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```EXP(3)		      DragonFly Library Functions Manual		EXP(3)

NAME
exp, expf, expl, exp2, exp2f, exp2l, expm1, expm1f, expm1l, log, logf,
logl, log2, log2f, log2l, log10, log10f, log10l, log1p, log1pf, log1pl,
pow, powf, powl -- exponential, logarithm, power functions

SYNOPSIS
#include <math.h>

double
exp(double x);

float
expf(float x);

long double
expl(long double x);

double
exp2(double x);

float
exp2f(float x);

long double
exp2l(long double x);

double
expm1(double x);

float
expm1f(float x);

long double
expm1l(long double x);

double
log(double x);

float
logf(float x);

long double
logl(long double x);

double
log2(double x);

float
log2f(float x);

long double
log2l(long double x);

double
log10(double x);

float
log10f(float x);

long double
log10l(long double x);

double
log1p(double x);

float
log1pf(float x);

long double
log1pl(long double x);

double
pow(double x, double y);

float
powf(float x, float y);

long double
powl(long double x, long double y);

DESCRIPTION
The exp() function computes the base e exponential value of the given
argument x.  The expf() function is a single precision version of exp().
The expl() function is an extended precision version of exp().

The exp2() function computes the base 2 exponential of the given argument
x.  The exp2f() function is a single precision version of exp2().	The
exp2l() function is an extended precision version of exp2().

The expm1() function computes the value exp(x)-1 accurately even for tiny
argument x.  The expm1f() function is a single precision version of
expm1().  The expm1l() function is an extended precision version of
expm1().

The log() function computes the value of the natural logarithm of argu-
ment x.  The logf() function is a single precision version of log().  The
logl() function is an extended precision version of log().

The log2() function computes the value of the logarithm of argument x to
base 2.  The log2f() function is a single precision version of log2().
The log2l() function is an extended precision version of log2().

The log10() function computes the value of the logarithm of argument x to
base 10.  The log10f() function is a single precision version of log10().
The log10l() function is an extended precision version of log10().

The log1p() function computes the value of log(1+x) accurately even for
tiny argument x.  The log1pf() function is a single precision version of
log1p().  The log1pl() function is an extended precision version of
log1p().

The pow() function computes the value of x to the exponent y.  The powf()
function is a single precision version of pow().  The powl() function is
an extended precision version of pow().

RETURN VALUES
These functions will return the appropriate computation unless an error
occurs or an argument is out of range.  The functions exp(), expm1() and
pow() detect if the computed value will overflow, set the global variable
errno to ERANGE and cause a reserved operand fault on a VAX or Tahoe.
The function pow(x, y) checks to see if x < 0 and y is not an integer, in
the event this is true, the global variable errno is set to EDOM and on
the VAX and Tahoe generate a reserved operand fault.  On a VAX and Tahoe,
errno is set to EDOM and the reserved operand is returned by log unless x
> 0, by log1p() unless x > -1.

ERRORS (due to Roundoff etc.)
exp(x), log(x), expm1(x) and log1p(x) are accurate to within an ulp, and
log10(x) to within about 2 ulps; an ulp is one Unit in the Last Place.
The error in pow(x, y) is below about 2 ulps when its magnitude is moder-
ate, but increases as pow(x, y) approaches the over/underflow thresholds
until almost as many bits could be lost as are occupied by the float-
ing-point format's exponent field; that is 8 bits for ``VAX D'' and 11
bits for IEEE 754 Double.	No such drastic loss has been exposed by test-
ing; the worst errors observed have been below 20 ulps for ``VAX D'', 300
ulps for IEEE 754 Double.	Moderate values of pow() are accurate enough
that pow(integer, integer) is exact until it is bigger than 2**56 on a
VAX, 2**53 for IEEE 754.

NOTES
The functions exp(x)-1 and log(1+x) are called expm1 and logp1 in BASIC
on the Hewlett-Packard HP-71B and APPLE Macintosh, EXP1 and LN1 in Pas-
cal, exp1 and log1 in C on APPLE Macintoshes, where they have been pro-
vided to make sure financial calculations of ((1+x)**n-1)/x, namely
expm1(n*log1p(x))/x, will be accurate when x is tiny.  They also provide
accurate inverse hyperbolic functions.

The function pow(x, 0) returns x**0 = 1 for all x including x = 0, infin-
ity (not found on a VAX), and NaN (the reserved operand on a VAX).  Pre-
vious implementations of pow() may have defined x**0 to be undefined in
some or all of these cases.  Here are reasons for returning x**0 = 1
always:

1.      Any program that already tests whether x is zero (or infinite or
NaN) before computing x**0 cannot care whether 0**0 = 1 or not.
Any program that depends upon 0**0 to be invalid is dubious any-
way since that expression's meaning and, if invalid, its conse-
quences vary from one computer system to another.

2.      Some Algebra texts (e.g., Sigler's) define x**0 = 1 for all x,
including x = 0.  This is compatible with the convention that
accepts a[0] as the value of polynomial

p(x) = a[0]*x**0 + a[1]*x**1 + a[2]*x**2 +...+ a[n]*x**n

at x = 0 rather than reject a[0]*0**0 as invalid.

3.      Analysts will accept 0**0 = 1 despite that x**y can approach any-
thing or nothing as x and y approach 0 independently.  The reason
for setting 0**0 = 1 anyway is this:

If x(z) and y(z) are any functions analytic (expandable  in
power series) in z around z = 0, and if there x(0) = y(0) =
0, then x(z)**y(z) -> 1 as z -> 0.

4.      If 0**0 = 1, then infinity**0 = 1/0**0 = 1 too; and then NaN**0 =
1 too because x**0 = 1 for all finite and infinite x, i.e., inde-
pendently of x.