# math(3)

```NAME
math - introduction to mathematical library functions

DESCRIPTION
These functions constitute the C math  library,  libm.  This  library  is
based  Sun  Microsystems  FDLIBM  (Freely  Distributable  LIBM), a C math
library for machines that support  IEEE  754  floating-point  arithmetic.
Declarations  for  the functions in this library may be obtained from the
include file <math.h>.

LIST OF FUNCTIONS

Name       Appears on Page   Description

acos         sin(3)      inverse trigonometric function
acosh        asinh(3)    inverse hyperbolic function
asin         sin(3)      inverse trigonometric function
asinh        asinh(3)    inverse hyperbolic function
atan         sin(3)      inverse trigonometric function
atanh        asinh(3)    inverse hyperbolic function
atan2        sin(3)      inverse trigonometric function
cbrt         sqrt(3)     cube root
ceil         floor(3)    integer no less than
copysign     ieee(3)     copy sign bit
cos          sin(3)      trigonometric function
cosh         sinh(3)     hyperbolic function
remainder    ieee(3)     remainder
erf          erf(3)      error function
erfc         erf(3)      complementary error function
exp          exp(3)      exponential
expm1        exp(3)      exp(x)-1
fabs         floor(3)    absolute value
floor        floor(3)    integer no greater than
hypot        hypot(3)    Euclidean distance
ilogb        ieee(3)     exponent extraction
j0           j0(3)       bessel function
j1           j0(3)       bessel function
jn           j0(3)       bessel function
log          exp(3)      natural logarithm
logb         ieee(3)     exponent extraction
log10        exp(3)      logarithm to base 10
log1p        exp(3)      log(1+x)
pow          exp(3)      exponential x**y
rint         floor(3)    round to nearest integer
sin          sin(3)      trigonometric function
sinh         sinh(3)     hyperbolic function
sqrt         sqrt(3)     square root
tan          sin(3)      trigonometric function
tanh         sinh(3)     hyperbolic function
y0           j0(3)       bessel function
y1           j0(3)       bessel function
yn           j0(3)       bessel function

NOTES

FDLIBM (double precision version) assumes:
a.  IEEE 754 style (if not precise compliance) arithmetic;
b.  32 bit 2's complement integer arithmetic;
c.  Each double precision floating-point number must be in IEEE 754
double format, and that each number can be retrieved as two 32-bit
integers;

Example: let y = 2.0
double fp number y:     2.0
IEEE double format:     0x4000000000000000

Referencing y as two integers:
*(int*)&y,*(1+(int*)&y) =       {0x40000000,0x0} (on sparc)
{0x0,0x40000000} (on 386)

Note: FDLIBM will detect, at run time, the correct ordering of
the high and low part of a floating-point number.

d. IEEE exceptions may trigger "signals" as is common in Unix
implementations.

-------------------
2. EXCEPTION CASES
-------------------
All exception cases in the FDLIBM functions will be mapped
to one of the following four exceptions:

+huge*huge, +tiny*tiny,     +1.0/0.0,       +0.0/0.0
(overflow)  (underflow)  (divided-by-zero)  (invalid)

For example, log(0) is a singularity and is thus mapped to
-1.0/0.0 = -infinity.
That is, FDLIBM's log will compute -one/zero and return the
computed value.  On an IEEE machine, this will trigger the
divided-by-zero exception and a negative infinity is returned by
default.

Similarly, exp(-huge) will be mapped to tiny*tiny to generate
an underflow signal.

IEEE STANDARD 754 Floating-Point Arithmetic:

Properties of IEEE 754 Double-Precision:
Wordsize: 64 bits, 8 bytes.  Radix: Binary.
Precision: 53 significant bits, roughly 16 significant 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 Inf.
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 Inf = 1/(x-y) != -1/(y-x) = -Inf.
Inf is signed.
it persists when added to itself or to any finite number.   Its
sign  transforms correctly through multiplication and division,
and (finite)/+Inf = +0 (nonzero)/0 = +Inf.  But Inf-Inf,  Inf*0
and Inf/Inf 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  are  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,
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  +Inf  or  towards  -Inf  at the programmer's
option.  And the same  kinds  of  rounding  are  specified  for
Binary-Decimal  Conversions,  at  least  for magnitudes between
roughly 1.0e-10 and 1.0e37.
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               +Inf
Divide by Zero         +Inf
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 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  4.3  BSD'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.

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 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.

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.

The math manual pages have been adapted from the 4.3BSD 3M  manual  pages
for  FDLIBM  by  Kees  J. Bot <kjb@cs.vu.nl> who normally avoids floating
point like the plague.  Some text may not apply to FDLIBM, but KJB didn't
know  whether  to  remove  it  or  not.  Don't blame the original authors
mentioned on these pages for inaccuracies introduced.

asinh(3), erf(3), exp(3), floor(3),  hypot(3),  ieee(3),  j0(3),  sin(3),
sinh(3), sqrt(3).

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.  The manuals for Pascal, C and BASIC on
the Apple Macintosh document  the  features  of  IEEE  754  pretty  well.
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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