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# 9. Numbers

XEmacs supports two to five numeric data types. Fixnums and floating point numbers are always supported. As a build-time option, bignums, ratios, and bigfloats may be enabled on some platforms.

Fixnums (called just integers in GNU Emacs and older versions of XEmacs) are whole numbers such as -3, 0, #b0111, #xFEED, #o744. Their values are exact, and their range is limited. The number prefixes ‘#b’, ‘#o’, and ‘#x’ are supported to represent numbers in binary, octal, and hexadecimal notation (or radix). Floating point numbers are numbers with fractional parts, such as -4.5, 0.0, or 2.71828. They can also be expressed in exponential notation: 1.5e2 equals 150; in this example, ‘e2’ stands for ten to the second power, and is multiplied by 1.5. Floating point values are not exact; they have a fixed, limited amount of precision.

Bignums are arbitrary precision integers. When supported, XEmacs can handle any integral calculations you have enough virtual memory to store. (More precisely, on current architectures the representation allows integers whose storage would exhaust the address space.) They are notated in the same way as other integers (fixnums). XEmacs automatically converts results of computations from fixnum to bignum, and back, depending on the storage required to represent the number. Thus use of bignums are entirely transparent to the user, except for a few special applications that expect overflows. Ratios are rational numbers with arbitrary precision. They are notated in the usual way with the solidus, for example 5/3 or -22/7.

Bigfloats are floating point numbers with arbitrary precision, which may be specified by the user (and may be different for different bigfloats at the same time). Unlike integers, which are always infinitely precise if they can be represented, floating point numbers are inherently imprecise. This means that choice of precision can be a very delicate issue. XEmacs automatically converts from float to bigfloat when floats and bigfloats are mixed in an expression, but a bigfloat will never be converted to a float unless the user explicitly coerces the value. Nor will the result of a float operation be converted to bigfloat, except for “contagion” from another operand that is already a bigfloat. However, when bigfloats of differing precision are mixed, the result will always have the larger precision. The exact rules are more carefully explained elsewhere (see section Canonicalization and Contagion).

Common Lisp terminology and historical Emacs terminology conflict here, to an extent. We attempt to use “fixnum” and “integer” consistently, but older XEmacs and GNU Emacs code and documentation use the latter to mean the former. “Float” is used in Emacs documentation to mean “fixed precision floating point number”, and the Common Lisp distinctions among short-floats, long-floats, etc., and bigfloats (which are not standardized in Common Lisp) are not reflected in XEmacs terminology. We’re working on this, but volunteers to fix it in the XEmacs manuals would be heartily welcomed.

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## 9.1 Integer Basics

The range of values for an integer depends on the machine. If a multiple-precision arithmetic library is available on your platform, support for bignums, that is, integers with arbitrary precision, may be compiled in to your XEmacs. The rest of this section assumes that the bignum extension is not available. The bignum extension and the user-visible differences in normal integer arithmetic are discussed in a separate section The Bignum Extension.

The minimum range is -1073741824 to 1073741823 (31 bits; i.e., to but some machines may provide a wider range. Many examples in this chapter assume an integer has 31 bits.

The range of fixnums is available to Lisp programs:

Variable: most-positive-fixnum

The fixed-precision integer closest in value to positive infinity.

Variable: most-negative-fixnum

The fixed-precision integer closest in value to negative infinity.

Here is a common idiom to temporarily suppress garbage collection:

 ```(garbage-collect) (let ((gc-cons-threshold most-positive-fixnum)) ;; allocation-intensive computation ) (garbage-collect) ```

The Lisp reader reads an integer as a sequence of digits with optional initial sign and optional final period.

 ``` 1 ; The integer 1. 1. ; The integer 1. +1 ; Also the integer 1. -1 ; The integer -1. 2147483648 ; Read error, due to overflow. 0 ; The integer 0. -0 ; The integer 0. ```

To understand how various functions work on integers, especially the bitwise operators (see section Bitwise Operations on Integers), it is often helpful to view the numbers in their binary form.

In 31-bit binary, the decimal integer 5 looks like this:

 ```000 0000 0000 0000 0000 0000 0000 0101 ```

(We have inserted spaces between groups of 4 bits, and two spaces between groups of 8 bits, to make the binary integer easier to read.)

The integer -1 looks like this:

 ```111 1111 1111 1111 1111 1111 1111 1111 ```

-1 is represented as 31 ones. (This is called two’s complement notation.)

The negative integer, -5, is creating by subtracting 4 from -1. In binary, the decimal integer 4 is 100. Consequently, -5 looks like this:

 ```111 1111 1111 1111 1111 1111 1111 1011 ```

In this implementation, the largest 31-bit binary integer is the decimal integer 1,073,741,823. In binary, it looks like this:

 ```011 1111 1111 1111 1111 1111 1111 1111 ```

Since the arithmetic functions do not check whether integers go outside their range, when you add 1 to 1,073,741,823, the value is the negative integer -1,073,741,824:

 ```(+ 1 1073741823) ⇒ -1073741824 ⇒ 100 0000 0000 0000 0000 0000 0000 0000 ```

Many of the arithmetic functions accept markers for arguments as well as integers. (See section Markers.) More precisely, the actual arguments to such functions may be either integers or markers, which is why we often give these arguments the name int-or-marker. When the argument value is a marker, its position value is used and its buffer is ignored.

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## 9.2 Rational Basics

Ratios (built-in rational numbers) are available only when the bignum extension is built into your XEmacs. This facility is new and experimental. It is discussed in a separate section for convenience of updating the documentation The Bignum Extension. The following functions are defined regardless of the presence of the extension, but have trivial results for integers.

Function: numerator rational

Return the numerator of the canonical form of rational. If rational is an integer, rational is returned. rational must be an integer or a ratio.

Function: denominator rational

Return the denominator of the canonical form of rational. If rational is an integer, 1 is returned. rational must be an integer or a ratio.

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## 9.3 Floating Point Basics

XEmacs supports floating point numbers. The precise range of floating point numbers is machine-specific; it is the same as the range of the C data type `double` on the machine in question. If a multiple-precision arithmetic library is available on your platform, support for bigfloats, that is, floating point numbers with arbitrary precision, may be compiled in to your XEmacs. The rest of this section assumes that the bignum extension is not available. The bigfloat extension and the user-visible differences in normal float arithmetic are discussed in a separate section The Bignum Extension.

The printed representation for floating point numbers requires either a decimal point (with at least one digit following), an exponent, or both. For example, ‘1500.0’, ‘15e2’, ‘15.0e2’, ‘1.5e3’, and ‘.15e4’ are five ways of writing a floating point number whose value is 1500. They are all equivalent. You can also use a minus sign to write negative floating point numbers, as in ‘-1.0’.

Most modern computers support the IEEE floating point standard, which provides for 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, `(sqrt -1.0)` returns a NaN. For practical purposes, there’s no significant difference between different NaN values in XEmacs Lisp, and there’s no rule for precisely which NaN value should be used in a particular case, so this manual doesn’t try to distinguish them. XEmacs Lisp has no read syntax for NaNs or infinities; perhaps we should create a syntax in the future.

You can use `logb` to extract the binary exponent of a floating point number (or estimate the logarithm of an integer):

Function: logb number

This function returns the binary exponent of number. More precisely, the value is the logarithm of number base 2, rounded down to an integer.

The range of floats is available to Lisp programs:

Variable: most-positive-float

The fixed-precision floating-point-number closest in value to positive infinity.

Variable: most-negative-float

The fixed-precision floating point number closest in value to negative infinity.

Variable: least-positive-float

The positive float closest in value to 0. May not be normalized.

Variable: least-negative-float

The positive float closest in value to 0. Must be normalized.

Variable: least-positive-normalized-float

The negative float closest in value to 0. May not be normalized.

Variable: least-negative-normalized-float

The negative float closest in value to 0. Must be normalized.

Note that for floating point numbers there is an interesting limit on how small they can get, as well as a limit on how big they can get. In some representations, a floating point number is normalized if the leading digit is non-zero. This allows representing numbers smaller than the most-negative exponent can express, by having fractional mantissas. This means that the number is less precise than a normalized floating point number, so Lisp programs can detect loss of precision due to unnormalized floats by checking whether the number is between `least-positive-float` and `least-positive-normalized-float`.

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## 9.4 The Bignum Extension

In XEmacs 21.5.18, an extension was added by Jerry James to allow linking with arbitrary-precision arithmetic libraries if they are available on your platform. “Arbitrary” precision means precisely what it says. Your ability to work with large numbers is limited only by the amount of virtual memory (and time) you can throw at them.

As of 09 April 2004, support for the GNU Multiple Precision arithmetic library (GMP) is nearly complete, and support for the BSD Multiple Precision arithmetic library (MP) is being debugged. To enable bignum support using GMP (respectively MP), invoke configure with your usual options, and add ‘--use-number-lib=gmp’ (respectively ‘--use-number-lib=mp’). The default is to disable bignum support, but if you are using a script to automate the build process, it may be convenient to explicitly disable support by appending--use-number-lib=no’ to your invocation of configure. GMP has an MP compatibility mode, but it is not recommended, as there remain poorly understood bugs (even more so than for other vendors’ versions of MP).

With GMP, exact arithmetic with integers and ratios of arbitrary precision and approximate (“floating point”) arithmetic of arbitrary precision are implemented efficiently in the library. (Note that numerical implementations are quite delicate and sensitive to optimization. If the library was poorly optimized for your hardware, as is often the case with Linux distributions for 80x86, you may achieve gains of several orders of magnitude by rebuilding the MP library. See http://www.swox.com/gmp/gmp-speed.html.) The MP implementation provides arbitrary precision integers. Ratios and arbitrary precision floats are not available with MP.

If your code needs to run correctly whether or not the feature is provided, you may test for the features `bignum`, `ratio`, and `bigfloat`.

The XEmacs bignum facility implements the Common Lisp notions of canonicalization and contagion. Canonicalization means that in exact (integer and ratio) arithmetic, a result of an operation is always converted to the “smallest” type that can represent it exactly. For exact numbers, the user only cares if efficiency is extremely important; Lisp does not try to determine an order of computation that avoids conversion to bignum (or ratio) even if one is available. (Note that integers are never silently converted to ratios: the result of `(/ 1 2)` is the integer `0`. You can request that a ratio be used if needed with `(div 1 2)`.)

Since floating point arithmetic is inherently imprecise, numbers are implicitly coerced to bigfloats only if other operands in the expression are bigfloat, and bigfloats are only coerced to other numerical types by explicit calls to the function `coerce`.

Bignum support is incomplete. If you would like to help with bignum support, especially on BSD MP, please subscribe to the XEmacs Beta mailing list, and book up on ‘number-gmp.h’ and ‘number-mp.h’. Jerry has promised to write internals documentation eventually, but if your skills run more to analysis and documentation than to writing new code, feel free to fill in the gap!

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### 9.4.1 Bignum Basics

In most cases, bignum support should be transparent to users and Lisp programmers. A bignum-enabled XEmacs will automatically convert from fixnums to bignums and back in pure integer arithmetic, and for GNU MP, from floats to bigfloats. (Bigfloats must be explicitly coerced to other types, even if they are exactly representable by less precise types.) The Lisp reader and printer have been enhanced to handle bignums, as have the mathematical functions. Rationals (fixnums, bignums, and ratios) are printed using the ‘%d’, ‘%o’, ‘%x’, and ‘%u’ format conversions.

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### 9.4.2 Ratio Basics

Ratios, when available have the read syntax and print representation ‘3/5’. Like other rationals (fixnums and bignums), they are printed using the ‘%d’, ‘%o’, ‘%x’, and ‘%u’ format conversions.

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### 9.4.3 Bigfloat Basics

Bigfloats, when available, have the same read syntax and print representations as fixed-precision floats.

It is possible to make bigfloat the default floating point format by setting `default-float-precision` to a non-zero value. Precision is given in bits, with a maximum precision of `bigfloat-maximum-precision`. Bigfloats are created automatically when a number with yes

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### 9.4.4 Canonicalization and Contagion

Canonicalization is a rule intended to enhance the time and space efficiency of exact arithmetic. Because bignums and ratios are implemented as record objects, they take up much more space than fixnums, which are implemented as an immediate object. Conversions and calls to the MP library also take time. So the implementation always converts the result of exact arithmetic to the smallest representation that can exactly represent the quantity.

 ```(+ 3/4 5) ⇒ 23/4 (+ 3/4 1/4 2) ⇒ 3 ```

Conversely, if an integer (read or computed) cannot be represented as a fixnum, a bignum will be used. Integer division is a somewhat exceptional case. Because it is useful and is the historical meaning of the function `/`, a separate function `div` is provided. `div` is identical to `/` except that when the rational result is not an integer, it is represented exactly as a ratio. In both cases if a rational result is an integer, it is automatically converted to the appropriate integral representation.

Note that the efficiency gain from canonicalization is likely to be less than you might think. Experience with numerical analysis shows that in very precise calculations, the required precision tends to increase. Thus it is typically wasted effort to attempt to convert to smaller representations, as the number is often reused and requires a larger representation. However, XEmacs Lisp presumes that calculations using bignums are the exception, so it applies canonicalization.

Contagion is one way to address the requirement that an arithmetic operation should not fail because of differing types of the operands. Contagion is the idea that less precise operands are converted to the more precise type, and then the operation is performed. While changing precision is a delicate issue, contagion is so useful that XEmacs performs it automatically.

In XEmacs, the following rules of contagion are used:

1. If an expression mixes an integral type with a ratio, then the usual rules of rational arithmetic apply. (If the result of the expression happens to be an integer, it will be canonicalized to integer.)
2. If an expression mixes a rational type (fixnum, bignum, or ratio) with a float, the rational operand is converted to a float and the operation performed if the result would fit in a float, otherwise both operands are promoted to bigfloat, and the operation performed.
3. If an expression mixes any other type with a bigfloat, the other operand is converted to bigfloat and the operation performed.
4. If bigfloats of different precision are mixed, all are converted to the highest precision, and the operation performed.

Note that there are no rules to canonicalize floats or bigfloats. This might seem surprising, but in both cases information will be lost. Any floating point representation is implicitly approximate. A conversion to a rational type, even if it seems exact, loses this information. More subtly, demoting a bigfloat to a smaller bigfloat or to a float would lose information about the precision of the result, and thus some information about the accuracy. Thus floating point numbers are always already in canonical form.

Of course the programmer can explicitly request canonicalization, or more coercion to another type. Coercion uses the Common Lisp compatibility function `coerce` from the ‘cl-extra.el’ library. A number can be explicitly converted to canonical form according to the above rules using

Function: canonicalize-number number

Return the canonical form of number.

However, if we’ve done our job properly, this is always a no-op. That is, if you find a number in un-canonicalized form, please report it as a bug.

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### 9.4.5 Compatibility Issues

Surgeon General’s Warning: The automatic conversions cannot be disabled at runtime. Old functions will not produce ratios unless there is a ratio operand, so there should be few surprises with type conflicts (the contagion rules are quite natural for Lisp programmers used to the behavior of integers and floats in pre-21.5.18 XEmacsen), but they can’t be ruled out. Also, if you work with extremely large numbers, your machine may arbitrarily decide to hand you an unpleasant surprise rather than a bignum.

User-visible changes in behavior include (in probable order of annoyance)

• Arithmetic can cause a segfault, depending on your MP library.

GMP by default allocates temporaries on the stack. If you run out of stack space, you’re dead; there is no way that we know of to reliably detect this condition, because ‘alloca’ is typically implemented to be fast rather than robust. If you just need a little more oomph, use a bigger stack (e.g., the ‘ulimit -s’ command in bash(1)). If you want robustness at the cost of speed, configure GMP with ‘--disable-alloca’ and rebuild the GMP library.

We do not know whether BSD MP uses ‘alloca’ or not. Please send any information you have as a bug report (M-x report-xemacs-bug <RET>), which will give us platform information. (We do know that BSD MP implementations vary across vendors, but how much, we do not know yet.)

• Our documentation’s terminology, and our API terminology, is not always Common-Lisp-conforming. Many places use “integer” where “fixnum” better reflects what the code accepts or produces; there are similar issues for the varying types of floating point numbers. Since Emacs Lisp has not had a ratio type before, there are no problems there.
• An atom with ratio read syntax now returns a number, not a symbol.
• Many operations that used to cause a range error now succeed, with intermediate results and return values coerced to bignums as needed.
• The ‘%u’ format conversion will now give an error if its argument is negative. (Without MP, it prints a number which Lisp can’t read.)

This is not a compatibility issue in the sense of specification, but careless programmers who have taken advantage of the immediate representation for numbers and written `(eq x y)` are in for a surprise. This doesn’t work with bignums, even if both arguments are bignums! Arbitrary precision obviously requires consing new objects because the objects are “large” and of variable size, and the definition of ‘eq’ does not permit different objects to compare as equal. Instead of `eq`, use `eql`, in which numbers of the same type which have equal values compare equal, or `=`, which does any necessary type coercions before comparing for equality Comparison of Numbers.

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## 9.5 Type Predicates for Numbers

The functions in this section test whether the argument is a number or whether it is a certain sort of number. The functions which test for type can take any type of Lisp object as argument (the more general predicates would not be of much use otherwise). However, the `zerop` predicate requires a number as its argument, and the `evenp`, and `oddp` predicates require integers as their arguments. See also `integer-or-marker-p`, `integer-char-or-marker-p`, `number-or-marker-p` and `number-char-or-marker-p`, in Predicates on Markers.

Function: numberp object

This predicate tests whether its argument is a number (either integer or floating point), and returns `t` if so, `nil` otherwise.

Function: realp object

The `realp` predicate tests to see whether object is a rational or floating point number, and returns `t` if so, `nil` otherwise. Currently equivalent to `numberp`.

Function: zerop number

This predicate tests whether its argument is zero, and returns `t` if so, `nil` otherwise. The argument must be a number.

These two forms are equivalent: `(zerop x)``(= x 0)`.

Function: integerp object

This predicate tests whether its argument is an integer, and returns `t` if so, `nil` otherwise.

Function: oddp integer

The `oddp` predicate tests to see whether integer is odd, and returns `t` if so, `nil` otherwise. integer must be an integer.

Function: evenp integer

The `evenp` predicate tests to see whether integer is even, and returns `t` if so, `nil` otherwise. integer must be an integer.

Function: natnump object

The `natnump` predicate (whose name comes from the phrase “natural-number-p”) tests to see whether its argument is a nonnegative integer, and returns `t` if so, `nil` otherwise. 0 is considered non-negative.

Function: fixnump object

The predicate tests to see whether its argument is an integer represented as a fixnum, and returns `t` if so, `nil` otherwise.

Function: bignump object

The `bignump` predicate tests to see whether object is an integer represented as a bignum, and returns `t` if so, `nil` otherwise.

Function: rationalp object

The `rationalp` predicate tests to see whether object is a rational number, and returns `t` if so, `nil` otherwise.

Function: ratiop object

The `ratiop` predicate tests to see whether object is a number represented as a ratio, and returns `t` if so, `nil` otherwise.

Function: floatingp object

The `floatingp` predicate tests to see whether object is a floating point number represented as a float or a bigfloat, and returns `t` if so, `nil` otherwise.

Function: floatp object

This predicate tests whether its argument is a floating point number and returns `t` if so, `nil` otherwise.

`floatp` does not exist in Emacs versions 18 and earlier. If the bignum extension is present, it returns `nil` for a bigfloat.

Function: bigfloatp object

The `bigfloatp` predicate tests to see whether object is an floating point number represented as a bigfloat, and returns `t` if so, `nil` otherwise.

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## 9.6 Comparison of Numbers

To test numbers for numerical equality, you should normally use `=`, not `eq`. There can be many distinct floating point, bignum, and ratio number objects with the same numeric value. If you use `eq` to compare them, then you test whether two values are the same object. By contrast, `=` compares only the numeric values of the objects.

In versions before 21.5.18, each integer value had a unique Lisp object in XEmacs Lisp. Therefore, `eq` was equivalent to `=` where integers are concerned. Even with the introduction of bignums, it is sometimes convenient to use `eq` for comparing an unknown value with an integer, because `eq` does not report an error if the unknown value is not a number—it accepts arguments of any type. By contrast, `=` signals an error if the arguments are not numbers or markers. However, it is a good idea to use `=` if you can, even for comparing exact values, because two bignums or ratios with the same value will often not be the same object.

On the other hand, some functions, such as the string- and buffer-searching functions, will return an integer on success, but something else (usually `nil`) on failure. If it is known what the numerical subtype (float, bigfloat, or exact) of the returned object will be if it is a number, then the predicate `eql` can be used for comparison without signaling an error on some expected return values. Because of canonicalization, `eql` can be used to compare a fixnum value to something that might be a ratio; if the potential ratio value is representable as a fixnum, it will be canonicalized to fixnum before comparing. However, although floats and bigfloats are of different types for the purpose of comparisons via `eql`, two bigfloats of different precision that are `=` will always be `eql`.

 ```(eql 2 (string-match "ere" "there")) ⇒ t (eql 2 (string-match "ere" "three")) ⇒ nil (eql 2 2.0) ⇒ nil (= 2 (string-match "ere" "there")) ⇒ t (= 2 (string-match "ere" "three")) error--> Wrong type argument: number-char-or-marker-p, nil (= 2 2.0) ⇒ t ```

There is another wrinkle: because floating point arithmetic is not exact, it is often a bad idea to check for equality of two floating point values. Usually it is better to test for approximate equality. Here’s a function to do this:

 ```(defconst fuzz-factor 1.0e-6) (defun approx-equal (x y) (or (and (= x 0) (= y 0)) (< (/ (abs (- x y)) (max (abs x) (abs y))) fuzz-factor))) ```

Common Lisp note: Comparing numbers in Common Lisp always requires `=` because Common Lisp implements multi-word integers, and two distinct integer objects can have the same numeric value. XEmacs Lisp can have just one fixnum object for any given value because it has a limited range of fixnum values.

In addition to numbers, all of the following functions also accept characters and markers as arguments, and treat them as their number equivalents.

Function: = number &rest more-numbers

This function returns `t` if all of its arguments are numerically equal, `nil` otherwise.

 ```(= 5) ⇒ t (= 5 6) ⇒ nil (= 5 5.0) ⇒ t (= 5 5 6) ⇒ nil ```
Function: /= number &rest more-numbers

This function returns `t` if no two arguments are numerically equal, `nil` otherwise.

 ```(/= 5 6) ⇒ t (/= 5 5 6) ⇒ nil (/= 5 6 1) ⇒ t ```
Function: < number &rest more-numbers

This function returns `t` if the sequence of its arguments is monotonically increasing, `nil` otherwise.

 ```(< 5 6) ⇒ t (< 5 6 6) ⇒ nil (< 5 6 7) ⇒ t ```
Function: <= number &rest more-numbers

This function returns `t` if the sequence of its arguments is monotonically nondecreasing, `nil` otherwise.

 ```(<= 5 6) ⇒ t (<= 5 6 6) ⇒ t (<= 5 6 5) ⇒ nil ```
Function: > number &rest more-numbers

This function returns `t` if the sequence of its arguments is monotonically decreasing, `nil` otherwise.

Function: >= number &rest more-numbers

This function returns `t` if the sequence of its arguments is monotonically nonincreasing, `nil` otherwise.

Function: max number &rest more-numbers

This function returns the largest of its arguments.

 ```(max 20) ⇒ 20 (max 1 2.5) ⇒ 2.5 (max 1 3 2.5) ⇒ 3 ```
Function: min number &rest more-numbers

This function returns the smallest of its arguments.

 ```(min -4 1) ⇒ -4 ```

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## 9.7 Numeric Conversions

To convert an integer to floating point, use the function `float`.

Function: float number

This returns number converted to floating point. If number is already a floating point number, `float` returns it unchanged.

There are four functions to convert floating point numbers to integers; they differ in how they round. These functions accept integer arguments also, and return such arguments unchanged. They return multiple values, see (cl.info)Multiple values.

All these functions take optional divisor arguments, and if this argument is specified, the number argument is divided by divisor before the calculation is made. An `arith-error` results if divisor is 0.

Function: truncate number &optional divisor

This returns number, converted to an integer by rounding towards zero.

Function: floor number &optional divisor

This returns number, converted to an integer by rounding downward (towards negative infinity).

Function: ceiling number &optional divisor

This returns number, converted to an integer by rounding upward (towards positive infinity).

Function: round number &optional divisor

This returns number, converted to an integer by rounding towards the nearest integer.

Rounding a value equidistant between two integers chooses the even integer. GNU Emacs and older XEmacs did not guarantee this, and the direction of rounding depended on the underlying machine and the C implementation.

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## 9.8 Arithmetic Operations

XEmacs Lisp provides the traditional four arithmetic operations: addition, subtraction, multiplication, and division. Remainder and modulus functions supplement the division functions. The functions to add or subtract 1 are provided because they are traditional in Lisp and commonly used.

All of these functions except `%` return a floating point value if any argument is floating.

It is important to note that in XEmacs Lisp, arithmetic functions do not check for overflow. Thus `(1+ 134217727)` may evaluate to -134217728, depending on your hardware and whether your XEmacs supports bignums.

Function: 1+ number

This function returns number plus one. number may be a number, character or marker. Markers and characters are converted to integers.

For example,

 ```(setq foo 4) ⇒ 4 (1+ foo) ⇒ 5 ```

This function is not analogous to the C operator `++`—it does not increment a variable. It just computes a sum. Thus, if we continue,

 ```foo ⇒ 4 ```

If you want to increment the variable, you must use `setq`, like this:

 ```(setq foo (1+ foo)) ⇒ 5 ```

Now that the `cl` package is always available from lisp code, a more convenient and natural way to increment a variable is `(incf foo)`.

Function: 1- number

This function returns number minus one. number may be a number, character or marker. Markers and characters are converted to integers.

Function: abs number

This returns the absolute value of number.

Function: + &rest numbers

This function adds its arguments together. When given no arguments, `+` returns 0.

If any of the arguments are characters or markers, they are first converted to integers.

 ```(+) ⇒ 0 (+ 1) ⇒ 1 (+ 1 2 3 4) ⇒ 10 ```
Function: - &optional number &rest other-numbers

The `-` function serves two purposes: negation and subtraction. When `-` has a single argument, the value is the negative of the argument. When there are multiple arguments, `-` subtracts each of the other-numbers from number, cumulatively. If there are no arguments, an error is signaled.

If any of the arguments are characters or markers, they are first converted to integers.

 ```(- 10 1 2 3 4) ⇒ 0 (- 10) ⇒ -10 (-) ⇒ 0 ```
Function: * &rest numbers

This function multiplies its arguments together, and returns the product. When given no arguments, `*` returns 1.

If any of the arguments are characters or markers, they are first converted to integers.

 ```(*) ⇒ 1 (* 1) ⇒ 1 (* 1 2 3 4) ⇒ 24 ```
Function: / dividend &rest divisors

The `/` function serves two purposes: inversion and division. When `/` has a single argument, the value is the inverse of the argument. When there are multiple arguments, `/` divides dividend by each of the divisors, cumulatively, returning the quotient. If there are no arguments, an error is signaled.

If none of the arguments are floats, then the result is an integer. This means the result has to be rounded. On most machines, the result is rounded towards zero after each division, but some machines may round differently with negative arguments. This is because the Lisp function `/` is implemented using the C division operator, which also permits machine-dependent rounding. As a practical matter, all known machines round in the standard fashion.

If any of the arguments are characters or markers, they are first converted to integers.

If you divide by 0, an `arith-error` error is signaled. (See section Errors.)

 ```(/ 6 2) ⇒ 3 ``````(/ 5 2) ⇒ 2 (/ 25 3 2) ⇒ 4 (/ 3.0) ⇒ 0.3333333333333333 (/ -17 6) ⇒ -2 ```

The result of `(/ -17 6)` could in principle be -3 on some machines.

Function: % dividend divisor

This function returns the integer remainder after division of dividend by divisor. The arguments must be integers or markers.

For negative arguments, the remainder is in principle machine-dependent since the quotient is; but in practice, all known machines behave alike.

An `arith-error` results if divisor is 0.

 ```(% 9 4) ⇒ 1 (% -9 4) ⇒ -1 (% 9 -4) ⇒ 1 (% -9 -4) ⇒ -1 ```

For any two integers dividend and divisor,

 ```(+ (% dividend divisor) (* (/ dividend divisor) divisor)) ```

always equals dividend.

Function: mod dividend divisor

This function returns the value of dividend modulo divisor; in other words, the remainder after division of dividend by divisor, but with the same sign as divisor. The arguments must be numbers or markers.

Unlike `%`, `mod` returns a well-defined result for negative arguments. It also permits floating point arguments; it rounds the quotient downward (towards minus infinity) to an integer, and uses that quotient to compute the remainder.

An `arith-error` results if divisor is 0.

 ```(mod 9 4) ⇒ 1 ``````(mod -9 4) ⇒ 3 ``````(mod 9 -4) ⇒ -3 ``````(mod -9 -4) ⇒ -1 ``````(mod 5.5 2.5) ⇒ .5 ```

For any two numbers dividend and divisor,

 ```(+ (mod dividend divisor) (* (floor dividend divisor) divisor)) ```

always equals dividend, subject to rounding error if either argument is floating point. For `floor`, see Numeric Conversions.

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## 9.9 Rounding Operations

The functions `ffloor`, `fceiling`, `fround` and `ftruncate` take a floating point argument and return a floating point result whose value is a nearby integer. `ffloor` returns the nearest integer below; `fceiling`, the nearest integer above; `ftruncate`, the nearest integer in the direction towards zero; `fround`, the nearest integer.

All these functions take optional divisor arguments, and if this argument is specified, the number argument is divided by divisor before the calculation is made. An `arith-error` results if divisor is 0. Also, they return multiple values, see (cl.info)Multiple values; the second value is the remainder.

Function: ffloor number &optional divisor

This function rounds number to the next lower integral value, and returns that value as a floating point number.

Function: fceiling number &optional divisor

This function rounds number to the next higher integral value, and returns that value as a floating point number.

Function: ftruncate number &optional divisor

This function rounds number towards zero to an integral value, and returns that value as a floating point number.

Function: fround number &optional divisor

This function rounds number to the nearest integral value, and returns that value as a floating point number.

Rounding a value equidistant between two integral values chooses the even value. While this is specified by Common Lisp, GNU Emacs and older XEmacs did not make this guarantee, and the direction of rounding depended on the underlying machine and the C implementation.

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## 9.10 Bitwise Operations on Integers

In a computer, an integer is represented as a binary number, a sequence of bits (digits which are either zero or one). A bitwise operation acts on the individual bits of such a sequence. For example, shifting moves the whole sequence left or right one or more places, reproducing the same pattern “moved over”.

The bitwise operations in XEmacs Lisp apply only to integers.

Function: lsh integer1 count

`lsh`, which is an abbreviation for logical shift, shifts the bits in integer1 to the left count places, or to the right if count is negative, bringing zeros into the vacated bits. If count is negative, `lsh` shifts zeros into the leftmost (most-significant) bit, producing a positive result even if integer1 is negative. Contrast this with `ash`, below.

Here are two examples of `lsh`, shifting a pattern of bits one place to the left. We show only the low-order eight bits of the binary pattern; the rest are all zero.

 ```(lsh 5 1) ⇒ 10 ;; Decimal 5 becomes decimal 10. 00000101 ⇒ 00001010 (lsh 7 1) ⇒ 14 ;; Decimal 7 becomes decimal 14. 00000111 ⇒ 00001110 ```

As the examples illustrate, shifting the pattern of bits one place to the left produces a number that is twice the value of the previous number.

Shifting a pattern of bits two places to the left produces results like this (with 8-bit binary numbers):

 ```(lsh 3 2) ⇒ 12 ;; Decimal 3 becomes decimal 12. 00000011 ⇒ 00001100 ```

On the other hand, shifting one place to the right looks like this:

 ```(lsh 6 -1) ⇒ 3 ;; Decimal 6 becomes decimal 3. 00000110 ⇒ 00000011 ``````(lsh 5 -1) ⇒ 2 ;; Decimal 5 becomes decimal 2. 00000101 ⇒ 00000010 ```

As the example illustrates, shifting one place to the right divides the value of a positive integer by two, rounding downward.

The function `lsh`, like all XEmacs Lisp arithmetic functions, does not check for overflow, so shifting left can discard significant bits and change the sign of the number. For example, left shifting 134,217,727 produces -2 on a 28-bit machine:

 ```(lsh 134217727 1) ; left shift ⇒ -2 ```

In binary, in the 28-bit implementation, the argument looks like this:

 ```;; Decimal 134,217,727 0111 1111 1111 1111 1111 1111 1111 ```

which becomes the following when left shifted:

 ```;; Decimal -2 1111 1111 1111 1111 1111 1111 1110 ```
Function: ash integer1 count

`ash` (arithmetic shift) shifts the bits in integer1 to the left count places, or to the right if count is negative.

`ash` gives the same results as `lsh` except when integer1 and count are both negative. In that case, `ash` puts ones in the empty bit positions on the left, while `lsh` puts zeros in those bit positions.

Thus, with `ash`, shifting the pattern of bits one place to the right looks like this:

 ```(ash -6 -1) ⇒ -3 ;; Decimal -6 becomes decimal -3. 1111 1111 1111 1111 1111 1111 1010 ⇒ 1111 1111 1111 1111 1111 1111 1101 ```

In contrast, shifting the pattern of bits one place to the right with `lsh` looks like this:

 ```(lsh -6 -1) ⇒ 134217725 ;; Decimal -6 becomes decimal 134,217,725. 1111 1111 1111 1111 1111 1111 1010 ⇒ 0111 1111 1111 1111 1111 1111 1101 ```

Here are other examples:

 ``` ; 28-bit binary values (lsh 5 2) ; 5 = 0000 0000 0000 0000 0000 0000 0101 ⇒ 20 ; = 0000 0000 0000 0000 0000 0001 0100 ``````(ash 5 2) ⇒ 20 (lsh -5 2) ; -5 = 1111 1111 1111 1111 1111 1111 1011 ⇒ -20 ; = 1111 1111 1111 1111 1111 1110 1100 (ash -5 2) ⇒ -20 ``````(lsh 5 -2) ; 5 = 0000 0000 0000 0000 0000 0000 0101 ⇒ 1 ; = 0000 0000 0000 0000 0000 0000 0001 ``````(ash 5 -2) ⇒ 1 ``````(lsh -5 -2) ; -5 = 1111 1111 1111 1111 1111 1111 1011 ⇒ 4194302 ; = 0011 1111 1111 1111 1111 1111 1110 ``````(ash -5 -2) ; -5 = 1111 1111 1111 1111 1111 1111 1011 ⇒ -2 ; = 1111 1111 1111 1111 1111 1111 1110 ```
Function: logand &rest ints-or-markers

This function returns the “logical and” of the arguments: the nth bit is set in the result if, and only if, the nth bit is set in all the arguments. (“Set” means that the value of the bit is 1 rather than 0.)

For example, using 4-bit binary numbers, the “logical and” of 13 and 12 is 12: 1101 combined with 1100 produces 1100. In both the binary numbers, the leftmost two bits are set (i.e., they are 1’s), so the leftmost two bits of the returned value are set. However, for the rightmost two bits, each is zero in at least one of the arguments, so the rightmost two bits of the returned value are 0’s.

Therefore,

 ```(logand 13 12) ⇒ 12 ```

If `logand` is not passed any argument, it returns a value of -1. This number is an identity element for `logand` because its binary representation consists entirely of ones. If `logand` is passed just one argument, it returns that argument.

 ``` ; 28-bit binary values (logand 14 13) ; 14 = 0000 0000 0000 0000 0000 0000 1110 ; 13 = 0000 0000 0000 0000 0000 0000 1101 ⇒ 12 ; 12 = 0000 0000 0000 0000 0000 0000 1100 ``````(logand 14 13 4) ; 14 = 0000 0000 0000 0000 0000 0000 1110 ; 13 = 0000 0000 0000 0000 0000 0000 1101 ; 4 = 0000 0000 0000 0000 0000 0000 0100 ⇒ 4 ; 4 = 0000 0000 0000 0000 0000 0000 0100 ``````(logand) ⇒ -1 ; -1 = 1111 1111 1111 1111 1111 1111 1111 ```
Function: logior &rest ints-or-markers

This function returns the “inclusive or” of its arguments: the nth bit is set in the result if, and only if, the nth bit is set in at least one of the arguments. If there are no arguments, the result is zero, which is an identity element for this operation. If `logior` is passed just one argument, it returns that argument.

 ``` ; 28-bit binary values (logior 12 5) ; 12 = 0000 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0000 0101 ⇒ 13 ; 13 = 0000 0000 0000 0000 0000 0000 1101 ``````(logior 12 5 7) ; 12 = 0000 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0000 0101 ; 7 = 0000 0000 0000 0000 0000 0000 0111 ⇒ 15 ; 15 = 0000 0000 0000 0000 0000 0000 1111 ```
Function: logxor &rest ints-or-markers

This function returns the “exclusive or” of its arguments: the nth bit is set in the result if, and only if, the nth bit is set in an odd number of the arguments. If there are no arguments, the result is 0, which is an identity element for this operation. If `logxor` is passed just one argument, it returns that argument.

 ``` ; 28-bit binary values (logxor 12 5) ; 12 = 0000 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0000 0101 ⇒ 9 ; 9 = 0000 0000 0000 0000 0000 0000 1001 ``````(logxor 12 5 7) ; 12 = 0000 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0000 0101 ; 7 = 0000 0000 0000 0000 0000 0000 0111 ⇒ 14 ; 14 = 0000 0000 0000 0000 0000 0000 1110 ```
Function: lognot integer

This function returns the logical complement of its argument: the nth bit is one in the result if, and only if, the nth bit is zero in integer, and vice-versa.

 ```(lognot 5) ⇒ -6 ;; 5 = 0000 0000 0000 0000 0000 0000 0101 ;; becomes ;; -6 = 1111 1111 1111 1111 1111 1111 1010 ```

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## 9.11 Standard Mathematical Functions

These mathematical functions are available if floating point is supported (which is the normal state of affairs). They allow integers as well as floating point numbers as arguments.

Function: sin number
Function: cos number
Function: tan number

These are the ordinary trigonometric functions, with argument measured in radians.

Function: asin number

The value of `(asin number)` is a number between -pi/2 and pi/2 (inclusive) whose sine is number; if, however, number is out of range (outside [-1, 1]), then the result is a NaN.

Function: acos number

The value of `(acos number)` is a number between 0 and pi (inclusive) whose cosine is number; if, however, number is out of range (outside [-1, 1]), then the result is a NaN.

Function: atan number &optional number2

The value of `(atan number)` is a number between -pi/2 and pi/2 (exclusive) whose tangent is number.

If optional argument number2 is supplied, the function returns `atan2(number,number2)`.

Function: sinh number
Function: cosh number
Function: tanh number

These are the ordinary hyperbolic trigonometric functions.

Function: asinh number
Function: acosh number
Function: atanh number

These are the inverse hyperbolic trigonometric functions.

Function: exp number

This is the exponential function; it returns e to the power number. e is a fundamental mathematical constant also called the base of natural logarithms.

Function: log number &optional base

This function returns the logarithm of number, with base base. If you don’t specify base, the base `e` is used. If number is negative, the result is a NaN.

Function: log10 number

This function returns the logarithm of number, with base 10. If number is negative, the result is a NaN. `(log10 x)``(log x 10)`, at least approximately.

Function: expt x y

This function returns x raised to power y. If both arguments are integers and y is positive, the result is an integer; in this case, it is truncated to fit the range of possible integer values.

Function: sqrt number

This returns the square root of number. If number is negative, the value is a NaN.

Function: cube-root number

This returns the cube root of number.

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