Cross browser Javascript number precision

Question

Within JavaScript, numbers are defined as 64bit double-precision. I have a specific use in mind for a distributed web application, which would only work if I can rely on consistent results across all browsers.

Despite the spec using the IEEE standard, I naturally have a suspicion that there may be tiny differences in implementations of the maths library or even the underlying hardware, which could cause compound errors.

Is there any source of compatibility data, or a reliable test suite to verify double precision calculations in the browser? In particular, I also need to consider mobile browsers (usually ARM based).

Clarification -

This is a question about browser compatibility. I'm trying to understand whether all browsers can be relied upon to treat numbers in a reliable, consistent and repeatable way as defined for IEEE floating point. In most languages this is a safe assumption, but it's interesting that there's a little uncertainty about this in the browser.

There's been some great advice on how to avoid floating point problems due to lack of precision and rounding errors. In most cases, if you require accuracy you should follow this advice!

For this question, I'm not trying to avoid the problem but understand it. Floating point numbers are inherently inaccurate by design, but as long as some care is taken with how builds are made that inaccuracy can be completely predictable and consistent. IEEE-754 describes this to a level of detail that only a standards body could.

I've decided to offer a small bounty if anyone can cite,

• Genuine compatibility data relating to the implementation of IEEE numbers in mainstream browsers.
• A test suite intended to verify the implementation within the browsers, including verifying the correct internal use of a 64 bit floating point number (53 bit mantissa).

In this question I'm not looking for alternative options, workarounds or ways to avoid the problem. Thank you for the suggestions.

Answer 1

This is just for fun, as you already stated and I created a new answer because this one is in a different vein. But I still feel like there are a few random passerby's who are ignoring the futility of the problem. So let's start by addressing your points:

Firstly:

Genuine compatibility data relating to the implementation of IEEE numbers in mainstream browsers.

doesn't exist, and for that matter doesn't even make any sense, IEEE is just a standards body...? I am not sure if this vague on purpose or on accident, I will assume you were trying to say IEEE 754, but there in lies the rub... there are technically 2 versions of this standard IEEE 754-2008 AND IEEE 754-1985. Basically the former is newer and addresses the latter's oversights. Any sane person would assume that any maintained JavaScript implementation would update to the latest and greatest standard, but any sane person should know JavaScript better than that, and even if JavaScript wasn't crazy, there is no specification saying that the implementation has to be/stay up to date (check the ECMA spec yourself if you don't believe me, they don't even talk "versions"). To compound the matters further the IEEE Standard 754-2008 for Floating-Point Arithmetic supports two encoding formats: the decimal encoding format, and the binary encoding format. Which as would be expected are compatible with each other in the sense that you can go back and forth without loss of data, but that's assuming we have access to the binary representation of the number, which we don't (without attaching a debugger and looking at the store the old school way)

However, from what I can tell it seems it is general practice to "back" a JavaScript Number with an old fashioned double which of course means that we are at the mercy of the compiler used to actually build the browser. But even in that realm, we can't and shouldn't be assuming equality even if all the compilers were on the same version of the standard (they aren't) and even if all the compilers implemented the standard in its entirety (they don't). Here's an excerpt from this paper, which I have deemed an interesting, worthwhile and relevant-to-this-dialog read...

Many programmers like to believe that they can understand the behavior of a program and prove that it will work correctly without reference to the compiler that compiles it or the computer that runs it. In many ways, supporting this belief is a worthwhile goal for the designers of computer systems and programming languages. Unfortunately, when it comes to floating-point arithmetic, the goal is virtually impossible to achieve. The authors of the IEEE standards knew that, and they didn't attempt to achieve it. As a result, despite nearly universal conformance to (most of) the IEEE 754 standard throughout the computer industry, programmers of portable software must continue to cope with unpredictable floating-point arithmetic.

While finding that I also found this reference implementation done completely in JavaScript (note: I haven't actually verified the validity of the implementation).

All that said, let's move on to your second request:

A test suite intended to verify the implementation within the browsers, including verifying the correct internal use of a 64 bit floating point number (53 bit mantissa).

Since JavaScript is an interpreted platform you should see now that there is no way to test the set of script + compiler (VM/engine) + compiler that compiled the compiler + machine in an absolute and reliable way from the point of JavaScript. So unless you want to build a test suite that acts as a browser host and actually "peeks" into the private memory of the process to ensure a valid representation, which would be fruitless most likely anyway since the number are most likely "backed" by a double and that is going to conform as it does in the C or C++ that browser was built in. There is no absolute way to do this from JavaScript since all we have access to is the "object" and even when we view the Number in a console we are looking at a .toString version. For that matter I would posit that this is the only form that matters since it will be determined from the binary and would only become a point of failure if for the statement: n1 === n2 && n1.toString() !== n2.toString() you could find an n1, n2 that is relevant...

That said, we can test the string version and in reality it is just as good as testing the binary as long as we keep a few oddities in mind. Especially since nothing outside the JavaScript engine/VM ever touches the binary version. However this puts you at the mercy of an oddly specific, possibly very finicky and poised to be changed point of failure. Just for reference, here is an excerpt from webkit's JavaScriptCore's Number Prototype (NumberPrototype.cpp) displaying the complexity of the conversion:

    // The largest finite floating point number is 1.mantissa * 2^(0x7fe-0x3ff).     // Since 2^N in binary is a one bit followed by N zero bits. 1 * 2^3ff requires     // at most 1024 characters to the left of a decimal point, in base 2 (1025 if     // we include a minus sign). For the fraction, a value with an exponent of 0     // has up to 52 bits to the right of the decimal point. Each decrement of the     // exponent down to a minimum of -0x3fe adds an additional digit to the length     // of the fraction. As such the maximum fraction size is 1075 (1076 including     // a point). We pick a buffer size such that can simply place the point in the     // center of the buffer, and are guaranteed to have enough space in each direction     // fo any number of digits an IEEE number may require to represent.     typedef char RadixBuffer[2180];      // Mapping from integers 0..35 to digit identifying this value, for radix 2..36.     static const char* const radixDigits = "0123456789abcdefghijklmnopqrstuvwxyz";      static char* toStringWithRadix(RadixBuffer& buffer, double number, unsigned radix)     {         ASSERT(isfinite(number));         ASSERT(radix >= 2 && radix <= 36);          // Position the decimal point at the center of the string, set         // the startOfResultString pointer to point at the decimal point.         char* decimalPoint = buffer + sizeof(buffer) / 2;         char* startOfResultString = decimalPoint;          // Extract the sign.         bool isNegative = number < 0;         if (signbit(number))             number = -number;         double integerPart = floor(number);          // We use this to test for odd values in odd radix bases.         // Where the base is even, (e.g. 10), to determine whether a value is even we need only         // consider the least significant digit. For example, 124 in base 10 is even, because '4'         // is even. if the radix is odd, then the radix raised to an integer power is also odd.         // E.g. in base 5, 124 represents (1 * 125 + 2 * 25 + 4 * 5). Since each digit in the value         // is multiplied by an odd number, the result is even if the sum of all digits is even.         //         // For the integer portion of the result, we only need test whether the integer value is         // even or odd. For each digit of the fraction added, we should invert our idea of whether         // the number is odd if the new digit is odd.         //         // Also initialize digit to this value; for even radix values we only need track whether         // the last individual digit was odd.         bool integerPartIsOdd = integerPart <= static_cast<double>(0x1FFFFFFFFFFFFFull) && static_cast<int64_t>(integerPart) & 1;         ASSERT(integerPartIsOdd == static_cast<bool>(fmod(integerPart, 2)));         bool isOddInOddRadix = integerPartIsOdd;         uint32_t digit = integerPartIsOdd;          // Check if the value has a fractional part to convert.         double fractionPart = number - integerPart;         if (fractionPart) {             // Write the decimal point now.             *decimalPoint = '.';              // Higher precision representation of the fractional part.             Uint16WithFraction fraction(fractionPart);              bool needsRoundingUp = false;             char* endOfResultString = decimalPoint + 1;              // Calculate the delta from the current number to the next & previous possible IEEE numbers.             double nextNumber = nextafter(number, std::numeric_limits<double>::infinity());             double lastNumber = nextafter(number, -std::numeric_limits<double>::infinity());             ASSERT(isfinite(nextNumber) && !signbit(nextNumber));             ASSERT(isfinite(lastNumber) && !signbit(lastNumber));             double deltaNextDouble = nextNumber - number;             double deltaLastDouble = number - lastNumber;             ASSERT(isfinite(deltaNextDouble) && !signbit(deltaNextDouble));             ASSERT(isfinite(deltaLastDouble) && !signbit(deltaLastDouble));              // We track the delta from the current value to the next, to track how many digits of the             // fraction we need to write. For example, if the value we are converting is precisely             // 1.2345, so far we have written the digits "1.23" to a string leaving a remainder of             // 0.45, and we want to determine whether we can round off, or whether we need to keep             // appending digits ('4'). We can stop adding digits provided that then next possible             // lower IEEE value is further from 1.23 than the remainder we'd be rounding off (0.45),             // which is to say, less than 1.2255. Put another way, the delta between the prior             // possible value and this number must be more than 2x the remainder we'd be rounding off             // (or more simply half the delta between numbers must be greater than the remainder).             //             // Similarly we need track the delta to the next possible value, to dertermine whether             // to round up. In almost all cases (other than at exponent boundaries) the deltas to             // prior and subsequent values are identical, so we don't need track then separately.             if (deltaNextDouble != deltaLastDouble) {                 // Since the deltas are different track them separately. Pre-multiply by 0.5.                 Uint16WithFraction halfDeltaNext(deltaNextDouble, 1);                 Uint16WithFraction halfDeltaLast(deltaLastDouble, 1);                  while (true) {                     // examine the remainder to determine whether we should be considering rounding                     // up or down. If remainder is precisely 0.5 rounding is to even.                     int dComparePoint5 = fraction.comparePoint5();                     if (dComparePoint5 > 0 || (!dComparePoint5 && (radix & 1 ? isOddInOddRadix : digit & 1))) {                         // Check for rounding up; are we closer to the value we'd round off to than                         // the next IEEE value would be?                         if (fraction.sumGreaterThanOne(halfDeltaNext)) {                             needsRoundingUp = true;                             break;                         }                     } else {                         // Check for rounding down; are we closer to the value we'd round off to than                         // the prior IEEE value would be?                         if (fraction < halfDeltaLast)                             break;                     }                      ASSERT(endOfResultString < (buffer + sizeof(buffer) - 1));                     // Write a digit to the string.                     fraction *= radix;                     digit = fraction.floorAndSubtract();                     *endOfResultString++ = radixDigits[digit];                     // Keep track whether the portion written is currently even, if the radix is odd.                     if (digit & 1)                         isOddInOddRadix = !isOddInOddRadix;                      // Shift the fractions by radix.                     halfDeltaNext *= radix;                     halfDeltaLast *= radix;                 }             } else {                 // This code is identical to that above, except since deltaNextDouble != deltaLastDouble                 // we don't need to track these two values separately.                 Uint16WithFraction halfDelta(deltaNextDouble, 1);                  while (true) {                     int dComparePoint5 = fraction.comparePoint5();                     if (dComparePoint5 > 0 || (!dComparePoint5 && (radix & 1 ? isOddInOddRadix : digit & 1))) {                         if (fraction.sumGreaterThanOne(halfDelta)) {                             needsRoundingUp = true;                             break;                         }                     } else if (fraction < halfDelta)                         break;                      ASSERT(endOfResultString < (buffer + sizeof(buffer) - 1));                     fraction *= radix;                     digit = fraction.floorAndSubtract();                     if (digit & 1)                         isOddInOddRadix = !isOddInOddRadix;                     *endOfResultString++ = radixDigits[digit];                      halfDelta *= radix;                 }             }              // Check if the fraction needs rounding off (flag set in the loop writing digits, above).             if (needsRoundingUp) {                 // Whilst the last digit is the maximum in the current radix, remove it.                 // e.g. rounding up the last digit in "12.3999" is the same as rounding up the                 // last digit in "12.3" - both round up to "12.4".                 while (endOfResultString[-1] == radixDigits[radix - 1])                     --endOfResultString;                  // Radix digits are sequential in ascii/unicode, except for '9' and 'a'.                 // E.g. the first 'if' case handles rounding 67.89 to 67.8a in base 16.                 // The 'else if' case handles rounding of all other digits.                 if (endOfResultString[-1] == '9')                     endOfResultString[-1] = 'a';                 else if (endOfResultString[-1] != '.')                     ++endOfResultString[-1];                 else {                     // One other possibility - there may be no digits to round up in the fraction                     // (or all may be been rounded off already), in which case we may need to                     // round into the integer portion of the number. Remove the decimal point.                     --endOfResultString;                     // In order to get here there must have been a non-zero fraction, in which case                     // there must be at least one bit of the value's mantissa not in use in the                     // integer part of the number. As such, adding to the integer part should not                     // be able to lose precision.                     ASSERT((integerPart + 1) - integerPart == 1);                     ++integerPart;                 }             } else {                 // We only need to check for trailing zeros if the value does not get rounded up.                 while (endOfResultString[-1] == '0')                     --endOfResultString;             }              *endOfResultString = '\0';             ASSERT(endOfResultString < buffer + sizeof(buffer));         } else             *decimalPoint = '\0';          BigInteger units(integerPart);          // Always loop at least once, to emit at least '0'.         do {             ASSERT(buffer < startOfResultString);              // Read a single digit and write it to the front of the string.             // Divide by radix to remove one digit from the value.             digit = units.divide(radix);             *--startOfResultString = radixDigits[digit];         } while (!!units);          // If the number is negative, prepend '-'.         if (isNegative)             *--startOfResultString = '-';         ASSERT(buffer <= startOfResultString);          return startOfResultString;     } 

... as you can see, the number here is backed by a traditional double and the conversion is anything but simple and straightforward. So what I devised was this: since I conjecture that the only spot that these implementations will differ are their "rendering" to strings. I built a test generator that is three fold:

1. tests the "string result" against a reference string result
2. tests their parsed equivalents (ignoring any epsilon, I mean exact!)
3. tests a special version of the strings that solely adjusts for the rounding "interpretation"

To accomplish this we need access to a reference build, my first thought was to use one from a native language but with that I found that the numbers produced seemed to have a higher precision than JavaScript in general leading to far more errors. So then I thought, what if I just used an implementation already inside a JavaScript engine. WebKit/JavaScriptCore seemed like a really good choice but it would have also been a lot of work to get the reference build up and running so I opted for the simplicity of .NET since it has access to "jScript" while not ideal seemed upon initial examination to produce closer results than the native counterpart. I didn't really want to code in jScript since the language is all but deprecated so I opted for C# bootstrapping jScript through a CodeDomProvider.... After a little tinkering here's what it produced: http://jsbin.com/afiqil (finally demo sauce!!!!1!), so now you can run it in all browsers and compile your own data, which upon my personal inspection it seems string rounding interpretation varies in EVERY browser I tried, however I've yet to find a major browser that handled the numbers behind the scenes (other that the stringify-ing) differently...

now for the C# sauce:

    using System;     using System.Collections.Generic;     using System.ComponentModel;     using System.Data;     using System.Drawing;     using System.Linq;     using System.Text;     using System.Windows.Forms;     using System.CodeDom.Compiler;     using System.Reflection;      namespace DoubleFloatJs     {         public partial class Form1 : Form         {              private static string preamble = @"      var successes = [];     var failures = [];      function fpu_test_add(v1, v2) {         return '' + (v1 + v2);       }      function fpu_test_sub(v1, v2) {         return '' + (v1 - v2);     }      function fpu_test_mul(v1, v2) {         return '' + (v1 * v2);     }      function fpu_test_div(v1, v2) {         return '' + (v1 / v2);     }      function format(name, result1, result2, result3, received, expected) {         return '<span style=""display:inline-block;width:350px;"">' + name + '</span>' +             '<span style=""display:inline-block;width:60px;text-align:center;font-weight:bold; color:' + (result1 ? 'green;"">OK' : 'red;"">NO') + '</span>' +              '<span style=""display:inline-block;width:60px;text-align:center;font-weight:bold; color:' + (result2 ? 'green;"">OK' : 'red;"">NO') + '</span>' +              '<span style=""display:inline-block;width:60px;text-align:center;font-weight:bold; color:' + (result3 ? 'green;"">OK' : 'red;"">NO') + '</span>' +              '<span style=""display:inline-block;width:200px;vertical-align:top;"">' + received + '<br />' + expected + '</span>';     }      function check_ignore_round(received, expected) {         return received.length > 8 &&             received.length == expected.length &&              received.substr(0, received.length - 1) === expected.substr(0, expected.length - 1);     }      function check_parse_parity_no_epsilon(received, expected) {         return parseFloat(received) === parseFloat(expected);     }      function fpu_test_result(v1, v2, textFn, received, expected) {         var result = expected === received,             resultNoRound = check_ignore_round(received, expected),             resultParse = check_parse_parity_no_epsilon(received, expected),             resDiv = document.createElement('div');          resDiv.style.whiteSpace = 'nowrap';         resDiv.style.fontFamily = 'Courier New, Courier, monospace';         resDiv.style.fontSize = '0.74em';         resDiv.style.background = result ? '#aaffaa' : '#ffaaaa';         resDiv.style.borderBottom = 'solid 1px #696969';         resDiv.style.padding = '2px';          resDiv.innerHTML = format(textFn + '(' + v1 + ', ' + v2 + ')', result, resultNoRound, resultParse, received, expected);          document.body.appendChild(resDiv);         (result ? successes : failures).push(resDiv);         return resDiv;     }      function fpu_test_run(v1, v2, addRes, subRes, mulRes, divRes) {         var i, res,              fnLst = [fpu_test_add, fpu_test_sub, fpu_test_mul, fpu_test_div],             fnNam = ['add', 'sub', 'mul', 'div'];          for (i = 0; i < fnLst.length; i++) {             res = fnLst[i].call(null, v1, v2);             fpu_test_result(v1, v2, fnNam[i], res, arguments[i + 2]);         }     }      function setDisplay(s, f) {         var i;         for (i = 0; i < successes.length; i++) {             successes[i].style.display = s;         }         for (i = 0; i < failures.length; i++) {             failures[i].style.display = f;         }     }      var test_header = fpu_test_result('value1', 'value2', 'func', 'received', 'expected'),         test_header_cols = test_header.getElementsByTagName('span');      test_header_cols[1].innerHTML = 'string';     test_header_cols[2].innerHTML = 'rounded';     test_header_cols[3].innerHTML = 'parsed';     test_header.style.background = '#aaaaff';      failures.length = successes.length = 0;      ";              private static string summation = @"      var bs = document.createElement('button');     var bf = document.createElement('button');     var ba = document.createElement('button');      bs.innerHTML = 'show successes (' + successes.length + ')';     bf.innerHTML = 'show failures (' + failures.length + ')';     ba.innerHTML = 'show all (' + (successes.length + failures.length) + ')';      ba.style.width = bs.style.width = bf.style.width = '200px';     ba.style.margin = bs.style.margin = bf.style.margin = '4px';     ba.style.padding = bs.style.padding = bf.style.padding = '4px';      bs.onclick = function() { setDisplay('block', 'none'); };     bf.onclick = function() { setDisplay('none', 'block'); };     ba.onclick = function() { setDisplay('block', 'block'); };      document.body.insertBefore(bs, test_header);     document.body.insertBefore(bf, test_header);     document.body.insertBefore(ba, test_header);     document.body.style.minWidth = '700px';      ";              private void buttonGenerate_Click(object sender, EventArgs e)             {                 var numberOfTests = this.numericNumOfTests.Value;                 var strb = new StringBuilder(preamble);                 var rand = new Random();                  for (int i = 0; i < numberOfTests; i++)                 {                     double v1 = rand.NextDouble();                     double v2 = rand.NextDouble();                      strb.Append("fpu_test_run(")                         .Append(v1)                         .Append(", ")                         .Append(v2)                         .Append(", '")                         .Append(JsEval("" + v1 + '+' + v2))                         .Append("', '")                         .Append(JsEval("" + v1 + '-' + v2))                         .Append("', '")                         .Append(JsEval("" + v1 + '*' + v2))                         .Append("', '")                         .Append(JsEval("" + v1 + '/' + v2))                         .Append("');")                         .AppendLine();                 }                  strb.Append(summation);                  this.textboxOutput.Text = strb.ToString();                 Clipboard.SetText(this.textboxOutput.Text);             }              public Form1()             {                 InitializeComponent();                  Type evalType = CodeDomProvider                     .CreateProvider("JScript")                     .CompileAssemblyFromSource(new CompilerParameters(), "package e{class v{public static function e(e:String):String{return eval(e);}}}")                     .CompiledAssembly                     .GetType("e.v");                  this.JsEval = s => (string)evalType.GetMethod("e").Invoke(null, new[] { s });             }              private readonly Func<string, string> JsEval;          }     } 

or a pre-compiled version if you should choose: http://uploading.com/files/ad4a85md/DoubleFloatJs.exe/ this is an executable, download at your own risk

I should mention that the purpose of the program is just to produce a JavaScript file in a text box and copy it to the clipboard for convenience for pasting wherever you choose, you could easily turn this around and put it on an asp.net server and add reporting to results to ping the server and keep track in some massive database... which is what I would do to it if I desired the information..

...and, ...I'm, ...spent I hope this helps you -ck

Answer 2

Summarizing everything below, you can expect compliance on the majority of systems save for a few IE's glitches, but ought to use a sanity check as a precaution (proposition is included).

• ECMAScript specification (both rev. 3 and 5) is very precise on IEEE 754's peculiarities such as conversions, rounding mode, overflow/underflow and even signed zeros (sec. 5.2, 8.5, 9.*, 11.5.*, 11.8.5; 15.7 and 15.8 deal with floats too). It doesn't just "leave things to the underlying implementation". There are no apparent differences between vv. 3 and 5 and all major browsers support v.3 at least. So its rules are honored by everyone, at least nominally. Let's see...

  • No browser passes test262 ECMAScript compliance tests completely (WP#Conformance tests). However, no test262 errors found on google are float-related.

  • IE5.5+ ([MS-ES3]) reports discrepancies in Number.toFixed, Number.toExponential, Number.toPrecision. Other differences aren't float-related. I couldn't run test262 in IE8;

  • FF uses special types for number and mandatory conversion fns between them and C types (see e.g. JSAPI user guide) which suggests that things are in fact not handed down to C. Test on FF10 didn't show any float-related errors;
  • Opera users did report float-related errors half a year ago;
  • Webkit's test262 failures don't appear to be float-related.

To validate a system, you can use float-related tests from test262. They are located at http://test262.ecmascript.org/json/ch<2-digit # of spec chapter>.json; test code can be extracted with (python 2.6+):

ch="05";  #substitute chapter # import urllib,json,base64 j=json.load(urllib.urlopen("http://test262.ecmascript.org/json/ch%s.json"%ch)) tt=j['testsCollection']['tests'] f=open('ch%s.js'%ch,'w') for t in tt:   print >>f   print >>f,base64.b64decode(t['code']) f.close() 

Another opportunity is IEEE 754 compliance tests in C.

Relevant sections from test262 (ones that compare floating point numbers) are as follows:

{ "S11": "5.1.A4: T1-T8", "S15": {     "7": "3: 2.A1 & 3.A1",     "8": {         "1": "1-8: A1",         "2": {             "4": "A4 & A5",             "5": "A: 3,6,7,10-13,15,17-19",             "7": "A6 & A7",             "13": "A24",             "16": "A6 & A7",             "17": "A6",             "18": "A6"         }     } }, "S8": "5.A2: 1 & 2" } 

this list and the concatenated source of all the relevant test files (as of 3/9/2012, no files from the harness) can be found here: http://pastebin.com/U6nX6sKL