Internally Derive stores all numbers as integers or reduced ratios of integers (fractions).  All numerical operations are performed using rational arithmetic.  Either exact, approximate, or mixed precision can be used.


Use the Precision field of the Options > Mode Settings > Simplification command19_L5FP to determine and/or change the precision mode and digits of precision.  The current precision mode is the highlighted in the mode field.  The minimum number of significant decimal digits used in approximate or mixed modes is displayed in the digits field.


When a new precision mode is selected, the command generates an expression of the form

Precision := mode

where mode is Approximate, Exact, or Mixed.  When a new number of digits of precision is entered, the command generates an expression of the form

PrecisionDigits := digits

where digits is a positive integer.  The precision mode and digits of precision can also be changed by entering on the expression entry line expressions of the above form using upper and lower case exactly as shown.  Exact is the factory default precision mode setting, and 10 is the factory default digits of precision.


If Exact is selected, irrational numbers are not approximated and as much memory as necessary is used to store rational numbers.  Exact precision mode is appropriate when you want an exact answer and must be sure that you have an exact answer.


Irrational numbers can not be represented exactly as ratios of integers.  Thus, in exact mode, radicals (surds) are left in symbolic form when simplified rather than being approximated.  For example, √4 simplifies to 2, but √3 remains symbolic.  In exact mode various techniques are used to simplify expressions involving nested radicals.  For example,

SQRT(4 - 2·SQRT(3))

simplifies to √3 - 1.


If Approximate is selected, irrational numbers and large rational numbers simplify to the simplest rational number that approximates the original number accurate to the current precision.


The number of significant digits of precision can be set in the Precision field to any desired positive integer.  However, the time required to perform rational operations roughly quadruples when the number of digits of precision is doubled.  In addition, huge rational numbers will quickly exhaust available memory.


Approximate mode can be significantly faster and require less memory than exact or mixed mode when rational operations result in numbers having large numerators and denominators.


Approximate mode is appropriate when you want to save time and space at the expense of some roundoff errors.  This may be acceptable if the expected roundoff error is negligible compared to errors in experimental data or compared to the accuracy required of the result.


If Mixed is selected, irrational numbers are approximated but rational numbers are not.  As with approximate precision mode, irrational numbers simplify to the simplest rational number that approximates the original number accurate to the current precision.


Mixed precision mode is appropriate when you are willing to approximate irrational operations to force numeric results but want all rational operations done exactly.


For example, in approximate mode with 6 digits of precision

SQRT(3422357/2313 - 1140443/771)

simplifies to 0.666622 an approximate result because rounding the rational numbers before they are subtracted results in significant roundoff error.  However, in mixed mode it simplifies to 2/3 an exact result because the difference of the two rational numbers is computed exactly.


Regardless of the number in the digits field, exact mode uses as many digits as necessary for exactness.  Thus, set the digits field to the precision you want for approximate calculations.


If you change the precision and issue another Options > Mode Settings > Simplification command19_L5FP, the number of digits displayed in the Precision field may exceed by one the precision you requested.  Derive stores rational numbers using enough bytes of memory to provide the current precision.  Since each byte provides more than one decimal digit of precision, the precision may be slightly more than you requested.


Most individual roundoff errors are a few digits more accurate than the indicated precision.  However, roundoff errors may accumulate in a sequence of operations.  Moreover, subtracting two nearly equal approximate numbers may yield a result having few or no significant digits.  This is called catastrophic cancellation.


For example, if n is a large positive integer, a difference of the form √(n²+1) - n leads to catastrophic cancellation since √(n² + 1) and n approximate to nearly equal numbers.  Thus, in approximate mode with 6 digits of precision

SQRT(10001) - 100

simplifies to 0.00499722.  However, only the first three digits of this result are correct.  This can be verified by increasing the digits of precision to 10 and resimplifying the above difference.


As opposed to the floating-point arithmetic used by most calculators and mathematical software, approximate rational arithmetic has the useful property that simple exact rational results are often obtained despite intermediate roundings or irrational approximations.


If an irrational expression is equivalent to a simple fraction, then approximate mode is more likely than mixed mode to yield that simple fraction.  Otherwise, mixed mode is likely to be more accurate.


In exact mode Derive tries to denest expressions involving nested radicals.  For example,

SQRT(5 + 2·SQRT(6))

simplifies to √3 + √2, and 

(243·SQRT(5) - 294·SQRT(3))^(1/3)

simplifies to 3√5 - 2√3.  However, it is difficult if not impossible to fully simplify all such expressions involving nested radicals (but we're working on it!).


Other Options > Mode Settings > Simplification command19_L5FP fields

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