### 3.2 Floating-Point Basics

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 almost 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 systems, notably 32-bit x86.

On some old computer systems, Emacs may not use IEEE floating-point. We know of one such system on which Emacs runs correctly, but does not follow IEEE-754: the VAX running NetBSD using GCC 10.4.0, where the VAX ‘D_Floating’ format is used instead. IBM System/370-derived mainframes and their XL/C compiler are also capable of utilizing a hexadecimal floating point format, but Emacs has not yet been built in such a configuration.

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 a decimal point in a floating-point number that does not have an exponent; ‘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` and `0.0` are numerically equal even though other operations can distinguish them.

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.

NaNs are not available on systems which do not use IEEE floating-point arithmetic; if the read syntax for a NaN is used on a VAX, for example, the reader signals an error.

When NaNs and signed zeros are involved, non-numeric functions like `eql`, `equal`, `sxhash-eql`, `sxhash-equal` and `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 `t` whereas `(= x y)` uses numeric comparison and returns `nil`; conversely, `(equal 0.0 -0.0)` returns `nil` whereas `(= 0.0 -0.0)` returns `t`.

Here are read syntaxes for these special floating-point values:

infinity

1.0e+INF’ and ‘-1.0e+INF

not-a-number

0.0e+NaN’ and ‘-0.0e+NaN

The following functions are specialized for handling floating-point numbers:

Function: isnan x

This predicate returns `t` if its floating-point argument is a NaN, `nil` otherwise.

Function: frexp x

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.

Function: ldexp s e

Given a numeric significand s and an integer exponent e, this function returns the floating point number s * 2**e.

Function: copysign x1 x2

This function copies the sign of x2 to the value of x1, and returns the result. x1 and x2 must be floating point.

Function: logb x

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
```