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Common Lisp Extensions

9.2 Numerical Functions

These functions perform various arithmetic operations on numbers.

Function: gcd &rest integers

This function returns the Greatest Common Divisor of the arguments. For one argument, it returns the absolute value of that argument. For zero arguments, it returns zero.

Function: lcm &rest integers

This function returns the Least Common Multiple of the arguments. For one argument, it returns the absolute value of that argument. For zero arguments, it returns one.

Function: isqrt integer

This function computes the "integer square root" of its integer argument, i.e., the greatest integer less than or equal to the true square root of the argument.

Function: floor* number &optional divisor

This function implements the Common Lisp floor function. It is called floor* to avoid name conflicts with the simpler floor function built-in to Emacs.

With one argument, floor* returns a list of two numbers: The argument rounded down (toward minus infinity) to an integer, and the "remainder" which would have to be added back to the first return value to yield the argument again. If the argument is an integer x, the result is always the list (x 0). If the argument is a floating-point number, the first result is a Lisp integer and the second is a Lisp float between 0 (inclusive) and 1 (exclusive).

With two arguments, floor* divides number by divisor, and returns the floor of the quotient and the corresponding remainder as a list of two numbers. If (floor* x y) returns (q r), then q*y + r = x, with r between 0 (inclusive) and r (exclusive). Also, note that (floor* x) is exactly equivalent to (floor* x 1).

This function is entirely compatible with Common Lisp's floor function, except that it returns the two results in a list since Emacs Lisp does not support multiple-valued functions.

Function: ceiling* number &optional divisor

This function implements the Common Lisp ceiling function, which is analogous to floor except that it rounds the argument or quotient of the arguments up toward plus infinity. The remainder will be between 0 and minus r.

Function: truncate* number &optional divisor

This function implements the Common Lisp truncate function, which is analogous to floor except that it rounds the argument or quotient of the arguments toward zero. Thus it is equivalent to floor* if the argument or quotient is positive, or to ceiling* otherwise. The remainder has the same sign as number.

Function: round* number &optional divisor

This function implements the Common Lisp round function, which is analogous to floor except that it rounds the argument or quotient of the arguments to the nearest integer. In the case of a tie (the argument or quotient is exactly halfway between two integers), it rounds to the even integer.

Function: mod* number divisor

This function returns the same value as the second return value of floor.

Function: rem* number divisor

This function returns the same value as the second return value of truncate.

These definitions are compatible with those in the Quiroz `cl.el' package, except that this package appends `*' to certain function names to avoid conflicts with existing Emacs functions, and that the mechanism for returning multiple values is different.