exp, expf, expm1, expm1f, log, logf, log10, log10f, log1p, log1pf, pow,
powf - exponential, logarithm, power functions
Math Library (libm, -lm)
pow(double x, double y);
powf(float x, float y);
The exp() function computes the exponential value of the given argument
The expm1() function computes the value exp(x)-1 accurately even for tiny
The log() function computes the value of the natural logarithm of argument
The log10() function computes the value of the logarithm of argument x to
The log1p() function computes the value of log(1+x) accurately even for
tiny argument x.
The pow() computes the value of x to the exponent y.
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. 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 generate a reserved operand fault. On a VAX, errno is set to EDOM
and the reserved operand is returned by log unless x > 0, by log1p()
unless x > -1.
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 moderate,
but increases as pow(x, y) approaches the over/underflow thresholds
until almost as many bits could be lost as are occupied by the floating-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 testing;
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.
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 Pascal,
exp1 and log1 in C on APPLE Macintoshes, where they have been provided
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, Infinity
(not found on a VAX), and NaN (the reserved operand on a VAX). Previous
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
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 anyway
since that expression's meaning and, if invalid, its consequences
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 as the value of polynomial
p(x) = a*x**0 + a*x**1 + a*x**2 +...+ a[n]*x**n
at x = 0 rather than reject a*0**0 as invalid.
3. Analysts will accept 0**0 = 1 despite that x**y can approach anything
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., independently
The exp(), log(), log10() and pow() functions conform to ANSI X3.159-1989
A exp(), log() and pow() functions appeared in Version 6 AT&T UNIX. A
log10() function appeared in Version 7 AT&T UNIX. The log1p() and
expm1() functions appeared in 4.3BSD.
BSD July 31, 1991 BSD
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