float(3) BSD Library Functions Manual float(3)
NAME
float -- description of floating-point types available on OS X and iOS
DESCRIPTION
This page describes the available C floating-point types. For a list of
math library functions that operate on these types, see the page on the
math library, "man math".
TERMINOLOGY
Floating point numbers are represented in three parts: a sign, a mantissa
(or significand), and an exponent. Given such a representation with sign
s, mantissa m, and exponent e, the corresponding numerical value is
s*m*2**e.
Floating-point types differ in the number of bits of accuracy in the man-
tissa (called the precision), and set of available exponents (the expo-
nent range).
Floating-point numbers with the maximum available exponent are reserved
operands, denoting an infinity if the significand is precisely zero, and
a Not-a-Number, or NaN, otherwise.
Floating-point numbers with the minimum available exponent are either
zero if the significand is precisely zero, and denormal otherwise. Note
that zero is signed: +0 and -0 are distinct floating point numbers.
Floating-point numbers with exponents other than the maximum and minimum
available are called normal numbers.
PROPERTIES OF IEEE-754 FLOATING-POINT
Basic arithmetic operations in IEEE-754 floating-point are correctly
rounded: this means that the result delivered is the same as the result
that would be achieved by computing the exact real-number operation on
the operands, then rounding the real-number result to a floating-point
value.
Overflow occurs when the value of the exact result is too large in magni-
tude to be represented in the floating-point type in which the computa-
tion is being performed; doing so would require an exponent outside of
the exponent range of the type. By default, computations that result in
overflow return a signed infinity.
Underflow occurs when the value of the exact result is too small in mag-
nitude to be represented as a normal number in the floating-point type in
which the computation is being performed. By default, underflow is grad-
ual, and produces a denormal number or a zero.
All floating-points number of a given type are integer multiples of the
smallest non-zero floating-point number of that type; however, the con-
verse is not true. This means that, in the default mode, (x-y) = 0 only
if x = y.
The sign of zero transforms correctly through multiplication and divi-
sion, and is preserved by addition of zeros with like signs, but x - x
yields +0 for every finite floating-point number x. The only operations
that reveal the sign of a zero are x/(+-0) and copysign(x,+-0). In par-
ticular, comparisons (x > y, x != y, etc) are not affected by the sign of
zero.
The sign of infinity transforms correctly through multiplication and
division, and infinities are unaffected by addition or subtraction of any
finite floating-point number. But Inf-Inf, Inf*0, and Inf/Inf are, like
0/0 or sqrt(-3), invalid operations that produce NaN.
NaNs are the default results of invalid operations, and they propagate
through subsequent arithmetic operations. If x is a NaN, then x != x is
TRUE, and every other comparison predicate (x > y, x = y, x <= y, etc)
evaluates to FALSE, regardless of the value of y. Additionally, predi-
cates that entail an ordered comparison (rather than mere equality or
inequality) signal Invalid Operation when one of the arguments is NaN.
IEEE-754 provides five kinds of floating-point exceptions, listed below:
Exception Default Result
__________________________________________
Invalid Operation NaN or FALSE
Overflow +-Infinity
Divide by Zero +-Infinity
Underflow Gradual Underflow
Inexact Rounded Value
NOTE: An exception is not an error unless it is handled incorrectly.
What makes a class of exceptions exceptional is that no single default
response can be satisfactory in every instance. On the other hand,
because a default response will serve most instances of the exception
satisfactorily, simply aborting the computation cannot be justified.
For each kind of floating-point exception, IEEE-754 provides a flag that
is raised each time its exception is signaled, and remains raised until
the program resets it. Programs may test, save, and restore the flags,
or a subset thereof.
PRECISION AND EXPONENT RANGE OF SPECIFIC FLOATING-POINT TYPES
On both OS X and iOS, the type float corresponds to IEEE-754 single pre-
cision. A single-precision number is represented in 32 bits, and has a
precision of 24 significant bits, roughly like 7 significant decimal dig-
its. 8 bits are used to encode the exponent, which gives an exponent
range from -126 to 127, inclusive.
The header <float.h> defines several useful constants for the float type:
FLT_MANT_DIG - The number of binary digits in the significand of a float.
FLT_MIN_EXP - One more than the smallest exponent available in the float
type.
FLT_MAX_EXP - One more than the largest exponent available in the float
type.
FLT_DIG - the precision in decimal digits of a float. A decimal value
with this many digits, stored as a float, always yields the same value up
to this many digits when converted back to decimal notation.
FLT_MIN_10_EXP - the smallest n such that 10**n is a non-zero normal num-
ber as a float.
FLT_MAX_10_EXP - the largest n such that 10**n is finite as a float.
FLT_MIN - the smallest positive normal float.
FLT_MAX - the largest finite float.
FLT_EPSILON - the difference between 1.0 and the smallest float bigger
than 1.0.
On both OS X and iOS, the type double corresponds to IEEE-754 double pre-
cision. A double-precision number is represented in 64 bits, and has a
precision of 53 significant bits, roughly like 16 significant decimal
digits. 11 bits are used to encode the exponent, which gives an exponent
range from -1022 to 1023, inclusive.
The header <float.h> defines several useful constants for the double
type:
DBL_MANT_DIG - The number of binary digits in the significand of a dou-
ble.
DBL_MIN_EXP - One more than the smallest exponent available in the double
type.
DBL_MAX_EXP - One more than the exponent available in the double type.
DBL_DIG - the precision in decimal digits of a double. A decimal value
with this many digits, stored as a double, always yields the same value
up to this many digits when converted back to decimal notation.
DBL_MIN_10_EXP - the smallest n such that 10**n is a non-zero normal num-
ber as a double.
DBL_MAX_10_EXP - the largest n such that 10**n is finite as a double.
DBL_MIN - the smallest positive normal double.
DBL_MAX - the largest finite double.
DBL_EPSILON - the difference between 1.0 and the smallest double bigger
than 1.0.
On Intel macs, the type long double corresponds to IEEE-754 double
extended precision. A double extended number is represented in 80 bits,
and has a precision of 64 significant bits, roughly like 19 significant
decimal digits. 15 bits are used to encode the exponent, which gives an
exponent range from -16383 to 16384, inclusive.
The header <float.h> defines several useful constants for the long double
type:
LDBL_MANT_DIG - The number of binary digits in the significand of a long
double.
LDBL_MIN_EXP - One more than the smallest exponent available in the long
double type.
LDBL_MAX_EXP - One more than the exponent available in the long double
type.
LDBL_DIG - the precision in decimal digits of a long double. A decimal
value with this many digits, stored as a long double, always yields the
same value up to this many digits when converted back to decimal nota-
tion.
LDBL_MIN_10_EXP - the smallest n such that 10**n is a non-zero normal
number as a long double.
LDBL_MAX_10_EXP - the largest n such that 10**n is finite as a long dou-
ble.
LDBL_MIN - the smallest positive normal long double.
LDBL_MAX - the largest finite long double.
LDBL_EPSILON - the difference between 1.0 and the smallest long double
bigger than 1.0.
On ARM iOS devices, the type long double corresponds to IEEE-754 double
precision. Thus, the values of the LDBL_* macros are identical to those
of the corresponding DBL_* macros.
SEE ALSO
math(3), complex(3)
STANDARDS
Floating-point arithmetic conforms to the ISO/IEC 9899:2011 standard.
BSD March 28, 2007 BSD
Mac OS X 10.9.1 - Generated Tue Jan 7 19:42:11 CST 2014