actually is assigned the nearest representable double-precision value, which is 714341252076979072 (0x9e9d9958274c380). When you print out the value, it is being rounded to 15 significant decimal digits, which gives 14341252076979100.
It is not caused by this json parser. Just try to enter 714341252076979033 to fbug's console. You'll see the same 714341252076979100.
See this blog post for details:
http://www.exploringbinary.com/print-precision-of-floating-point-integers-varies-too
JavaScript uses double precision floating point values, ie a total precision of 53 bits, but you need
ceil(lb 714341252076979033) = 60
bits to exactly represent the value.
The nearest exactly representable number is 714341252076979072 (write the original number in binary, replace the last 7 digits with 0 and round up because the highest replaced digit was 1).
You'll get 714341252076979100 instead of this number because ToString() as described by ECMA-262, §9.8.1 works with powers of ten and in 53 bit precision all these numbers are equal.
The problem is that your number requires a greater precision than JavaScript has.
Can you send the number as a string? Separated in two parts?
JavaScript can only handle exact whole numbers up to about 9000 million million (that's 9 with 15 zeros). Higher than that and you get garbage. Work around this by using strings to hold the numbers. If you need to do math with these numbers, write your own functions or see if you can find a library for them: I suggest the former as I don't like the libraries I've seen. To get you started, see two of my functions at another answer.
See this code:
var jsonString = '{"id":714341252076979033,"type":"FUZZY"}';
var jsonParsed = JSON.parse(jsonString);
console.log(jsonString, jsonParsed);
When I see my console in Firefox 3.5, the value of jsonParsed is the number rounded:
Object id=714341252076979100 type=FUZZY
Tried different values, the same outcome (number rounded).
I also don't get its rounding rules. 714341252076979136 is rounded to 714341252076979200, whereas 714341252076979135 is rounded to 714341252076979100.
Why is this happening?
You're overflowing the capacity of JavaScript's number type, see §8.5 of the spec for details. Those IDs will need to be strings.
IEEE-754 double-precision floating point (the kind of number JavaScript uses) can't precisely represent all numbers (of course). Famously, 0.1 + 0.2 == 0.3 is false. That can affect whole numbers just like it affects fractional numbers; it starts once you get above 9,007,199,254,740,991 (Number.MAX_SAFE_INTEGER).
Beyond Number.MAX_SAFE_INTEGER + 1 (9007199254740992), the IEEE-754 floating-point format can no longer represent every consecutive integer. 9007199254740991 + 1 is 9007199254740992, but 9007199254740992 + 1 is also 9007199254740992 because 9007199254740993 cannot be represented in the format. The next that can be is 9007199254740994. Then 9007199254740995 can't be, but 9007199254740996 can.
The reason is we've run out of bits, so we no longer have a 1s bit; the lowest-order bit now represents multiples of 2. Eventually, if we keep going, we lose that bit and only work in multiples of 4. And so on.
Your values are well above that threshold, and so they get rounded to the nearest representable value.
As of ES2020, you can use BigInt for integers that are arbitrarily large, but there is no JSON representation for them. You could use strings and a reviver function:
const jsonString = '{"id":"714341252076979033","type":"FUZZY"}';
// Note it's a string −−−−^−−−−−−−−−−−−−−−−−−^
const obj = JSON.parse(jsonString, (key, value) => {
if (key === "id" && typeof value === "string" && value.match(/^\d+$/)) {
return BigInt(value);
}
return value;
});
console.log(obj);
(Look in the real console, the snippets console doesn't understand BigInt.)
If you're curious about the bits, here's what happens: An IEEE-754 binary double-precision floating-point number has a sign bit, 11 bits of exponent (which defines the overall scale of the number, as a power of 2 [because this is a binary format]), and 52 bits of significand (but the format is so clever it gets 53 bits of precision out of those 52 bits). How the exponent is used is complicated (described here), but in very vague terms, if we add one to the exponent, the value of the significand is doubled, since the exponent is used for powers of 2 (again, caveat there, it's not direct, there's cleverness in there).
So let's look at the value 9007199254740991 (aka, Number.MAX_SAFE_INTEGER):
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110011 1111111111111111111111111111111111111111111111111111
= 9007199254740991 (Number.MAX_SAFE_INTEGER)
That exponent value, 10000110011, means that every time we add one to the significand, the number represented goes up by 1 (the whole number 1, we lost the ability to represent fractional numbers much earlier).
But now that significand is full. To go past that number, we have to increase the exponent, which means that if we add one to the significand, the value of the number represented goes up by 2, not 1 (because the exponent is applied to 2, the base of this binary floating point number):
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110100 0000000000000000000000000000000000000000000000000000
= 9007199254740992 (Number.MAX_SAFE_INTEGER + 1)
Well, that's okay, because 9007199254740991 + 1 is 9007199254740992 anyway. But! We can't represent 9007199254740993. We've run out of bits. If we add just 1 to the significand, it adds 2 to the value:
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110100 0000000000000000000000000000000000000000000000000001
= 9007199254740994 (Number.MAX_SAFE_INTEGER + 3)
The format just cannot represent odd numbers anymore as we increase the value, the exponent is too big.
Eventually, we run out of significand bits again and have to increase the exponent, so we end up only being able to represent multiples of 4. Then multiples of 8. Then multiples of 16. And so on.
What you're seeing here is actually the effect of two roundings. Numbers in ECMAScript are internally represented double-precision floating-point. When id is set to 714341252076979033 (0x9e9d9958274c359 in hex), it actually is assigned the nearest representable double-precision value, which is 714341252076979072 (0x9e9d9958274c380). When you print out the value, it is being rounded to 15 significant decimal digits, which gives 14341252076979100.
It is not caused by this json parser. Just try to enter 714341252076979033 to fbug's console. You'll see the same 714341252076979100.
See this blog post for details:
http://www.exploringbinary.com/print-precision-of-floating-point-integers-varies-too
JavaScript uses double precision floating point values, ie a total precision of 53 bits, but you need
ceil(lb 714341252076979033) = 60
bits to exactly represent the value.
The nearest exactly representable number is 714341252076979072 (write the original number in binary, replace the last 7 digits with 0 and round up because the highest replaced digit was 1).
You'll get 714341252076979100 instead of this number because ToString() as described by ECMA-262, §9.8.1 works with powers of ten and in 53 bit precision all these numbers are equal.
The problem is that your number requires a greater precision than JavaScript has.
Can you send the number as a string? Separated in two parts?
JavaScript can only handle exact whole numbers up to about 9000 million million (that's 9 with 15 zeros). Higher than that and you get garbage. Work around this by using strings to hold the numbers. If you need to do math with these numbers, write your own functions or see if you can find a library for them: I suggest the former as I don't like the libraries I've seen. To get you started, see two of my functions at another answer.
See this code:
var jsonString = '{"id":714341252076979033,"type":"FUZZY"}';
var jsonParsed = JSON.parse(jsonString);
console.log(jsonString, jsonParsed);
When I see my console in Firefox 3.5, the value of jsonParsed is the number rounded:
Object id=714341252076979100 type=FUZZY
Tried different values, the same outcome (number rounded).
I also don't get its rounding rules. 714341252076979136 is rounded to 714341252076979200, whereas 714341252076979135 is rounded to 714341252076979100.
Why is this happening?
You're overflowing the capacity of JavaScript's number type, see §8.5 of the spec for details. Those IDs will need to be strings.
IEEE-754 double-precision floating point (the kind of number JavaScript uses) can't precisely represent all numbers (of course). Famously, 0.1 + 0.2 == 0.3 is false. That can affect whole numbers just like it affects fractional numbers; it starts once you get above 9,007,199,254,740,991 (Number.MAX_SAFE_INTEGER).
Beyond Number.MAX_SAFE_INTEGER + 1 (9007199254740992), the IEEE-754 floating-point format can no longer represent every consecutive integer. 9007199254740991 + 1 is 9007199254740992, but 9007199254740992 + 1 is also 9007199254740992 because 9007199254740993 cannot be represented in the format. The next that can be is 9007199254740994. Then 9007199254740995 can't be, but 9007199254740996 can.
The reason is we've run out of bits, so we no longer have a 1s bit; the lowest-order bit now represents multiples of 2. Eventually, if we keep going, we lose that bit and only work in multiples of 4. And so on.
Your values are well above that threshold, and so they get rounded to the nearest representable value.
As of ES2020, you can use BigInt for integers that are arbitrarily large, but there is no JSON representation for them. You could use strings and a reviver function:
const jsonString = '{"id":"714341252076979033","type":"FUZZY"}';
// Note it's a string −−−−^−−−−−−−−−−−−−−−−−−^
const obj = JSON.parse(jsonString, (key, value) => {
if (key === "id" && typeof value === "string" && value.match(/^\d+$/)) {
return BigInt(value);
}
return value;
});
console.log(obj);
(Look in the real console, the snippets console doesn't understand BigInt.)
If you're curious about the bits, here's what happens: An IEEE-754 binary double-precision floating-point number has a sign bit, 11 bits of exponent (which defines the overall scale of the number, as a power of 2 [because this is a binary format]), and 52 bits of significand (but the format is so clever it gets 53 bits of precision out of those 52 bits). How the exponent is used is complicated (described here), but in very vague terms, if we add one to the exponent, the value of the significand is doubled, since the exponent is used for powers of 2 (again, caveat there, it's not direct, there's cleverness in there).
So let's look at the value 9007199254740991 (aka, Number.MAX_SAFE_INTEGER):
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110011 1111111111111111111111111111111111111111111111111111
= 9007199254740991 (Number.MAX_SAFE_INTEGER)
That exponent value, 10000110011, means that every time we add one to the significand, the number represented goes up by 1 (the whole number 1, we lost the ability to represent fractional numbers much earlier).
But now that significand is full. To go past that number, we have to increase the exponent, which means that if we add one to the significand, the value of the number represented goes up by 2, not 1 (because the exponent is applied to 2, the base of this binary floating point number):
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110100 0000000000000000000000000000000000000000000000000000
= 9007199254740992 (Number.MAX_SAFE_INTEGER + 1)
Well, that's okay, because 9007199254740991 + 1 is 9007199254740992 anyway. But! We can't represent 9007199254740993. We've run out of bits. If we add just 1 to the significand, it adds 2 to the value:
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110100 0000000000000000000000000000000000000000000000000001
= 9007199254740994 (Number.MAX_SAFE_INTEGER + 3)
The format just cannot represent odd numbers anymore as we increase the value, the exponent is too big.
Eventually, we run out of significand bits again and have to increase the exponent, so we end up only being able to represent multiples of 4. Then multiples of 8. Then multiples of 16. And so on.
What you're seeing here is actually the effect of two roundings. Numbers in ECMAScript are internally represented double-precision floating-point. When id is set to 714341252076979033 (0x9e9d9958274c359 in hex), it actually is assigned the nearest representable double-precision value, which is 714341252076979072 (0x9e9d9958274c380). When you print out the value, it is being rounded to 15 significant decimal digits, which gives 14341252076979100.
It is not caused by this json parser. Just try to enter 714341252076979033 to fbug's console. You'll see the same 714341252076979100.
See this blog post for details:
http://www.exploringbinary.com/print-precision-of-floating-point-integers-varies-too
JavaScript uses double precision floating point values, ie a total precision of 53 bits, but you need
ceil(lb 714341252076979033) = 60
bits to exactly represent the value.
The nearest exactly representable number is 714341252076979072 (write the original number in binary, replace the last 7 digits with 0 and round up because the highest replaced digit was 1).
You'll get 714341252076979100 instead of this number because ToString() as described by ECMA-262, §9.8.1 works with powers of ten and in 53 bit precision all these numbers are equal.
The problem is that your number requires a greater precision than JavaScript has.
Can you send the number as a string? Separated in two parts?
JavaScript can only handle exact whole numbers up to about 9000 million million (that's 9 with 15 zeros). Higher than that and you get garbage. Work around this by using strings to hold the numbers. If you need to do math with these numbers, write your own functions or see if you can find a library for them: I suggest the former as I don't like the libraries I've seen. To get you started, see two of my functions at another answer.
See this code:
var jsonString = '{"id":714341252076979033,"type":"FUZZY"}';
var jsonParsed = JSON.parse(jsonString);
console.log(jsonString, jsonParsed);
When I see my console in Firefox 3.5, the value of jsonParsed is the number rounded:
Object id=714341252076979100 type=FUZZY
Tried different values, the same outcome (number rounded).
I also don't get its rounding rules. 714341252076979136 is rounded to 714341252076979200, whereas 714341252076979135 is rounded to 714341252076979100.
Why is this happening?
You're overflowing the capacity of JavaScript's number type, see §8.5 of the spec for details. Those IDs will need to be strings.
IEEE-754 double-precision floating point (the kind of number JavaScript uses) can't precisely represent all numbers (of course). Famously, 0.1 + 0.2 == 0.3 is false. That can affect whole numbers just like it affects fractional numbers; it starts once you get above 9,007,199,254,740,991 (Number.MAX_SAFE_INTEGER).
Beyond Number.MAX_SAFE_INTEGER + 1 (9007199254740992), the IEEE-754 floating-point format can no longer represent every consecutive integer. 9007199254740991 + 1 is 9007199254740992, but 9007199254740992 + 1 is also 9007199254740992 because 9007199254740993 cannot be represented in the format. The next that can be is 9007199254740994. Then 9007199254740995 can't be, but 9007199254740996 can.
The reason is we've run out of bits, so we no longer have a 1s bit; the lowest-order bit now represents multiples of 2. Eventually, if we keep going, we lose that bit and only work in multiples of 4. And so on.
Your values are well above that threshold, and so they get rounded to the nearest representable value.
As of ES2020, you can use BigInt for integers that are arbitrarily large, but there is no JSON representation for them. You could use strings and a reviver function:
const jsonString = '{"id":"714341252076979033","type":"FUZZY"}';
// Note it's a string −−−−^−−−−−−−−−−−−−−−−−−^
const obj = JSON.parse(jsonString, (key, value) => {
if (key === "id" && typeof value === "string" && value.match(/^\d+$/)) {
return BigInt(value);
}
return value;
});
console.log(obj);
(Look in the real console, the snippets console doesn't understand BigInt.)
If you're curious about the bits, here's what happens: An IEEE-754 binary double-precision floating-point number has a sign bit, 11 bits of exponent (which defines the overall scale of the number, as a power of 2 [because this is a binary format]), and 52 bits of significand (but the format is so clever it gets 53 bits of precision out of those 52 bits). How the exponent is used is complicated (described here), but in very vague terms, if we add one to the exponent, the value of the significand is doubled, since the exponent is used for powers of 2 (again, caveat there, it's not direct, there's cleverness in there).
So let's look at the value 9007199254740991 (aka, Number.MAX_SAFE_INTEGER):
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110011 1111111111111111111111111111111111111111111111111111
= 9007199254740991 (Number.MAX_SAFE_INTEGER)
That exponent value, 10000110011, means that every time we add one to the significand, the number represented goes up by 1 (the whole number 1, we lost the ability to represent fractional numbers much earlier).
But now that significand is full. To go past that number, we have to increase the exponent, which means that if we add one to the significand, the value of the number represented goes up by 2, not 1 (because the exponent is applied to 2, the base of this binary floating point number):
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110100 0000000000000000000000000000000000000000000000000000
= 9007199254740992 (Number.MAX_SAFE_INTEGER + 1)
Well, that's okay, because 9007199254740991 + 1 is 9007199254740992 anyway. But! We can't represent 9007199254740993. We've run out of bits. If we add just 1 to the significand, it adds 2 to the value:
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110100 0000000000000000000000000000000000000000000000000001
= 9007199254740994 (Number.MAX_SAFE_INTEGER + 3)
The format just cannot represent odd numbers anymore as we increase the value, the exponent is too big.
Eventually, we run out of significand bits again and have to increase the exponent, so we end up only being able to represent multiples of 4. Then multiples of 8. Then multiples of 16. And so on.
What you're seeing here is actually the effect of two roundings. Numbers in ECMAScript are internally represented double-precision floating-point. When id is set to 714341252076979033 (0x9e9d9958274c359 in hex), it actually is assigned the nearest representable double-precision value, which is 714341252076979072 (0x9e9d9958274c380). When you print out the value, it is being rounded to 15 significant decimal digits, which gives 14341252076979100.
It is not caused by this json parser. Just try to enter 714341252076979033 to fbug's console. You'll see the same 714341252076979100.
See this blog post for details:
http://www.exploringbinary.com/print-precision-of-floating-point-integers-varies-too
JavaScript uses double precision floating point values, ie a total precision of 53 bits, but you need
ceil(lb 714341252076979033) = 60
bits to exactly represent the value.
The nearest exactly representable number is 714341252076979072 (write the original number in binary, replace the last 7 digits with 0 and round up because the highest replaced digit was 1).
You'll get 714341252076979100 instead of this number because ToString() as described by ECMA-262, §9.8.1 works with powers of ten and in 53 bit precision all these numbers are equal.
The problem is that your number requires a greater precision than JavaScript has.
Can you send the number as a string? Separated in two parts?
JavaScript can only handle exact whole numbers up to about 9000 million million (that's 9 with 15 zeros). Higher than that and you get garbage. Work around this by using strings to hold the numbers. If you need to do math with these numbers, write your own functions or see if you can find a library for them: I suggest the former as I don't like the libraries I've seen. To get you started, see two of my functions at another answer.
See this code:
var jsonString = '{"id":714341252076979033,"type":"FUZZY"}';
var jsonParsed = JSON.parse(jsonString);
console.log(jsonString, jsonParsed);
When I see my console in Firefox 3.5, the value of jsonParsed is the number rounded:
Object id=714341252076979100 type=FUZZY
Tried different values, the same outcome (number rounded).
I also don't get its rounding rules. 714341252076979136 is rounded to 714341252076979200, whereas 714341252076979135 is rounded to 714341252076979100.
Why is this happening?
You're overflowing the capacity of JavaScript's number type, see §8.5 of the spec for details. Those IDs will need to be strings.
IEEE-754 double-precision floating point (the kind of number JavaScript uses) can't precisely represent all numbers (of course). Famously, 0.1 + 0.2 == 0.3 is false. That can affect whole numbers just like it affects fractional numbers; it starts once you get above 9,007,199,254,740,991 (Number.MAX_SAFE_INTEGER).
Beyond Number.MAX_SAFE_INTEGER + 1 (9007199254740992), the IEEE-754 floating-point format can no longer represent every consecutive integer. 9007199254740991 + 1 is 9007199254740992, but 9007199254740992 + 1 is also 9007199254740992 because 9007199254740993 cannot be represented in the format. The next that can be is 9007199254740994. Then 9007199254740995 can't be, but 9007199254740996 can.
The reason is we've run out of bits, so we no longer have a 1s bit; the lowest-order bit now represents multiples of 2. Eventually, if we keep going, we lose that bit and only work in multiples of 4. And so on.
Your values are well above that threshold, and so they get rounded to the nearest representable value.
As of ES2020, you can use BigInt for integers that are arbitrarily large, but there is no JSON representation for them. You could use strings and a reviver function:
const jsonString = '{"id":"714341252076979033","type":"FUZZY"}';
// Note it's a string −−−−^−−−−−−−−−−−−−−−−−−^
const obj = JSON.parse(jsonString, (key, value) => {
if (key === "id" && typeof value === "string" && value.match(/^\d+$/)) {
return BigInt(value);
}
return value;
});
console.log(obj);
(Look in the real console, the snippets console doesn't understand BigInt.)
If you're curious about the bits, here's what happens: An IEEE-754 binary double-precision floating-point number has a sign bit, 11 bits of exponent (which defines the overall scale of the number, as a power of 2 [because this is a binary format]), and 52 bits of significand (but the format is so clever it gets 53 bits of precision out of those 52 bits). How the exponent is used is complicated (described here), but in very vague terms, if we add one to the exponent, the value of the significand is doubled, since the exponent is used for powers of 2 (again, caveat there, it's not direct, there's cleverness in there).
So let's look at the value 9007199254740991 (aka, Number.MAX_SAFE_INTEGER):
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110011 1111111111111111111111111111111111111111111111111111
= 9007199254740991 (Number.MAX_SAFE_INTEGER)
That exponent value, 10000110011, means that every time we add one to the significand, the number represented goes up by 1 (the whole number 1, we lost the ability to represent fractional numbers much earlier).
But now that significand is full. To go past that number, we have to increase the exponent, which means that if we add one to the significand, the value of the number represented goes up by 2, not 1 (because the exponent is applied to 2, the base of this binary floating point number):
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110100 0000000000000000000000000000000000000000000000000000
= 9007199254740992 (Number.MAX_SAFE_INTEGER + 1)
Well, that's okay, because 9007199254740991 + 1 is 9007199254740992 anyway. But! We can't represent 9007199254740993. We've run out of bits. If we add just 1 to the significand, it adds 2 to the value:
+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− sign bit
/ +−−−−−−−+−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− exponent
/ / | +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−+− significand
/ / | / |
0 10000110100 0000000000000000000000000000000000000000000000000001
= 9007199254740994 (Number.MAX_SAFE_INTEGER + 3)
The format just cannot represent odd numbers anymore as we increase the value, the exponent is too big.
Eventually, we run out of significand bits again and have to increase the exponent, so we end up only being able to represent multiples of 4. Then multiples of 8. Then multiples of 16. And so on.
What you're seeing here is actually the effect of two roundings. Numbers in ECMAScript are internally represented double-precision floating-point. When id is set to 714341252076979033 (0x9e9d9958274c359 in hex), it actually is assigned the nearest representable double-precision value, which is 714341252076979072 (0x9e9d9958274c380). When you print out the value, it is being rounded to 15 significant decimal digits, which gives 14341252076979100.
It is not caused by this json parser. Just try to enter 714341252076979033 to fbug's console. You'll see the same 714341252076979100.
See this blog post for details:
http://www.exploringbinary.com/print-precision-of-floating-point-integers-varies-too
JavaScript uses double precision floating point values, ie a total precision of 53 bits, but you need
ceil(lb 714341252076979033) = 60
bits to exactly represent the value.
The nearest exactly representable number is 714341252076979072 (write the original number in binary, replace the last 7 digits with 0 and round up because the highest replaced digit was 1).
You'll get 714341252076979100 instead of this number because ToString() as described by ECMA-262, §9.8.1 works with powers of ten and in 53 bit precision all these numbers are equal.
The problem is that your number requires a greater precision than JavaScript has.
Can you send the number as a string? Separated in two parts?
JavaScript can only handle exact whole numbers up to about 9000 million million (that's 9 with 15 zeros). Higher than that and you get garbage. Work around this by using strings to hold the numbers. If you need to do math with these numbers, write your own functions or see if you can find a library for them: I suggest the former as I don't like the libraries I've seen. To get you started, see two of my functions at another answer.