EXP

Section: C Library Functions (3)
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BSD mandoc
BSD 4

NAME

exp expm1 log log10 log1p pow - exponential, logarithm, power functions

SYNOPSIS

Fd #include <math.h> Ft double Fn exp double x Ft double Fn expm1 double x Ft double Fn log double x Ft double Fn log10 double x Ft double Fn log1p double x Ft double Fn pow double x double y

DESCRIPTION

The Fn exp function computes the exponential value of the given argument Fa x .

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

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

The Fn log10 function computes the value for the logarithm of argument Fa x to base 10.

The Fn log1p function computes the value of log(1+x) accurately even for tiny argument Fa x .

The Fn 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 Fn pow x y is below about 2 ups when its magnitude is moderate, but increases as Fn 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 Fn pow are accurate enough that Fn 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 approprate computation unless an error occurs or an argument is out of range. The functions Fn exp , Fn expm1 and Fn pow detect if the computed value will overflow, set the global variable errno to Er RANGE and cause a reserved operand fault on a VAX or Tahoe The function Fn pow x y checks to see if Fa x < 0 and Fa y is not an integer, in the event this is true, the global variable errno is set to Er EDOM and on the VAX and Tahoe generate a reserved operand fault. On a VAX and Tahoe errno is set to Er EDOM and the reserved operand is returned by log unless Fa x > 0, by Fn log1p unless Fa 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 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 Fn 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 always:

Any program that already tests whether x is zero (or infinite or ) 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.
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.
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.
If 0**0 = 1, then infinity**0 = 1/0**0 = 1 too; and then **0 = 1 too because x**0 = 1 for all finite and infinite x, i.e., independently of x.

HISTORY

A Fn exp , Fn log and Fn pow function appeared in AT&T System v6 . A Fn log10 function appeared in AT&T System v7 . The Fn log1p and Fn expm1 functions appeared in BSD 4.3

This document was created by man2html, using the manual pages.
Time: 00:35:29 GMT, March 30, 2022

EXP