Floats
======

A floating-point number or "float" is a number stored in scientific
notation.  The number of significant digits in the fractional part is
governed by the current floating precision (See Precision).  The
range of acceptable values is from `10^-3999999' (inclusive) to
`10^4000000' (exclusive), plus the corresponding negative values and
zero.

Calculations that would exceed the allowable range of values (such as
`exp(exp(20))') are left in symbolic form by Calc.  The messages
"floating-point overflow" or "floating-point underflow" indicate that
during the calculation a number would have been produced that was too
large or too close to zero, respectively, to be represented by Calc.
This does not necessarily mean the final result would have overflowed,
just that an overflow occurred while computing the result.  (In fact,
it could report an underflow even though the final result would have
overflowed!)

If a rational number and a float are mixed in a calculation, the
result will in general be expressed as a float.  Commands that require
an integer value (such as `k g' [`gcd']) will also accept
integer-valued floats, i.e., floating-point numbers with nothing after
the decimal point.

Floats are identified by the presence of a decimal point and/or an
exponent.  In general a float consists of an optional sign, digits
including an optional decimal point, and an optional exponent
consisting of an `e', an optional sign, and up to seven exponent
digits.  For example, `23.5e-2' is 23.5 times ten to the minus-second
power, or 0.235.

Floating-point numbers are normally displayed in decimal notation with
all significant figures shown.  Exceedingly large or small numbers are
displayed in scientific notation.  Various other display options are
available.  See Float Formats.

Floating-point numbers are stored in decimal, not binary.  The result
of each operation is rounded to the nearest value representable in the
number of significant digits specified by the current precision,
rounding away from zero in the case of a tie.  Thus (in the default
display mode) what you see is exactly what you get.  Some operations
such as square roots and transcendental functions are performed with
several digits of extra precision and then rounded down, in an effort
to make the final result accurate to the full requested precision.
However, accuracy is not rigorously guaranteed.  If you suspect the
validity of a result, try doing the same calculation in a higher
precision.  The Calculator's arithmetic is not intended to be
IEEE-conformant in any way.

While floats are always *stored* in decimal, they can be entered and
displayed in any radix just like integers and fractions.  The notation
`RADIX#DDD.DDD' is a floating-point number whose digits are in the
specified radix.  Note that the `.'  is more aptly referred to as a
"radix point" than as a decimal point in this case.  The number
`8#123.4567' is defined as `8#1234567 * 8^-4'.  If the radix is 14 or
less, you can use `e' notation to write a non-decimal number in
scientific notation.  The exponent is written in decimal, and is
considered to be a power of the radix: `8#1234567e-4'.  If the radix
is 15 or above, the letter `e' is a digit, so scientific notation must
be written out, e.g., `16#123.4567*16^2'.  The first two exercises of
the Modes Tutorial explore some of the properties of non-decimal
floats.