Mac OS X Manual Page For float(3)

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FLOAT(3) BSD Library Functions Manual FLOAT(3)

NAME
float -- description of floating-point types available on OS X

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 mantissa
tissa (called the precision), and set of available exponents (the expo-nent exponent
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 magnitude
tude to be represented in the floating-point type in which the computa-tion computation
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 magnitude
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, gradual,
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 converse
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, division,
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, particular,
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 Intel and PPC macs, the type float corresponds to IEEE-754 single
precision. A single-precision number is represented in 32 bits, and has
a precision of 24 significant bits, roughly like 7 significant decimal
digits. 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 number
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 Intel and PPC macs, the type double corresponds to IEEE-754 dou-ble double
ble precision. A double-precision number is represented in 64 bits, and
has a precision of 53 significant bits, roughly like 16 significant deci-mal decimal
mal 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. double.
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 number
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. notation.
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. double.
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.

LONG DOUBLE ON POWERPC MACS
On PowerPC macs, by default the type long double is mapped to IEEE-754
double precision, described above. If additional precision is required,
a non-IEEE-754 128 bit long double format is also available. To use this
format, compile with the -mlong-double-128 flag. If you wish to use the
<math.h> functions, you must include the linker flag -lmx as well as the
usual -lm. The -mlong-double-128 flag is only valid when the target
architecture is ppc or ppc64.

Each 128-bit long double is made up of two IEEE doubles (head and tail).
The value of the long double is the sum of the values of the two parts
(unless the head double has value -0.0, in which case the value of the
long double is -0.0). The value of the head is required to be the value
of the long double rounded to the nearest double. If the head is an
infinity, the value of the long double is the value of the head, and the
tail must be +-0.0. The tail of a NaN can be any double value. There
are many 128-bit bit patterns that are not valid as long doubles. These
do not represet any value.

The 128-bit long double format provides 106 significant bits, which is
roughly like 31 significant decimal digits. It has the same exponent
range as double, from -1022 to 1023, inclusive. The usual constants are
provided from <float.h>, as described above.

In the 128-bit long double format, addition and subtraction have a rela-tive relative
tive error bound of one ulp, or 2**-106. Multiplication has a relative
error within 2 ulps, and division a relative error within 3 ulps.