IEEE 754‐1985, Binary 64, Floating‐Point Examiner

Chris W. Johnson
2022‐10‐02

Imprecision is inevitable in numeric data types, because none can represent every possible number. It is especially common in floating‐point types, like IEEE 754, which are designed to preserve magnitude by sacrificing precision. (Conversely, integer data types preserve precision by sacrificing magnitude.) Loss of precision, for whatever reason, can have important and surprising effects, especially when people are unaware of the issue. Lack of that awareness is a common problem (even among computer programmers) in part because decimal representations of these numbers are pervasively created with the intent of hiding inaccuracies.

It is hoped this tool will aid understanding of the causes and effects of those issues.

The “IEEE 754‐1985, binary 64” format is a very common floating‐point format used by Java ¹ to represent numbers in its double primitives and Double objects, and for all floating‐point calculations. By extension, any other languages executed in the Java Virtual Machine (JVM) will use it. It is also used by languages unrelated to Java and the JVM. Notably, C and C++ use it wherever it is the host CPU’s native representation (prime examples are Intel’s CPUs, so it is a very common case), JavaScript uses it for essentially all numbers, C# float and double types use it, Python uses it on most platforms, and so on.

TL;DR

Examine values 0.1 (⅒) and 0.125 (⅛). Observe the former is approximated, while the latter is exact. Why? 0.1 is not a power of two, while 0.125 is 2⁻³, and this numeric format’s values are quotients from divisions by 2⁵², multiplied by powers of two. (Hence the format’s “binary” designation.) Didn’t know all that? Don’t feel bad; most computer programmers don’t. Now, be one of the exceptions. (It’s important.)

Number: ℹ️ ️

Input Formats

General rules:

Leading and trailing whitespace is ignored.
Sign, if explicit, is first character. Supported signs: ‘+’, ‘-’ (U+002D, HYPHEN-MINUS), and ‘−’ (U+2212, MINUS SIGN).
Input letter case is irrelevant. Uppercase, lowercase, and mixed‐case are accepted.

Input Example	Interpretation
INFINITY	Infinity.
INF	Infinity.
∞	Infinity (U+221E).
NaN	Not‐a‐Number. (Result of “`0.0 / 0.0`”, “`0.0 * ∞`”, and other problematic calculations.)
?	Decimal number. Locale aware.
?	Decimal number with digit grouping. Locale aware, and grouping with ‘_’ (U+005F) works in all locales.
0x0123456789AbCdEf	Hexadecimal with “0x” prefix.² Interpreted as big-endian representation of IEEE 754-1985 binary 64 format (not an integer). 1–16 hexadecimal digits, optionally grouped with ‘_’ (U+005F). Optional “L” suffix.
10602c1299	Hexadecimal, probably. Guess based on character ‘A’–‘F’, fraction separator’s absence, and ≤ 16 digits. Interpreted as big-endian representation of IEEE 754-1985 binary 64 format (not an integer). Optional grouping with ‘_’ (U+005F). Optional “L” suffix.
04	Hexadecimal, potentially. Guess based on leading zero‘s presence, fraction separator’s absence, and ≤ 16 digits. Interpreted as big-endian representation of IEEE 754-1985 binary 64 format (not an integer). Optional grouping with ‘_’ (U+005F). Optional “L” suffix.

Browser Compatibility

Portions of this page require MathML support. As of May, 2013, Firefox and Safari had MathML support, but Chrome did not (it was in version 24, but no other versions).

Floating‐Point Format		IEEE 754‐1985, binary 64
Value, Typical Representation, Decimal
Value, Actual, Binary	Bits 63–0	$_{2}$
Value, Actual, Hexadecimal	Bits 63–0	$_{16}$
Sign	Bit 63
Exponent	Bits 62–52
Mantissa	Bits 51–0
Value Calculation		= Full‐Precision Result ⩰ (low‐precision)

Change the Bits, See What Happens…

Click on the bits to change their values. This page updates immediately.

±	Exponent											Mantissa
—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—	—

Conventions

All hexadecimal and binary representations are big‐endian.
Binary (base 2) representations are suffixed with a subscripted “2”. Their bits are identified using numbers 63 to zero, assigned from left to right.
Hexadecimal (base 16) representations are suffixed with a subscripted “16”.
Hexadecimal representations are used when shorter than decimal.

Footnotes

Java Virtual Machine versions prior to 15 use IEEE 754‐1985. Versions 15 and above use IEEE 754‐2019, which changes neither the binary format, nor its interpretation, as examined here.
In Java, for example, such values are obtained using “System.out.printf("0x%x", Double.doubleToLongBits(double));”.