Question Engine 2:Numerical tolerances

Analysis of the various tolerances for numerical and calculated question type answer (MDL-31837)

This page is currently built and NOT completed Pierre Pichet 22:59, 14 August 2012 (WST)

In grading a numerical response, the (student) numerical value is compared to the answer numerical value.

This comparison allow a tolerance that is associated with the answer and that can be expressed in various ways.

For calculated question answers the tolerance can be of 3 different types: relative, nominal or geometric.

The following is from http://www.php.net/manual/en/language.types.float.php

The size of a float is platform-dependent, although a maximum of ~1.8e308 with a precision of roughly 14 decimal digits is a common value (the 64 bit IEEE format).

Warning Floating point precision

Floating point numbers have limited precision. Although it depends on the system, PHP typically uses the IEEE 754 double precision format, which will give a maximum relative error due to rounding in the order of 1.11e-16. Non elementary arithmetic operations may give larger errors, and, of course, error propagation must be considered when several operations are compounded.

Additionally, rational numbers that are exactly representable as floating point numbers in base 10, like 0.1 or 0.7, do not have an exact representation as floating point numbers in base 2, which is used internally, no matter the size of the mantissa. Hence, they cannot be converted into their internal binary counterparts without a small loss of precision. This can lead to confusing results: for example, floor((0.1+0.7)*10) will usually return 7 instead of the expected 8, since the internal representation will be something like 7.9999999999999991118....

So never trust floating number results to the last digit, and do not compare floating point numbers directly for equality. If higher precision is necessary, the arbitrary precision math functions and gmp functions are available.

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