After pasting the number t=3.7333333258105216E16 in jsconsole.com or in Web Inspector, I get 37333333258105220.
parseFloat(3.7333333258105216E16) gives the same result.
What is the reason ?
JavaScript represents numbers as floats. This storage format consists of 64 bits. One bit is for the sign, 11 bits are for the power of 10 to multiply the number by, and 52 bits are for the number.
Because of the above, numbers can be acurate to the 1/2^52, or 1 / 4,503,599,627,370,496. Thus, numbers are accurate to within this fraction. Check out this wikipedia page for more information on floating point numbers.
I tested this out by trying to add one to 4,503,599,627,370,495. It gets to 4,503,599,627,370,496, but does not get past it. Here's the fiddle for testing.
You are encountering floating-point roundoff. JavaScript numbers are implemented as double precision, 64-bit floats according to the IEEE 754 standard.
You can't always accurately represent a floating point decimal number in binary. It is losing precision at the end of the number so it can fit in 64 bits.
Related
An algorithm I'm using needs to squeeze as many levels of precision as possible from a float number in Javascript. I don't mind whether the precision comes from a number that is very large or with a lot of numbers after the decimal point, I just literally need as many numerals in it as possible.
(If you care why, it is for a drag n' drop ranking algorithm which has to deal with a lot of halvings before rebalancing itself. I do also know there are better string-based algorithms but the numerical approach suits my purposes)
The MDN Docs say that:
The JavaScript Number type is a double-precision 64-bit binary format IEEE 754 value, like double in Java or C#. This means it can represent fractional values, but there are some limits to what it can store. A Number only keeps about 17 decimal places of precision; arithmetic is subject to rounding.
How should I best use the "17 decimal places of precision"?
Does the 17 decimal places mean "17 numerals in total, inclusive of those before and after the decimal place"
e.g. (adding underscores to represent thousand-separators for readability)
# 17 numerals: safe
111_222_333_444_555_66
# 17 numerals + decimal point: safe
111_222_333_444_555_6.6
1.11_222_333_444_555_66
# 18 numerals: unsafe
111_222_333_444_555_666
# 18 numerals + decimal point: unsafe
1.11_222_333_444_555_666
111_222_333_444_555_66.6
I assume that the precision of the number determines the number of numerals that you can use and that the position of the decimal point in those numerals is effectively academic.
Am I thinking about the problem correctly?
Does the presence of the decimal point have any bearing on the calculation or is it simply a matter of the number of numerals present
Should I assume that 17 numerals is safe / 18 is unsafe?
Does this vary by browser (not just today but over say, a 10 year window, should one assume that browser precision may increase)?
Short answer: you can probably squeeze out 15 "safe" digits, and it doesn't matter where you place your decimal point.
It's anyone's guess how the JavaScript standard is going to evolve and use other number representations.
Notice how the MDN doc says "about 17 decimals"? Right, it's because sometimes you can represent that many digits, and sometimes less. It's because the floating point representation doesn't map 1-to-1 to our decimal system.
Even numbers with seemingly less information will give rounding errors.
For example
0.1 + 0.2 => 0.30000000000000004
console.log(0.1 + 0.2);
However, in this case we have a lot of margin in the precision, so you can just ask for the precision you want to get rid of the rounding error
console.log((0.1 + 0.2).toPrecision(1));
For a larger illustration of this, consider the following snippet:
for(let i=0;i<22;i++) {
console.log(Number.MAX_SAFE_INTEGER / (10 ** i));
}
You will see a lot of rounding errors on digit 16. However, there would be cases where even the 16th decimal shows a rounding error. If you look here
https://en.wikipedia.org/wiki/IEEE_754
it states that binary 64 has 15.95 decimal digits. That's why I'd guess that 15 digits is the max precision you will get out of this.
You'd have to do your operations, and before you save back the number to any representational form, you'd have to do .toPrecision(15).
Finally this has some good explanations. https://floating-point-gui.de/formats/fp/
BTW, I got curious by reading this question so I read up as I wrote this answer. There are many people with better knowledge of this than me.
Does the presence of the decimal point have any bearing on the calculation or is it simply a matter of the number of numerals present
Kinda. To answer that, you'll need to look into how 64bit "double precision" floating point numbers are represented in memory. The "number of numerals" roughly translates into "length of the mantissa", which is indeed fixed and independent from the position of the point. However: it's binary digits and a binary point, not decimal digits and the decimal point. They do not correspond to each other directly. And then there's stuff like subnormal numbers.
Should I assume that 17 numerals is safe / 18 is unsafe?
No. In fact, only 15 decimal numerals would be "safe" if that's the representation you're starting with and want to exactly represent as a double.
Does this vary by browser (not just today but over say, a 10 year window, should one assume that browser precision may increase)?
No, it doesn't vary. The JavaScript number type will always be 64bit doubles.
Am I thinking about the problem correctly?
No.
You say you're considering this in the context of a drag'n'drop ranking algorithm, and you don't want do this string-based. However, thinking about decimal places in numbers is essentially thinking about string representation of numbers. Don't do that - either go all the way to strings, or treat numbers as binary.
Since you also mention "rebalancing", I assume you want to use numbers to encode the position of each item in a binary tree. That's a reasonable approach, but you really need to consider the binary representation of the number for that. And you really should use integers there, not floating-point numbers, as the logic would be much more complex otherwise. Start by deciding how many bits you want to use. There are some limitations for each, so choose wisely:
31/32 bit are what JS bitwise operators for numbers work on. Supported by all browsers easily.
53 bit are the range of integers you can exactly represent with floating-point numbers. Integer arithmetic will work as expected up to that size. Bitwise operations require extra code.
Fixed multiples of 8 (say, 64 bit) are what you can represent with typed arrays. Bitwise operations can be done part-wise, arithmetic operations require extra code. Or use a BigUint64Array that gives you 64 bits as a bigint to calculate with/operate on, but is not supported in old browsers.
Arbitrary precision can be achieved with bigint numbers, which support both bitwise and arithmetic operations, but again don't work in old browsers. Polyfills and bigint libraries are available though.
I am looking to describe how numbers are stored in javascript to a lay person. Would the following statement be accurate:
Very large numbers in javascript are often approximated.
However,precision should be guaranteed to 16 digits.
For example, 123455.373849 can always be represented accurately,
but the number 9,007,199,254,740,991,293 may not be.
Is there a better way to explain it, or any inaccuracies in the above statement?
16 digits? No, not really. Up to 53bit integers can be represented accurately and every number that can be represented as (53bit) * 2 ** (10bit).
Also, there are no 64bit integers in JavaScript, there are 64bit floating point numbers (and only 53bit of that hold the integer part), and BigInts that can have far more bits.
Very large numbers in javascript are often approximated.
Kind of, very large integers can only be approximated (or you use BigInts), however even small non integers, e.g. 0.1 can also not be represented exactly.
For example, 123455.373849 can always be represented accurately
No, probably not.
but the number 9,007,199,254,740,991,293 may not be.
Yup, thats far beyond 2 ** 53 - 1.
For (lossy) compression purposes, I'd like to be able to convert Javascript numbers into 16-bit float representation to be stored in Uint16Arrays (or Uint8Arrays, whichever's easiest.) Then I'd like to be able to convert back from the 2 bytes to a number. I don't need to perform any arithmetic on the 16-bit numbers, it's just for compact storage.
I'm having trouble finding an algorithm to do this. It doesn't need to be a IEEE standard, just something accurate to a few decimal places.
Floats are preferable to fixed point because I'd rather not have to pre-determine the range of values.
Encoding
The first step is to extract the exponent and normalized fraction of the number. In C, this is done using the frexp function which isn't available in JavaScript. Googling for frexp javascript yields a couple of implementations.
For example, here's a straight-forward implementation that extracts the bits directly from the IEEE representation using typed arrays. Here's a quick-and-dirty, inexact version using only Math functions. Here's an exact (?) version.
The second step is to create your 16-bit FP number from the obtained exponent and mantissa using bit operations. Make sure to check the exponent range and to round the lower-precision mantissa for better accuracy.
Decoding
Extract exponent and mantissa from your 16-bit FP number. Convert them to a JavaScript number either with
// The value of 'adjust' depends on the size of the mantissa.
Math.pow(2, exponent - adjust) * mantissa
or by directly creating an IEEE bit pattern with typed arrays.
Subnormal numbers, infinity, NaN
Subnormal JavaScript numbers can be simply rounded to zero. You'll have to decide whether you want to support subnormal numbers in your 16-bit format. This will complicate the conversion process in exchange for better accuracy of numbers near zero.
You'll also have to decide whether to support infinity and NaN in your format. These values could be handled similarly to the IEEE format.
I was wondering what the floating point limitations of JavaScript were. I have the following code that doesn't return a detailed floating point result as in:
2e-2 is the same as 0.02
var numero = 1E-12;
document.write(numero);
returns 1e-12.
What is the max exponential result that JavaScript can handle?
JavaScript is an implementation of ECMAScript, specified in Ecma-262 and ISO/IEC 16262. Ecma-262 specifies that IEEE 754 64-bit binary floating point is used.
In this format, the smallest positive number is 2–1074 (slightly over 4.94e–324), and the largest finite number is 21024–2971 (slightly under 1.798e308). Infinity can be represented, so, in this sense, there is no upper limit to the value of a number in JavaScript.
Numbers in this format have at most 53 bits in their significands (fraction parts). (Numbers under 2–1022 are subnormal and have fewer bits.) The limited number of bits means that many numbers are not exactly representable, including the .02 in your example. Consequently, the results of arithmetic operations are rounded to the nearest representable values, and errors in chains of calculations may cancel or may accumulate, even catastrophically.
The format also includes some special entities called NaNs (for Not a Number). NaNs may be used to indicate that a number has not been initialized, for special debugging purposes, or to represent the result of an operation for which no number is suitable (such as the square root of –1).
The maximum that is less than zero..?
There is a detailed discussion here. Basically, a 0.00000x will be displayed in exponential (5 zeroes after the decimal).
Of course, you could test this for yourself ;) particularly to see if this behaviour is reliable cross-browser.
Personally, I don't think you should be concerned, or rely, on this behaviour. When you come to display a number just format it appropriately.
JavaScript stores all numbers in double-precision floating-point format, with a 52-bit mantissa and an 11-bit exponent (the IEEE 754 Standard for storing numeric values), and therefore its Number-to-String conversions are often inaccurate. For instance,
111111111*111111111===12345678987654321
is correct, but
(111111111*111111111).toString()
returns "12345678987654320" instead of "12345678987654321". Likewise, 0.362*100 yields 36.199999999999996.
Is there a simple way to accurately convert numbers to strings?
It is NOT the number to string conversion that is inaccurate. It is the storage of the number itself in floating point as all values cannot be represented precisely in floating point and there is a limit to the number of significant digits that can be stored (about 16). You simply can't use floating point if you need perfect precision and you certainly can't be using a number that has this many significant digits. Alternatives are integers, binary coded decimals, big decimal libraries, etc...
Here's a simple demo of the problem representing certain values in floating point: http://jsfiddle.net/jfriend00/nv4MJ/
Depending upon what problem you are actually trying to solve, decimal precision issues can often be worked around by using toFixed(n) when converting to display.
Other references to read for more explanation:
How to deal with floating point number precision in JavaScript?
Why can't decimal numbers be represented exactly in binary?
What range of numbers can be represented in a 16-, 32- and 64-bit IEEE-754 systems?
Why Floating-Point Numbers May Lose Precision - MSDN - Microsoft
Is floating point math broken?
Accurate floating point arithmetic in JavaScript
https://stackoverflow.com/questions/744099/is-there-a-good-javascript-bigdecimal-library