MATH(3M) MATH(3M)
NAME
math - introduction to mathematical library functions
DESCRIPTION
These functions constitute the C math library libm. There are four
versions of the math library libm.a, libmx.a, libm43.a and libfastm.a
The first, libm.a, contains routines newly implemented (1994) using
algorithms which take advantage of the Mips architecture and includes
many routines for the float data type.
For the -64 and -n32 versions of libm.a, a second version of the math
library, libmx.a, contains functions which give identical results to
those in libm.a, but which use System V error handling.
See matherr(3M) for a description of error handling for libmx.a
functions.
The third version of the math library, libm43.a, contains routines all
based on the original codes in the 4.3BSD release. The difference
between the error bounds for libm.a and libm43.a is typically around 1
unit in the last place, whereas the performance difference may be a
factor of two or more.
The link editor searches this library under the "-lm", "-lmx", or "-lm43"
option. Declarations for these functions may be obtained from the
include file <math.h>.
The fourth library, libfastm.a, contains faster, lower-precision versions
of various routines from libm.a.
LIST OF FUNCTIONS
Error bounds listed below apply only to the -64 and -n32 versions of
libm.a and libmx.a The error bound sometimes applies only to the primary
range.
Error Bound (ULPs)
Name Appears on Page Description libm.a libm43.a
acos sin(3M) inverse trigonometric function 2 3
acosf sin(3M) inverse trigonometric function 1
acosh asinh(3M) inverse hyperbolic function 3 3
asin sin(3M) inverse trigonometric function 2 3
asinf sin(3M) inverse trigonometric function 1
asinh asinh(3M) inverse hyperbolic function 3 3
atan sin(3M) inverse trigonometric function 1.5 1
atanf sin(3M) inverse trigonometric function 1
atanh asinh(3M) inverse hyperbolic function 3 3
atan2 sin(3M) inverse trigonometric function 2 2
atan2f sin(3M) inverse trigonometric function 1
cabs hypot(3M) complex absolute value 1 1
cabsf hypot(3M) complex absolute value 1
cbrt sqrt(3M) cube root 1 1
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ceil floor(3M) integer no less than 0 0
ceilf floor(3M) integer no less than 0 0
copysign ieee(3M) copy sign bit 0 0
cos sin(3M) trigonometric function 2 1
cosf sin(3M) trigonometric function 1
cosh sinh(3M) hyperbolic function 2 3
coshf sinh(3M) hyperbolic function 1
drem ieee(3M) remainder 0 0
erf erf(3M) error function ? ?
erfc erf(3M) complementary error function ? ?
exp exp(3M) exponential 1 1
expf exp(3M) exponential 1
expm1 exp(3M) exp(x)-1 1 1
expm1f exp(3M) exp(x)-1 1
fabs floor(3M) absolute value 0 0
fabsf floor(3M) absolute value 0 0
finite ieee(3M) floating point arithmetic (N/A)
floor floor(3M) integer no greater than 0 0
floorf floor(3M) integer no greater than 0 0
fmod floor(3M) remainder function 0
fmodf floor(3M) remainder function 0
hypot hypot(3M) Euclidean distance 1 1
hypotf hypot(3M) Euclidean distance 1 1
j0 j0(3M) bessel function ? ?
j1 j0(3M) bessel function ? ?
jn j0(3M) bessel function ? ?
lgamma lgamma(3M) log gamma function ? ?
log exp(3M) natural logarithm 1 1
logf exp(3M) natural logarithm 1
logb ieee(3M) exponent extraction 0 0
log10 exp(3M) logarithm to base 10 2 3
log10f exp(3M) logarithm to base 10 1.5
log1p exp(3M) log(1+x) 1 1
log1pf exp(3M) log(1+x) 1 1
pow exp(3M) exponential x**y 2 60-500
powf exp(3M) exponential x**y 1
rint floor(3M) round to nearest integer 0 0
sin sin(3M) trigonometric function 2 1
sinf sin(3M) trigonometric function 1
sinh sinh(3M) hyperbolic function 2 3
sinhf sinh(3M) hyperbolic function 1
sqrt sqrt(3M) square root 1 1
sqrtf sqrt(3M) square root 1
tan sin(3M) trigonometric function 2 3
tanf sin(3M) trigonometric function 1
tanh sinh(3M) hyperbolic function 2 3
tanhf sinh(3M) hyperbolic function 1
trunc floor(3M) truncate to whole number 0 0
truncf floor(3M) truncate to whole number 0 0
y0 j0(3M) bessel function ? ?
y1 j0(3M) bessel function ? ?
yn j0(3M) bessel function ? ?
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VECTOR INTRINSICS
Beginning with IRIX 6.2, libm now supports the following vector
intrinsics:
/* single precision vector routines */
vacosf( float *x, float *y, long count, long stridex, long stridey )
vasinf( float *x, float *y, long count, long stridex, long stridey )
vatanf( float *x, float *y, long count, long stridex, long stridey )
vcosf( float *x, float *y, long count, long stridex, long stridey )
vexpf( float *x, float *y, long count, long stridex, long stridey )
vlogf( float *x, float *y, long count, long stridex, long stridey )
vlog10f( float *x, float *y, long count, long stridex, long stridey )
vsinf( float *x, float *y, long count, long stridex, long stridey )
vsqrtf( float *x, float *y, long count, long stridex, long stridey )
vtanf( float *x, float *y, long count, long stridex, long stridey )
/* double precision vector routines */
vacos( double *x, double *y, long count, long stridex, long stridey )
vasin( double *x, double *y, long count, long stridex, long stridey )
vatan( double *x, double *y, long count, long stridex, long stridey )
vcos( double *x, double *y, long count, long stridex, long stridey )
vexp( double *x, double *y, long count, long stridex, long stridey )
vlog( double *x, double *y, long count, long stridex, long stridey )
vlog10( double *x, double *y, long count, long stridex, long stridey )
vsin( double *x, double *y, long count, long stridex, long stridey )
vsqrt( double *x, double *y, long count, long stridex, long stridey )
vtan( double *x, double *y, long count, long stridex, long stridey )
Input and output arrays for the above routines should either be identical
or non-overlapping.
On Mips4 processors, these routines are software pipelined to take
advantage of the multiple execution units. On that machine, throughput
is up to several times greater than one gets by calling the scalar
intrinsics repeatedly. On processors other than the Mips4, these
routines are still available; although not software pipelined on those
processors, they still eliminate considerable call overhead when they can
be used. Note that the vector routines do not support denormals on the
Mips4 processors.
The single precision vector routines can also be called by the names
vfacos, vfasin, etc.
Semantics of these routines:
i=0, 1, ..., count-1: y[i*stridey] = f(x[i*stridex])
Example:
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double x[10000], y[10000];
for (i=0; i<1000; i++ ) y[2*i] = sin(x[3*i]);
Transform (by hand) into
vsin(x, y, 1000, 3, 2);
Vector and scalar routines may differ slightly, however none of the
results differ from the mathematically correct result by more than 2 ulps
(units in the last place). Note that the vector square root routines are
less accurate than the hardware versions; vsqrt and vsqrtf use the
reciprocal square root instruction and lose up to about 2 bits of
accuracy. vsqrt and vfsqrt give correct answers for zero and infinite
arguments.
LONG DOUBLE ARITHMETIC
Long double arithmetic is supported by the MIPSpro compiler. The
representation used is not IEEE compliant; long doubles are represented
on this system as the sum or difference of two doubles, normalized so
that the smaller double is <= .5 ulp of the larger. This is equivalent
to a 107 bit mantissa with an 11 bit biased exponent (bias = 1023), and 1
sign bit. In terms of decimal precision, this is approximately 34
decimal digits.
Long double constants are coded as double precision constants followed by
the letter 'l' (upper or lower case). The largest (finite) long double
constant is 1.797693134862315807937289714053023e308L .
The smallest long double precision constant is
4.940656458412465441765687928682213e-324L . Long doubles less than
1.805194375864829576069262081173746e-276L
may require a double denormal in their representation and therefore
contain less than 107 bits precision.
Long double NaNs and (signed) infinities are supported by the MIPSpro
compiler. Long double infinity is represented as the sum of a double
infinity and a double zero; similarly for NaNs.
In Fortran, long doubles are denoted by the term REAL *16.
In general, long double arithmetic operations (+, -, *, /) are not
precisely rounded, but are accurate to approximately 3 ulps.
Note that long double arithmetic operations are done in software by
MIPSpro compilers; results of these operations may vary slightly from
release to release due to improvements in the algorithms which implement
them.
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Long double operations on this system are only supported in round to
nearest rounding mode (the default). The system must be in round to
nearest rounding mode when issuing long double arithmetic operations or
calling any of the long double functions, or incorrect answers will
result.
DIFFERENCES BETWEEN -o32, -n32, -64
For the IRIX 6.2 release, faster and more accurate algorithms were
implemented, and vector functions were added to the math library. In
order to maintain numerical compatibility with older releases, these
changes were made only in the -n32 and -64 versions of the library and
not in the -o32 version. ( Where there are differences in accuracy, this
document describes the behavior of the -n32 and -64 versions of the
library. )
To take advantage of the new functions and algorithms, you need to
compile and link using either the -n32 or the -64 option.
Note however, that the -o32 version of libmx contains all routines
present in the -n32 and -64 versions of libmx except the quad precision
and vector routines, and gives results identical to the -n32 and -64
versions.
NOTES
Users concerned with portability to other computer systems should note
that the long double and float versions of these functions are optional
according to the ANSI C Programming Language Specification ISO/IEC 9899 :
1990 (E).
Long double functions have been renamed to be compliant with the ANSI-C
standard, however to be backward compatible, they may still be called
with the double precision function name prefixed with a q. (Exceptions:
functions fabsl and fmodl may be called with names qabs and qmod, resp.)
In 4.3BSD, distributed from the University of California in late 1985,
most of the foregoing functions come in two versions, one for the
double-precision "D" format in the DEC VAX-11 family of computers,
another for double-precision arithmetic conforming to the IEEE Standard
754 for Binary Floating-point Arithmetic. The two versions behave very
similarly, as should be expected from programs more accurate and robust
than was the norm when UNIX was born. For instance, the programs are
accurate to within the numbers of ulps tabulated above; an ulp is one
Unit in the Last Place. And the programs have been cured of anomalies
that afflicted the older math library libm in which incidents like the
following had been reported:
sqrt(-1.0) = 0.0 and log(-1.0) = -1.7e38.
cos(1.0e-11) > cos(0.0) > 1.0.
pow(x,1.0) != x when x = 2.0, 3.0, 4.0, ..., 9.0.
pow(-1.0,1.0e10) trapped on Integer Overflow.
sqrt(1.0e30) and sqrt(1.0e-30) were very slow.
This machine conforms to the IEEE Standard 754 for Binary Floating-point
Arithmetic, to which only the notes for IEEE floating-point apply and are
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included here.
(See however, the notes regarding long double precision below.)
IEEE STANDARD 754 Floating-point Arithmetic:
This standard is on its way to becoming more widely adopted than any
other design for computer arithmetic.
Properties of IEEE 754 Double-precision:
Wordsize: 64 bits, 8 bytes. Radix: Binary.
Precision: 53 sig. bits, roughly 16 sig. decimals.
If x and x' are consecutive positive Double-precision numbers
(they differ by 1 ulp), then
1.1e-16 < 0.5**53 < (x'-x)/x < 0.5**52 < 2.3e-16.
Range: Overflow threshold = 2.0**1024 = 1.8e308
Underflow threshold = 0.5**1022 = 2.2e-308
Overflow goes by default to a signed Infinity.
Underflow is Gradual, rounding to the nearest integer multiple
of 0.5**1074 = 4.9e-324.
Zero is represented ambiguously as +0 or -0.
Its sign transforms correctly through multiplication or
division, and is preserved by addition of zeros with like
signs; but x-x yields +0 for every finite x. The only
operations that reveal zero's sign are division by zero and
copysign(x,+0). In particular, comparison (x > y, x > y, etc.)
cannot be affected by the sign of zero; but if finite x = y
then Infinity = 1/(x-y) != -1/(y-x) = -Infinity.
Infinity is signed.
it persists when added to itself or to any finite number. Its
sign transforms correctly through multiplication and division,
and (finite)/+Infinity = +0 (nonzero)/0 = +Infinity. But
Infinity-Infinity, Infinity*0 and Infinity/Infinity are, like
0/0 and sqrt(-3), invalid operations that produce NaN.
Reserved operands:
there are 2**53-2 of them, all called NaN (Not a Number).
Some, called Signaling NaNs, trap any floating-point operation
performed upon them; they could be used to mark missing or
uninitialized values, or nonexistent elements of arrays. The
rest are Quiet NaNs; they are the default results of Invalid
Operations, and propagate through subsequent arithmetic
operations. If x != x then x is NaN; every other predicate (x
> y, x = y, x < y, ...) is FALSE if NaN is involved.
NOTE: Trichotomy is violated by NaN.
Besides being FALSE, predicates that entail ordered
comparison, rather than mere (in)equality, signal Invalid
Operation when NaN is involved.
Rounding:
Every algebraic operation (+, -, *, /, sqrt) is rounded by
default to within half an ulp, and when the rounding error is
exactly half an ulp then the rounded value's least significant
bit is zero. This kind of rounding is usually the best kind,
sometimes provably so; for instance, for every x = 1.0, 2.0,
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3.0, 4.0, ..., 2.0**52, we find (x/3.0)*3.0 == x and
(x/10.0)*10.0 == x and ... despite that both the quotients and
the products have been rounded. Only rounding like IEEE 754
can do that. But no single kind of rounding can be proved best
for every circumstance, so IEEE 754 provides rounding towards
zero or towards +Infinity or towards -Infinity at the
programmer's option.
Exceptions:
IEEE 754 recognizes five kinds of floating-point exceptions,
listed below in declining order of probable importance.
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 handled badly. What
makes a class of exceptions exceptional is that no single
default response can be satisfactory in every instance. On the
other hand, if a default response will serve most instances
satisfactorily, the unsatisfactory instances cannot justify
aborting computation every time the exception occurs.
For each kind of floating-point exception, IEEE 754 provides a Flag
that is raised each time its exception is signaled, and stays raised
until the program resets it. Programs may also test, save and
restore a flag. Thus, IEEE 754 provides three ways by which
programs may cope with exceptions for which the default result might
be unsatisfactory:
1) Test for a condition that might cause an exception later, and
branch to avoid the exception.
2) Test a flag to see whether an exception has occurred since the
program last reset its flag.
3) Test a result to see whether it is a value that only an
exception could have produced.
CAUTION: The only reliable ways to discover whether Underflow
has occurred are to test whether products or quotients lie
closer to zero than the underflow threshold, or to test the
Underflow flag. (Sums and differences cannot underflow in IEEE
754; if x != y then x-y is correct to full precision and
certainly nonzero regardless of how tiny it may be.) Products
and quotients that underflow gradually can lose accuracy
gradually without vanishing, so comparing them with zero (as one
might on a VAX) will not reveal the loss. Fortunately, if a
gradually underflowed value is destined to be added to something
bigger than the underflow threshold, as is almost always the
case, digits lost to gradual underflow will not be missed
because they would have been rounded off anyway. So gradual
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underflows are usually provably ignorable. The same cannot be
said of underflows flushed to 0.
At the option of an implementor conforming to IEEE 754, other ways
to cope with exceptions may be provided:
4) ABORT. This mechanism classifies an exception in advance as an
incident to be handled by means traditionally associated with
error-handling statements like "ON ERROR GO TO ...". Different
languages offer different forms of this statement, but most
share the following characteristics:
- No means is provided to substitute a value for the offending
operation's result and resume computation from what may be the
middle of an expression. An exceptional result is abandoned.
- In a subprogram that lacks an error-handling statement, an
exception causes the subprogram to abort within whatever program
called it, and so on back up the chain of calling subprograms
until an error-handling statement is encountered or the whole
task is aborted and memory is dumped.
5) STOP. This mechanism, requiring an interactive debugging
environment, is more for the programmer than the program. It
classifies an exception in advance as a symptom of a
programmer's error; the exception suspends execution as near as
it can to the offending operation so that the programmer can
look around to see how it happened. Quite often the first
several exceptions turn out to be quite unexceptionable, so the
programmer ought ideally to be able to resume execution after
each one as if execution had not been stopped.
6) ... Other ways lie beyond the scope of this document.
The crucial problem for exception handling is the problem of Scope, and
the problem's solution is understood, but not enough manpower was
available to implement it fully in time to be distributed in 4.3BSD's
libm. Ideally, each elementary function should act as if it were
indivisible, or atomic, in the sense that ...
i) No exception should be signaled that is not deserved by the data
supplied to that function.
ii) Any exception signaled should be identified with that function
rather than with one of its subroutines.
iii) The internal behavior of an atomic function should not be disrupted
when a calling program changes from one to another of the five or
so ways of handling exceptions listed above, although the
definition of the function may be correlated intentionally with
exception handling.
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Ideally, every programmer should be able conveniently to turn a debugged
subprogram into one that appears atomic to its users. But simulating all
three characteristics of an atomic function is still a tedious affair,
entailing hosts of tests and saves/restores; work is under way to
ameliorate the inconvenience.
Meanwhile, the functions in libm are only approximately atomic. They
signal no inappropriate exception except possibly ...
Over/Underflow
when a result, if properly computed, might have lain barely
within range, and
Inexact in cabs, cbrt, hypot, log10 and pow
when it happens to be exact, thanks to fortuitous cancellation
of errors.
Otherwise, ...
Invalid Operation is signaled only when
any result but NaN would probably be misleading.
Overflow is signaled only when
the exact result would be finite but beyond the overflow
threshold.
Divide-by-Zero is signaled only when
a function takes exactly infinite values at finite operands.
Underflow is signaled only when
the exact result would be nonzero but tinier than the underflow
threshold.
Inexact is signaled only when
greater range or precision would be needed to represent the
exact result.
Exceptions on this machine:
The exception enables and the flags that are raised when an
exception occurs (as well as the rounding mode) are in the
floating-point control and status register. This register can be
read or written by the routines described on the man page fpc(3C).
This register's layout is described in the file <sys/fpu.h>.
A useful set of ``user trap handlers'' is available. See the man
page sigfpe(3C).
The raw interface to the hardware registers is only intended to be
used by the code to implement IEEE user trap handlers. IEEE
floating-point exceptions are enabled by setting the enable bit for
that exception in the floating-point control and status register.
If an exception then occurs the UNIX signal SIGFPE is sent to the
process. It is up to the signal handler to determine the
instruction that caused the exception and to take the action
specified by the user. The instruction that caused the exception is
in one of two places. If the floating-point board is used (the
floating-point implementation revision register indicates this in
its implementation field) then the instruction that caused the
exception is in the floating-point exception instruction register.
In all other implementations the instruction that caused the
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exception is at the address of the program counter as modified by
the branch delay bit in the cause register. Both the program
counter and cause register are in the sigcontext structure passed to
the signal handler (see signal(2)). If the program is to be
continued past the instruction that caused the exception the program
counter in the signal context must be advanced. If the instruction
is in a branch delay slot then the branch must be emulated to
determine if the branch is taken and then the resulting program
counter can be calculated (see emulate_branch(3X) and signal(2)).
Note however, that on systems using the R8000 processor, floating
point exceptions are generally fatal when trapped unless the process
is being run in precise exception mode.
PLATFORM SPECIFIC LIBRARIES
When compiling -n32 or -64, each processor has specially tuned, hardware
specific, versions of libm and libfastm, that the run time linker will
use, by default, whenever available.
The R10000 tuned libraries are found in the directories:
/usr/lib32/mips4/r10000/
/usr/lib64/mips4/r10000/
The R8000 tuned libraries are found in the directories:
/usr/lib32/mips4/r8000/
/usr/lib64/mips4/r8000/
The R5000 tuned libraries are found in the directories:
/usr/lib32/mips4/
/usr/lib64/mips4/
And the R4000 tuned libraries are found in the directories:
/usr/lib32/mips3/
/usr/lib64/mips3/
At runtime, each program automatically uses the "best" library for the
system on which it is executing. For example, if the executing program is
a mip3 program designed to run on an r4000 processor, it will still use
the mips4 R1000-tuned math library when running on an r10000 system.
BUGS
When signals are appropriate, they are emitted by certain operations
within the codes, so a subroutine-trace may be needed to identify the
function with its signal in case method 5) above is in use. And the
codes all take the IEEE 754 defaults for granted; this means that a
decision to trap all divisions by zero could disrupt a code that would
otherwise get correct results despite division by zero.
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SEE ALSO
signal(2), fpc(3C), emulate_branch(3X), sigfpe(3C), matherr(3M)
R2010 Floating Point Coprocessor Architecture
R2360 Floating Point Board Product Description
An explanation of IEEE 754 and its proposed extension p854 was published
in the IEEE magazine MICRO in August 1984 under the title "A Proposed
Radix- and Word-length-independent Standard for Floating-point
Arithmetic" by W. J. Cody et al. Articles in the IEEE magazine COMPUTER
vol. 14 no. 3 (Mar. 1981), and in the ACM SIGNUM Newsletter Special
Issue of Oct. 1979, may be helpful although they pertain to superseded
drafts of the standard.
AUTHOR
W. Kahan, with the help of Z-S. Alex Liu, Stuart I. McDonald, Dr.
Kwok-Choi Ng, Peter Tang.
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