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##### 18.5.7.5 Computational Functions

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-prec n)))
>                   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 ‘`(normalize (list 'frac n d))`’, but more efficient.

— 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 ‘`(lessp y x)`’, 0 if ‘`(math-equal x y)`’, or 2 if the order is undefined or cannot be determined.

— 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 ‘`(imod x y)`’.

— Function: pow x y

> Compute x to the power y. In `defmath` code, this can also be written ‘`(^ x y)`’ or ‘`(expt x y)`’.

— 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.

— Function: pi

> 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`, `ln-10`, `phi` and `gamma-const`. 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.

— Function: full-circle symb

> 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 n is itself an integer, the computation is especially efficient.

— 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 f; ‘`(nil)`’ means it was found to be non-prime by table look-up (so no factors are known); ‘`(nil unknown)`’ means it is definitely non-prime but no factors are known because n was large enough that Fermat's probabilistic test had to be used; ‘`(t)`’ means the number is definitely prime; and ‘`(maybe i p)`’ means that Fermat's test, after i iterations, is p percent sure that the number is prime. The iters parameter is the number of Fermat iterations to use, in the case that this is necessary. If `prime-test` returns “maybe,” you can call it again with the same n to get a greater certainty; `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`.