EXP(3) MachTen Programmer’s Manual EXP(3)

NAME
exp, expm1, log, log10, log1p, pow - exponential, logarithm, power func-
tions

SYNOPSIS
#include <math.h>

double
exp(double x)

double
expm1(double x)

double
log(double x)

double
log10(double x)

double
log1p(double x)

double
pow(double x, double y)

DESCRIPTION
The exp() function computes the exponential value of the given argument
x.

The expm1() function computes the value exp(x)-1 accurately even for tiny
argument x.

The log() function computes the value for the natural logarithm of the
argument x.

The log10() function computes the value for the logarithm of argument x
to base 10.

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.

ERROR (due to Roundoff etc.)
exp(x), log(x), expm1(x) and log1p(x) are accurate to within an up, and
log10(x) to within about 2 ups; an up is one Unit in the Last Place. The
error in pow(x, y) is below about 2 ups 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 ups for VAX D, 300 ups 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.

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 RANGE 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.

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).
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 al-
ways:

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, in-
cluding x = 0. This is compatible with the convention that ac-
cepts 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.