The double value type represents a double-precision 64-bit number with values ranging from negative 1.79769313486232e308 to positive 1.79769313486232e308, as well as positive or negative zero, double.PositiveInfinity, double.NegativeInfinity, and not a number (double.NaN). It is intended to represent values that are extremely large (such as distances between planets or galaxies) or extremely small (the molecular mass of a substance in kilograms) and that often are imprecise (such as the distance from earth to another solar system), The double type complies with the IEC 60559:1989 (IEEE 754) standard for binary floating-point arithmetic.

This topic consists of the following sections:

• Floating-Point Representation and Precision

• Testing for Equality

• Floating-Point Values and Exceptions

• Floating-Point Functionality

Floating-Point Representation and Precision

The double data type stores double-precision floating-point values in a 64-bit binary format, as shown in the following table:

Significand or mantissa	0-51
Exponent	52-62
Sign (0 = Positive, 1 = Negative)	63

Just as decimal fractions are unable to precisely represent some fractional values (such as 1/3 or Math.PI), binary fractions are unable to represent some fractional values. For example, 1/10, which is represented precisely by .1 as a decimal fraction, is represented by .001100110011 as a binary fraction, with the pattern "0011" repeating to infinity. In this case, the floating-point value provides an imprecise representation of the number that it represents. Performing additional mathematical operations on the original floating-point value often tends to increase its lack of precision. For example, if we compare the result of multiplying .1 by 10 and adding .1 to .1 nine times, we see that addition, because it has involved eight more operations, has produced the less precise result. Note that this disparity is apparent only if we display the two double values by using the "R" standard numeric format string, which if necessary displays all 17 digits of precision supported by the double type.

code reference: System.Double.Structure#3

Because some numbers cannot be represented exactly as fractional binary values, floating-point numbers can only approximate real numbers.

All floating-point numbers also have a limited number of significant digits, which also determines how accurately a floating-point value approximates a real number. A double value has up to 15 decimal digits of precision, although a maximum of 17 digits is maintained internally. This means that some floating-point operations may lack the precision to change a floating point value. The following example provides an illustration. It defines a very large floating-point value, and then adds the product of double.Epsilon and one quadrillion to it. The product, however, is too small to modify the original floating-point value. Its least significant digit is thousandths, whereas the most significant digit in the product is 1.

code reference: System.Double.Structure#4

The limited precision of a floating-point number has several consequences:

• Two floating-point numbers that appear equal for a particular precision might not compare equal because their least significant digits are different. In the following example, a series of numbers are added together, and their total is compared with their expected total. Although the two values appear to be the same, a call to the Equals method indicates that they are not.

  code reference: System.Double.Structure#6

  If you change the format items in the Console.WriteLine(string, object, object) statement from {0} and {1} to {0:R} and {1:R} to display all significant digits of the two double values, it is clear that the two values are unequal because of a loss of precision during the addition operations. In this case, the issue can be resolved by calling the Math.Round(double, int) method to round the double values to the desired precision before performing the comparison.

• A mathematical or comparison operation that uses a floating-point number might not yield the same result if a decimal number is used, because the binary floating-point number might not equal the decimal number. A previous example illustrated this by displaying the result of multiplying .1 by 10 and adding .1 times.

  When accuracy in numeric operations with fractional values is important, you can use the decimal rather than the double type. When accuracy in numeric operations with integral values beyond the range of the long or ulong types is important, use the System.Numerics.BigInteger type.

• A value might not round-trip if a floating-point number is involved. A value is said to round-trip if an operation converts an original floating-point number to another form, an inverse operation transforms the converted form back to a floating-point number, and the final floating-point number is not equal to the original floating-point number. The roundtrip might fail because one or more least significant digits are lost or changed in a conversion. In the following example, three double values are converted to strings and saved in a file. As the output shows, however, even though the values appear to be identical, the restored values are not equal to the original values.

  code reference: System.Double.Structure#7

  In this case, the values can be successfully round-tripped by using the "R" standard numeric format string to preserve the full precision of double values, as the following example shows.

  code reference: System.Double.Structure#8

• float values have less precision than double values. A float value that is converted to a seemingly equivalent double often does not equal the double value because of differences in precision. In the following example, the result of identical division operations is assigned to a double and a float value. After the float value is cast to a double, a comparison of the two values shows that they are unequal.

  code reference: System.Double.Structure#5

  To avoid this problem, use either the double in place of the float data type, or use the Math.Round(double) method so that both values have the same precision.

In addition, the result of arithmetic and assignment operations with double values may differ slightly by platform because of the loss of precision of the double type. For example, the result of assigning a literal double value may differ in the 32-bit and 64-bit versions of the .NET Framework. The following example illustrates this difference when the literal value -4.42330604244772E-305 and a variable whose value is -4.42330604244772E-305 are assigned to a double variable. Note that the result of the double.Parse(string) method in this case does not suffer from a loss of precision.

code reference: System.Double.Class.Precision#1

Testing for Equality

To be considered equal, two double values must represent identical values. However, because of differences in precision between values, or because of a loss of precision by one or both values, floating-point values that are expected to be identical often turn out to be unequal because of differences in their least significant digits. As a result, calls to the double.Equals(double) method to determine whether two values are equal, or calls to the double.CompareTo(double) method to determine the relationship between two double values, often yield unexpected results. This is evident in the following example, where two apparently equal double values turn out to be unequal because the first has 15 digits of precision, while the second has 17.

code reference: System.Double.Structure#9

Calculated values that follow different code paths and that are manipulated in different ways often prove to be unequal. In the following example, one double value is squared, and then the square root is calculated to restore the original value. A second double is multiplied by 3.51 and squared before the square root of the result is divided by 3.51 to restore the original value. Although the two values appear to be identical, a call to the double.Equals(double) method indicates that they are not equal. Using the "R" standard format string to return a result string that displays all the significant digits of each Double value shows that the second value is .0000000000001 less than the first.

code reference: System.Double.Structure#10

In cases where a loss of precision is likely to affect the result of a comparison, you can adopt any of the following alternatives to calling the double.Equals(double) or double.CompareTo(double) method:

• Call the Math.Round(double) method to ensure that both values have the same precision. The following example modifies a previous example to use this approach so that two fractional values are equivalent.

  code reference: System.Double.Structure#11

  Note, though, that the problem of precision still applies to rounding of midpoint values. For more information, see the Math.Round(double, int, MidpointRounding) method.

• Test for approximate equality rather than equality. This requires that you define either an absolute amount by which the two values can differ but still be equal, or that you define a relative amount by which the smaller value can diverge from the larger value.

  The following example uses the latter approach to define an IsApproximatelyEqual method that tests the relative difference between two values. It also contrasts the result of calls to the IsApproximatelyEqual method and the double.Equals(double) method.

  code reference: System.Double.Structure#12

Floating-Point Values and Exceptions

Unlike operations with integral types, which throw exceptions in cases of overflow or illegal operations such as division by zero, operations with floating-point values do not throw exceptions. Instead, in exceptional situations, the result of a floating-point operation is zero, positive infinity, negative infinity, or not a number (NaN):

• If the result of a floating-point operation is too small for the destination format, the result is zero. This can occur when two very small numbers are multiplied, as the following example shows.

  code reference: System.Double.Structure#1

• If the magnitude of the result of a floating-point operation exceeds the range of the destination format, the result of the operation is double.PositiveInfinity or double.NegativeInfinity, as appropriate for the sign of the result. The result of an operation that overflows double.MaxValue is double.PositiveInfinity, and the result of an operation that overflows double.MinValue is double.NegativeInfinity, as the following example shows.

  code reference: System.Double.Structure#2

  double.PositiveInfinity also results from a division by zero with a positive dividend, and double.NegativeInfinity results from a division by zero with a negative dividend.

• If a floating-point operation is invalid, the result of the operation is double.NaN. For example, double.NaN results from the following operations:

• Any floating-point operation with an invalid input. For example, calling the Math.Sqrt(double) method with a negative value returns double.NaN, as does calling the Math.Acos(double) method with a value that is greater than one or less than negative one.

• Any operation with an argument whose value is double.NaN.

Floating-Point Functionality

The double structure and related types provide methods to perform operations in the following areas:

• Comparison of values. You can call the double.Equals(double) method to determine whether two double values are equal, or the double.CompareTo(double) method to determine the relationship between two values.

  The double structure also supports a complete set of comparison operators. For example, you can test for equality or inequality, or determine whether one value is greater than or equal to another. If one of the operands is a numeric type other than a double, it is converted to a double before performing the comparison.

  You can also call the double.IsNaN(double), double.IsInfinity(double), double.IsPositiveInfinity(double), and double.IsNegativeInfinity(double) methods to test for these special values.

• Mathematical operations. Common arithmetic operations, such as addition, subtraction, multiplication, and division, are implemented by language compilers and Common Intermediate Language (CIL) instructions, rather than by double methods. If one of the operands in a mathematical operation is a numeric type other than a double, it is converted to a double before performing the operation. The result of the operation is also a double value.

  Other mathematical operations can be performed by calling static (Shared in Visual Basic) methods in the Math class. It includes additional methods commonly used for arithmetic (such as Math.Abs(double), Math.Sign(double), and Math.Sqrt(double)), geometry (such as Math.Cos(double) and Math.Sin(double)), and calculus (such as Math.Log(double)).

  You can also manipulate the individual bits in a double value. The BitConverter.DoubleToInt64Bits(double) method preserves a double value's bit pattern in a 64-bit integer. The BitConverter.GetBytes(double) method returns its bit pattern in a byte array.

• Rounding. Rounding is often used as a technique for reducing the impact of differences between values caused by problems of floating-point representation and precision. You can round a double value by calling the Math.Round(double) method.

• Formatting. You can convert a double value to its string representation by calling the double.ToString method or by using the composite formatting feature. For information about how format strings control the string representation of floating-point values, see the Standard Numeric Format Strings and Custom Numeric Format Strings topics.

• Parsing strings. You can convert the string representation of a floating-point value to a double value by calling either the double.Parse(string) or double.TryParse(string, Double@) method. If the parse operation fails, the double.Parse(string) method throws an exception, whereas the double.TryParse(string, Double@) method returns false.

• Type conversion. The double structure provides an explicit interface implementation for the IConvertible interface, which supports conversion between any two standard .NET Framework data types. Language compilers also support the implicit conversion of values of all other standard numeric types to double values. Conversion of a value of any standard numeric type to a double is a widening conversion and does not require the user of a casting operator or conversion method,

  However, conversion of long and float values can involve a loss of precision. The following table lists the differences in precision for each of these types:

  The problem of precision most frequently affects float values that are converted to double values. In the following example, two values produced by identical division operations are unequal because one of the values is a a single-precision floating point value converted to a double.

  code reference: System.Double.Structure#3

This type is safe for multithreaded operations.

Namespace: System
Assembly: mscorlib (in mscorlib.dll)
Assembly Versions: 1.0.5000.0, 2.0.0.0, 4.0.0.0