The functions described here do the actual computational work of the
Calculator. In addition to these, note that any function described in
the main body of this manual may be called from Lisp; for example, if
the documentation refers to the `calc-sqrt`

[`sqrt`

] command,
this means `calc-sqrt`

is an interactive stack-based square-root
command and `sqrt`

(which `defmath`

expands to `calcFunc-sqrt`

)
is the actual Lisp function for taking square roots.

The functions `math-add`

, `math-sub`

, `math-mul`

,
`math-div`

, `math-mod`

, and `math-neg`

are not included
in this list, since `defmath`

allows you to write native Lisp
`+`

, `-`

, `*`

, `/`

, `%`

, and unary `-`

,
respectively, instead.

__Function:__ **normalize** *val*

(Full form: `math-normalize`

.)
Reduce the value `val` to standard form. For example, if `val`
is a fixnum, it will be converted to a bignum if it is too large, and
if `val` is a bignum it will be normalized by clipping off trailing
(i.e., most-significant) zero digits and converting to a fixnum if it is
small. All the various data types are similarly converted to their standard
forms. Variables are left alone, but function calls are actually evaluated
in formulas. For example, normalizing ``(+ 2 (calcFunc-abs -4))'` will
return 6.

If a function call fails, because the function is void or has the wrong
number of parameters, or because it returns `nil`

or calls
`reject-arg`

or `inexact-result`

, `normalize`

returns
the formula still in symbolic form.

If the current Simplification Mode is "none" or "numeric arguments
only," `normalize`

will act appropriately. However, the more
powerful simplification modes (like algebraic simplification) are
not handled by `normalize`

. They are handled by `calc-normalize`

,
which calls `normalize`

and possibly some other routines, such
as `simplify`

or `simplify-units`

. Programs generally will
never call `calc-normalize`

except when popping or pushing values
on the stack.

__Function:__ **evaluate-expr** *expr*

Replace all variables in `expr` that have values with their values,
then use `normalize`

to simplify the result. This is what happens
when you press the `=` key interactively.

__Macro:__ **with-extra-prec** *n body*

Evaluate the Lisp forms in `body` with precision increased by `n`
digits. This is a macro which expands to

(math-normalize (let ((calc-internal-prec (+ calc-internal-precn)))body))

The surrounding call to `math-normalize`

causes a floating-point
result to be rounded down to the original precision afterwards. This
is important because some arithmetic operations assume a number's
mantissa contains no more digits than the current precision allows.

__Function:__ **make-frac** *n d*

Build a fraction `` n:d'`. This is equivalent to calling

__Function:__ **make-float** *mant exp*

Build a floating-point value out of `mant` and `exp`, both
of which are arbitrary integers. This function will return a
properly normalized float value, or signal an overflow or underflow
if `exp` is out of range.

__Function:__ **make-sdev** *x sigma*

Build an error form out of `x` and the absolute value of `sigma`.
If `sigma` is zero, the result is the number `x` directly.
If `sigma` is negative or complex, its absolute value is used.
If `x` or `sigma` is not a valid type of object for use in
error forms, this calls `reject-arg`

.

__Function:__ **make-intv** *mask lo hi*

Build an interval form out of `mask` (which is assumed to be an
integer from 0 to 3), and the limits `lo` and `hi`. If
`lo` is greater than `hi`, an empty interval form is returned.
This calls `reject-arg`

if `lo` or `hi` is unsuitable.

__Function:__ **sort-intv** *mask lo hi*

Build an interval form, similar to `make-intv`

, except that if
`lo` is less than `hi` they are simply exchanged, and the
bits of `mask` are swapped accordingly.

__Function:__ **make-mod** *n m*

Build a modulo form out of `n` and the modulus `m`. Since modulo
forms do not allow formulas as their components, if `n` or `m`
is not a real number or HMS form the result will be a formula which
is a call to `makemod`

, the algebraic version of this function.

__Function:__ **float** *x*

Convert `x` to floating-point form. Integers and fractions are
converted to numerically equivalent floats; components of complex
numbers, vectors, HMS forms, date forms, error forms, intervals, and
modulo forms are recursively floated. If the argument is a variable
or formula, this calls `reject-arg`

.

__Function:__ **compare** *x y*

Compare the numbers `x` and `y`, and return *-1* if
``(lessp x y)'`, 1 if

__Function:__ **numdigs** *n*

Return the number of digits of integer `n`, effectively
``ceil(log10( n))'`, but much more efficient. Zero is
considered to have zero digits.

__Function:__ **scale-int** *x n*

Shift integer `x` left `n` decimal digits, or right *- n*
digits with truncation toward zero.

__Function:__ **scale-rounding** *x n*

Like `scale-int`

, except that a right shift rounds to the nearest
integer rather than truncating.

__Function:__ **fixnum** *n*

Return the integer `n` as a fixnum, i.e., a native Lisp integer.
If `n` is outside the permissible range for Lisp integers (usually
24 binary bits) the result is undefined.

__Function:__ **sqr** *x*

Compute the square of `x`; short for ``(* x x)'`.

__Function:__ **quotient** *x y*

Divide integer `x` by integer `y`; return an integer quotient
and discard the remainder. If `x` or `y` is negative, the
direction of rounding is undefined.

__Function:__ **idiv** *x y*

Perform an integer division; if `x` and `y` are both nonnegative
integers, this uses the `quotient`

function, otherwise it computes
``floor( x/y)'`. Thus the result is well-defined but
slower than for

`quotient`

.
__Function:__ **imod** *x y*

Divide integer `x` by integer `y`; return the integer remainder
and discard the quotient. Like `quotient`

, this works only for
integer arguments and is not well-defined for negative arguments.
For a more well-defined result, use ``(% x y)'`.

__Function:__ **idivmod** *x y*

Divide integer `x` by integer `y`; return a cons cell whose
`car`

is ``(quotient x y)'` and whose

`cdr`

is
__Function:__ **pow** *x y*

Compute `x` to the power `y`. In `defmath`

code, this can
also be written ``(^ x y)'` or

__Function:__ **abs-approx** *x*

Compute a fast approximation to the absolute value of `x`. For
example, for a rectangular complex number the result is the sum of
the absolute values of the components.

The function ``(pi)'` computes ``pi'` to the current precision.
Other related constant-generating functions are `two-pi`

,
`pi-over-2`

, `pi-over-4`

, `pi-over-180`

, `sqrt-two-pi`

,
`e`

, `sqrt-e`

, `ln-2`

, and `ln-10`

. Each function
returns a floating-point value in the current precision, and each uses
caching so that all calls after the first are essentially free.

__Macro:__ **math-defcache** `func` `initial` `form`

This macro, usually used as a top-level call like `defun`

or
`defvar`

, defines a new cached constant analogous to `pi`

, etc.
It defines a function `func`

which returns the requested value;
if `initial` is non-`nil`

it must be a ``(float ...)'`
form which serves as an initial value for the cache. If `func`
is called when the cache is empty or does not have enough digits to
satisfy the current precision, the Lisp expression `form` is evaluated
with the current precision increased by four, and the result minus its
two least significant digits is stored in the cache. For example,
calling ``(pi)'` with a precision of 30 computes ``pi'` to 34
digits, rounds it down to 32 digits for future use, then rounds it
again to 30 digits for use in the present request.

If the current angular mode is Degrees or HMS, this function returns the
integer 360. In Radians mode, this function returns either the
corresponding value in radians to the current precision, or the formula
``2*pi'`, depending on the Symbolic Mode. There are also similar
function `half-circle`

and `quarter-circle`

.

__Function:__ **power-of-2** *n*

Compute two to the integer power `n`, as a (potentially very large)
integer. Powers of two are cached, so only the first call for a
particular `n` is expensive.

__Function:__ **integer-log2** *n*

Compute the base-2 logarithm of `n`, which must be an integer which
is a power of two. If `n` is not a power of two, this function will
return `nil`

.

__Function:__ **div-mod** *a b m*

Divide `a` by `b`, modulo `m`. This returns `nil`

if
there is no solution, or if any of the arguments are not integers.

__Function:__ **pow-mod** *a b m*

Compute `a` to the power `b`, modulo `m`. If `a`,
`b`, and `m` are integers, this uses an especially efficient
algorithm. Otherwise, it simply computes ``(% (^ a b) m)'`.

__Function:__ **isqrt** *n*

Compute the integer square root of `n`. This is the square root
of `n` rounded down toward zero, i.e., ``floor(sqrt( n))'`.
If

__Function:__ **to-hms** *a ang*

Convert the argument `a` into an HMS form. If `ang` is specified,
it is the angular mode in which to interpret `a`, either `deg`

or `rad`

. Otherwise, the current angular mode is used. If `a`
is already an HMS form it is returned as-is.

__Function:__ **from-hms** *a ang*

Convert the HMS form `a` into a real number. If `ang` is specified,
it is the angular mode in which to express the result, otherwise the
current angular mode is used. If `a` is already a real number, it
is returned as-is.

__Function:__ **to-radians** *a*

Convert the number or HMS form `a` to radians from the current
angular mode.

__Function:__ **from-radians** *a*

Convert the number `a` from radians to the current angular mode.
If `a` is a formula, this returns the formula ``deg( a)'`.

__Function:__ **to-radians-2** *a*

Like `to-radians`

, except that in Symbolic Mode a degrees to
radians conversion yields a formula like `` a*pi/180'`.

__Function:__ **from-radians-2** *a*

Like `from-radians`

, except that in Symbolic Mode a radians to
degrees conversion yields a formula like `` a*180/pi'`.

__Function:__ **random-digit**

Produce a random base-1000 digit in the range 0 to 999.

__Function:__ **random-digits** *n*

Produce a random `n`-digit integer; this will be an integer
in the interval ``[0, 10^ n)'`.

__Function:__ **random-float**

Produce a random float in the interval ``[0, 1)'`.

__Function:__ **prime-test** *n iters*

Determine whether the integer `n` is prime. Return a list which has
one of these forms: ``(nil f)'` means the number is non-prime
because it was found to be divisible by

`prime-test`

returns "maybe,"
you can call it again with the same `prime-test`

remembers where it left off.
__Function:__ **to-simple-fraction** *f*

If `f` is a floating-point number which can be represented exactly
as a small rational number. return that number, else return `f`.
For example, 0.75 would be converted to 3:4. This function is very
fast.

__Function:__ **to-fraction** *f tol*

Find a rational approximation to floating-point number `f` to within
a specified tolerance `tol`; this corresponds to the algebraic
function `frac`

, and can be rather slow.

__Function:__ **quarter-integer** *n*

If `n` is an integer or integer-valued float, this function
returns zero. If `n` is a half-integer (i.e., an integer plus
*1:2* or 0.5), it returns 2. If `n` is a quarter-integer,
it returns 1 or 3. If `n` is anything else, this function
returns `nil`

.

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