Floating-point numbers are useful for representing numbers that are
not integral. The range of floating-point numbers is
the same as the range of the C data type
double on the machine
you are using. On all computers supported by Emacs, this is
IEEE binary64 floating point format, which is standardized by
IEEE Std 754-2019
and is discussed further in David Goldberg’s paper
“What Every Computer Scientist Should Know About Floating-Point Arithmetic”.
On modern platforms, floating-point operations follow the IEEE-754
standard closely; however, results are not always rounded correctly on
some obsolescent platforms, notably 32-bit x86.
The read syntax for floating-point numbers requires either a decimal point, an exponent, or both. Optional signs (‘+’ or ‘-’) precede the number and its exponent. For example, ‘1500.0’, ‘+15e2’, ‘15.0e+2’, ‘+1500000e-3’, and ‘.15e4’ are five ways of writing a floating-point number whose value is 1500. They are all equivalent. Like Common Lisp, Emacs Lisp requires at least one digit after any decimal point in a floating-point number; ‘1500.’ is an integer, not a floating-point number.
Emacs Lisp treats
-0.0 as numerically equal to ordinary zero
with respect to numeric comparisons like
=. This follows the
IEEE floating-point standard, which says
0.0 are numerically equal even though other operations can
The IEEE floating-point standard supports positive
infinity and negative infinity as floating-point values. It also
provides for a class of values called NaN, or “not a number”;
numerical functions return such values in cases where there is no
correct answer. For example,
(/ 0.0 0.0) returns a NaN.
A NaN is never numerically equal to any value, not even to itself.
NaNs carry a sign and a significand, and non-numeric functions treat
two NaNs as equal when their
signs and significands agree. Significands of NaNs are
machine-dependent, as are the digits in their string representation.
When NaNs and signed zeros are involved, non-numeric functions like
gethash determine whether values are indistinguishable, not
whether they are numerically equal. For example, when x and
y are the same NaN,
(equal x y) returns
(= x y) uses numeric comparison and returns
(equal 0.0 -0.0) returns
(= 0.0 -0.0) returns
Here are read syntaxes for these special floating-point values:
‘1.0e+INF’ and ‘-1.0e+INF’
‘0.0e+NaN’ and ‘-0.0e+NaN’
The following functions are specialized for handling floating-point numbers:
This predicate returns
t if its floating-point argument is a NaN,
This function returns a cons cell
(s . e),
where s and e are respectively the significand and
exponent of the floating-point number x.
If x is finite, then s is a floating-point number between 0.5 (inclusive) and 1.0 (exclusive), e is an integer, and x = s * 2**e. If x is zero or infinity, then s is the same as x. If x is a NaN, then s is also a NaN. If x is zero, then e is 0.
Given a numeric significand s and an integer exponent e, this function returns the floating point number s * 2**e.
This function copies the sign of x2 to the value of x1, and returns the result. x1 and x2 must be floating point.
This function returns the binary exponent of x. More precisely, if x is finite and nonzero, the value is the logarithm base 2 of |x|, rounded down to an integer. If x is zero or infinite, the value is infinity; if x is a NaN, the value is a NaN.
(logb 10) ⇒ 3 (logb 10.0e20) ⇒ 69 (logb 0) ⇒ -1.0e+INF