# The transcendentals Module#

The transcendentals module, exported from the common-dylan library, provides a set of generic functions for ANSI C-like behavior over real numbers. The restrictions and error cases described in this document are intended to be the same as they are in ANSI C.

Because implementation of these functions might be by a standard library for transcendentals accessed by a foreign function interface, the exact precision and algorithms (and hence, the exact results) for all of these functions is explicitly unspecified.

Note, however, that a program may expect the following, even in libraries that are implemented by calling foreign libraries:

• Domain and range errors should be signalled as Dylan errors.

• Floating point precision contagion must obey Dylan rules. That is, functions called on single precision values return single precision results, and functions on double precision values return double precision results. When a function (e.g., `^`, `atan2`, etc.) accepts two arguments, if either argument is a double precision value then the result is also double precision.

As a rule this module does not automatically convert integer values to floating point values. Callers should do so explicitly, so as to choose the appropriate floating point type for their needs.

Complex numbers are not implemented. If the result of calling any transcendentals function would be a complex number `<error>` is signalled.

## Reference#

This section contains a reference entry for each item exported from the transcendentals module.

### Constants#

\$single-e Constant#

The value of e, the base of natural logarithms, as a single precision floating point number.

Type:

`<single-float>`

\$double-e Constant#

The value of e, the base of natural logarithms, as a double precision floating point number.

Type:

`<double-float>`

\$single-pi Constant#

The value of π as a single precision floating point number.

Type:

`<single-float>`

\$double-pi Constant#

The value of π as a double precision floating point number.

Type:

`<double-float>`

### Functions#

^(<single-float>, <single-float>) Method#

Single precision floating point implementation of `^`.

Signature:

base ^ exponent => y

Returns base raised to the power exponent as a `<single-float>`. If base is `0` and exponent is not positive, an error is signalled. If base is negative and exponent is not an integer, an error is signalled.

^(<double-float>, <double-float>) Method#

Double precision floating point implementation of `^`.

Signature:

base ^ exponent => y

Returns base raised to the power exponent as a `<double-float>`. If base is `0` and exponent is not positive, an error is signalled. If base is negative and exponent is not an integer, an error is signalled.

^(<single-float>, <double-float>) Method#

Converts the first argument to `<double-float>` and calls `^(<double-float>, <double-float>)`.

^(<double-float>, <single-float>) Method#

Converts the second argument to `<double-float>` and calls `^(<double-float>, <double-float>)`.

acos Open Generic function#
Signature:

acos(x) => y

Parameters:
• x – An instance of type `<number>`. The angle, in radians. If x is not in the range `[-1,1]`, an error is signalled.

Values:

Returns the arc cosine of its argument. The floating point precision of the result is given by the precision of x.

acos(<single-float>) Method#

Single precision floating point implementation of `acos`. Returns a `<single-float>`.

acos(<double-float>) Method#

Double precision floating point implementation of `acos`. Returns a `<double-float>`.

acosh Open Generic function#
Signature:

acosh(x) => y

Parameters:
Values:

Returns the hyperbolic arc cosine of its argument. The floating point precision of the result is given by the precision of x.

acosh(<single-float>) Method#

Single precision floating point implementation of `acosh`. Returns a `<single-float>`.

acosh(<double-float>) Method#

Double precision floating point implementation of `acosh`. Returns a `<double-float>`.

asin Generic function#
Signature:

asin(x) => y

Parameters:
• x – An instance of type `<number>`. The angle, in radians. If x is not in the range [-1,+1], an error is signalled.

Values:

Returns the arc sine of its argument. The floating point precision of the result is given by the precision of x.

asin(<single-float>) Sealed Method#

Single precision floating point implementation of `asin`. Returns a `<single-float>`.

asin(<double-float>) Sealed Method#

Double precision floating point implementation of `asin`. Returns a `<double-float>`.

asinh Generic function#
Signature:

asinh(x) => y

Parameters:
Values:

Returns the hyperbolic arc sine of its argument. The floating point precision of the result is given by the precision of x.

asinh(<single-float>) Sealed Method#

Single precision floating point implementation of `asinh`. Returns a `<single-float>`.

asinh(<double-float>) Sealed Method#

Double precision floating point implementation of `asinh`. Returns a `<double-float>`.

atan Generic function#
Signature:

atan(x) => y

Parameters:
• x – An instance of type `<number>`. The angle, in radians. If x is not in the range [-1,+1], an error is signalled.

Values:

Returns the arc tangent of its argument. The floating point precision of the result is given by the precision of x.

atan(<single-float>) Sealed Method#

Single precision floating point implementation of `atan`. Returns a `<single-float>`.

atan(<double-float>) Sealed Method#

Double precision floating point implementation of `atan`. Returns a `<double-float>`.

atan2 Generic function#
Signature:

atan2(x, y) => z

Parameters:
Values:

Returns the arc tangent of x divided by y. x may be zero if y is not zero. The signs of x and y are used to derive what quadrant the angle falls in.

atan2(<single-float>, <single-float>) Sealed Method#

Single precision floating point implementation of `atan2`. Returns a `<single-float>`.

atan2(<double-float>, <double-float>) Sealed Method#

Double precision floating point implementation of `atan2`. Returns a `<double-float>`.

atan2(<single-float>, <double-float>) Sealed Method#

Converts the first argument to `<double-float>` and calls `atan2(<double-float>, <double-float>)`.

atan2(<double-float>, <single-float>) Sealed Method#

Converts the second argument to `<double-float>` and calls `atan2(<double-float>, <double-float>)`.

atanh Generic function#
Signature:

atanh(x) => y

Parameters:
Values:

Returns the hyperbolic arc tangent of its argument. The floating point precision of the result is given by the precision of x.

atanh(<single-float>) Sealed Method#

Single precision floating point implementation of `atanh`. Returns a `<single-float>`.

atanh(<double-float>) Sealed Method#

Double precision floating point implementation of `atanh`. Returns a `<double-float>`.

cos Generic function#
Signature:

cos(x) => y

Parameters:
Values:

Returns the cosine of its argument. The floating point precision of the result is given by the precision of x.

cos(<single-float>) Sealed Method#

Single precision floating point implementation of `cos`. Returns a `<single-float>`.

cos(<double-float>) Sealed Method#

Double precision floating point implementation of `cos`. Returns a `<double-float>`.

cosh Generic function#
Signature:

cosh(x) => y

Parameters:
Values:

Returns the hyperbolic cosine of its argument. The floating point precision of the result is given by the precision of x.

cosh(<single-float>) Sealed Method#

Single precision floating point implementation of `cosh`. Returns a `<single-float>`.

cosh(<double-float>) Sealed Method#

Double precision floating point implementation of `cosh`. Returns a `<double-float>`.

exp Generic function#
Signature:

exp(x) => y

Parameters:
Values:

Returns e, the base of natural logarithms, raised to the power x. The floating point precision is given by the precision of x.

exp(<single-float>) Sealed Method#

Single precision floating point implementation of `exp`. Returns a `<single-float>`.

exp(<double-float>) Sealed Method#

Double precision floating point implementation of `exp`. Returns a `<double-float>`.

hypot Generic function#
Signature:

hypot(x, y) => z

Parameters:
Values:

Returns the Euclidian distance without unnecessary overflow or underflow.

hypot(<single-float>, <single-float>) Method#

Returns the Euclidian distance as a `<single-float>` without unnecessary overflow or underflow.

hypot(<double-float>, <double-float>) Method#

Returns the Euclidian distance as a `<double-float>` without unnecessary overflow or underflow.

hypot(<single-float>, <double-float>) Method#

Converts the first argument to `<double-float>` and calls `hypot(<double-float>, <double-float>)`.

hypot(<double-float>, <single-float>) Method#

Converts the second argument to `<double-float>` and calls `hypot(<double-float>, <double-float>)`.

isqrt Function#
Signature:

isqrt(x) => y

Parameters:
Values:

Returns the integer square root of x, that is the greatest integer less than or equal to the exact positive square root of x. If `x < 0`, an error is signalled.

`sqrt`

log Generic function#

Returns the natural logarithm of its argument.

Signature:

log(x) => y

Parameters:
Values:

Returns the natural logarithm of x to the base e. If `x <= 0`, an error is signalled. The floating point precision of the result is given by the precision of x.

log(<single-float>) Method#
Signature:

log(x) => y

Parameters:
Values:

Returns the natural logarithm of x to the base e as a `<single-float>`.

log(<double-float>) Method#
Signature:

log(x) => y

Parameters:
Values:

Returns the natural logarithm of x to the base e as a `<single-float>`.

logn Function#

Returns the logarithm of its argument to the given base.

Signature:

logn(x, base) => y

Parameters:
Values:

Returns the logarithm of x to the base base. If `x <= 0` or ```base <= 1```, an error is signalled. The floating point precision of the result is given by the precision of x.

Note

In practice both x and base must be instances of `<float>` since they are passed directly to `log`, which only has methods on `<float>`.

ilog2 Function#
Signature:

ilog2(x) => y

Parameters:
Values:

Returns the integer base 2 logarithm of x, truncated to an `<integer>`. That is, it returns the greatest integer less than or equal to the exact base 2 logarithm of x.

sin Generic function#
Signature:

sin(x) => y

Parameters:
Values:

Returns the sine of its argument. The floating point precision of the result is given by the precision of x.

sin(<single-float>) Sealed Method#

Single precision floating point implementation of `sin`. Returns a `<single-float>`.

sin(<double-float>) Sealed Method#

Double precision floating point implementation of `sin`. Returns a `<double-float>`.

sincos Generic function#
Signature:

sincos(x) => (sin, cos)

Parameters:
Values:

Returns both the sine and the cosine of its argument. The floating point precision of the results is given by the precision of x. In some implementations `sincos` may have better performance than calling `sin(x)` and `cos(x)` separately.

sincos(<single-float>) Sealed Method#

Single precision floating point implementation of `sincos`. Returns a `<single-float>`.

sincos(<double-float>) Sealed Method#

Double precision floating point implementation of `sincos`. Returns a `<double-float>`.

sinh Generic function#
Signature:

sinh(x) => y

Parameters:
Values:

Returns the hyperbolic sine of its argument. The floating point precision of the result is given by the precision of x.

sinh(<single-float>) Sealed Method#

Single precision floating point implementation of `sinh`. Returns a `<single-float>`.

sinh(<double-float>) Sealed Method#

Double precision floating point implementation of `sinh`. Returns a `<double-float>`.

sqrt Generic function#
Signature:

sqrt(x) => y

Parameters:
Values:

Returns the square root of x. If x is less than zero an error is signalled. The floating point precision of the result is given by the precision of x.

`isqrt`

sqrt(<single-float>) Sealed Method#

Single precision floating point implementation of `sqrt`. Returns a `<single-float>`.

sqrt(<double-float>) Sealed Method#

Double precision floating point implementation of `sqrt`. Returns a `<double-float>`.

tan Generic function#
Signature:

tan(x) => y

Parameters:
Values:

Returns the tangent of x. The floating point precision of the result is given by the precision of x.

tan(<single-float>) Sealed Method#

Single precision floating point implementation of `tan`. Returns a `<single-float>`.

tan(<double-float>) Sealed Method#

Double precision floating point implementation of `tan`. Returns a `<double-float>`.

tanh Generic function#
Signature:

tanh(x) => y

Parameters:
Values:

Returns the hyperbolic tangent of x. The floating point precision of the result is given by the precision of x.

Single precision floating point implementation of `tanh`. Returns a `<single-float>`.
Double precision floating point implementation of `tanh`. Returns a `<double-float>`.