How many significant digits do floats and doubles have in java

Question

Does a float have 32 binary digits and a double have 64 binary digits  The documentation was too hard to make sense of   Do all of the bits translate to significant digits  Or does the location of the decimal point take up some of the bits

User · Answer

Floating point numbers are encoded using an exponential form, that is something like m * b ^ e, i.e. not like integers at all. The question you ask would be meaningful in the context of fixed point numbers. There are numerous fixed point arithmetic libraries available.

Regarding floating point arithmetic: The number of decimal digits depends on the presentation and the number system. For example there are periodic numbers (0.33333) which do not have a finite presentation in decimal but do have one in binary and vice versa.

Also it is worth mentioning that floating point numbers up to a certain point do have a difference larger than one, i.e. value + 1 yields value, since value + 1 can not be encoded using m * b ^ e, where m, b and e are fixed in length. The same happens for values smaller than 1, i.e. all the possible code points do not have the same distance.

Because of this there is no precision of exactly n digits like with fixed point numbers, since not every number with n decimal digits does have a IEEE encoding.

There is a nearly obligatory document which you should read then which explains floating point numbers: What every computer scientist should know about floating point arithmetic.

User · Answer

A normal math answer   Understanding that a floating point number is implemented as some bits representing the exponent and the rest  most for the digits  in the binary system   one has the following situation   With a high exponent  say 10     if the least significant bit is changed  a large difference between two adjacent distinghuishable numbers appear  Furthermore the base 2 decimal point makes that many base 10 numbers can only be approximated  1 5  1 10 being endless numbers   So in general  floating point numbers should not be used if you care about significant digits  For monetary amounts with calculation  e a  best use BigDecimal   For physics floating point doubles are adequate  floats almost never  Furthermore the floating point part of processors  the FPU  can even use a bit more precission internally

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A 32-bit float has about 7 digits of precision and a 64-bit double has about 16 digits of precision Long answer  Floating-point numbers have three components   A sign bit  to determine if the number is positive or negative  An exponent  to determine the magnitude of the number  A fraction  which determines how far between two exponent values the number is  This is sometimes called    the significand  mantissa  or coefficient     Essentially  this works out to sign   2 exponent    1   fraction   The    size    of the number  it   s exponent  is irrelevant to us  because it only scales the value of the fraction portion  Knowing that log10 n  gives the number of digits of n     we can determine the precision of a floating point number with log10 largest possible fraction   Because each bit in a float stores 2 possibilities  a binary number of n bits can store a number up to 2n - 1  a total of 2n values where one of the values is zero   This gets a bit hairier  because it turns out that floating point numbers are stored with one less bit of fraction than they can use  because zeroes are represented specially and all non-zero numbers have at least one non-zero binary bit     Combining this  the digits of precision for a floating point number is log10 2n   where n is the number of bits of the floating point number   s fraction  A 32-bit float has 24 bits of fraction for   7 22 decimal digits of precision  and a 64-bit double has 53 bits of fraction for   15 95 decimal digits of precision  For more on floating point accuracy  you might want to read about the concept of a machine epsilon       For n   1 at least     for other numbers your formula will look more like  log10  n      1         This rule is variously called the leading bit convention  the implicit bit convention  or the hidden bit convention      Wikipedia

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From java specification       The floating-point types are float and double  which are conceptually   associated with the single-precision 32-bit and double-precision   64-bit format IEEE 754 values and operations as specified in IEEE   Standard for Binary Floating-Point Arithmetic  ANSI IEEE Standard   754-1985  IEEE  New York     As it s hard to do anything with numbers without understanding IEEE754 basics  here s another link    It s important to understand that the precision isn t uniform and that this isn t an exact storage of the numbers as is done for integers   An example    double a   0 3 - 0 1  System out println a               prints  0 19999999999999998   If you need arbitrary precision  for example for financial purposes  you may need Big Decimal

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Look at Float intBitsToFloat and Double longBitsToDouble  which sort of explain how bits correspond to floating-point numbers   In particular  the bits of a normal float look something like   s   2 exp   1 ABCDEFGHIJKLMNOPQRSTUVW   where A   W are 23 bits -- 0s and 1s -- representing a fraction in binary -- s is   - 1  represented by a 0 or a 1 respectively  and exp is a signed 8-bit integer

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float  32 bits  4 bytes  where 23 bits are used for the mantissa  about 7 decimal digits   8 bits are used for the exponent  so a float can    move    the decimal point to the right or to the left using those 8 bits  Doing so avoids storing lots of zeros in the mantissa as in 0 0000003  3    10-7  or 3000000  3    107   There is 1 bit used as the sign bit   double  64 bits  8 bytes  where 52 bits are used for the mantissa  about 16 decimal digits   11 bits are used for the exponent and 1 bit is the sign bit   Since we are using binary  only 0 and 1   one bit in the mantissa is implicitly 1  both float and double use this trick  when the number is non-zero   Also  since everything is in binary  mantissa and exponents  the conversions to decimal numbers are usually not exact  Numbers like 0 5  0 25  0 75  0 125 are stored exactly  but 0 1 is not  As others have said  if you need to store cents precisely  do not use float or double  use int  long  BigInteger or BigDecimal   Sources   http   en wikipedia org wiki Floating point IEEE 754  floating point in modern computers  http   en wikipedia org wiki Binary64  http   en wikipedia org wiki Binary32

[java] How many significant digits do floats and doubles have in java?

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