gawkNext: MPFR Extension Reference [Contents][Index]
This file documents the MPFR numeric extension of GNU Awk (gawk)
version 4.1.1 and later.
Copyright (C) 2000–2002, 2004–2007, 2014 Free Software Foundation, Inc.
This is Edition 1.0 of MPFR Processing With gawk,
for the 1.0.3 (or later) version of the
MPFR extension of the GNU implementation of AWK.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with the Invariant Sections being “GNU General Public License”, with the Front-Cover Texts being “A GNU Manual”, and with the Back-Cover Texts as in (a) below. A copy of the license is included in the section entitled “GNU Free Documentation License”.
| • MPFR Extension Reference | ||
| • GNU Free Documentation License | ||
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MPFR is a portable library for arbitrary precision arithmetic on floating-point numbers. This means you can use the MPFR extension to perform calculations on numbers with a precision that is much higher (or lower, if you want) than the usual floating point numbers allow (as defined in the IEEE 754-1985 standard) .
| • Who Needs Arbitrary Precision Arithmetic ? | ||
| • Nullary Functions | ||
| • Unary Functions | ||
| • Binary Functions | ||
| • Nullary Predicates | ||
| • Unary Predicates | ||
| • Binary Predicates | ||
| • Input and Output of Numbers |
Next: Nullary Functions, Up: MPFR Extension Reference [Contents][Index]
To many users, it is not obvious why they should actually need this special kind of numbers with arbitrary precision. Two by two is four, who needs more ? For most calculations in everyday life (summing up prices or distances, calculating gas prices including VAT), the precision of your pocket calculator and your computer is indeed good enough. But if you go just a little further and evaluate the following polynomial, some doubts are cast on the capabilities of your software (example taken from the PASCAL-XSC Language Reference with Examples, page 188).
awk 'BEGIN {x=665857; y=470832; print x^4 - 4 * y^4 - 4 * y^2 }'
11885568
What is so surprising about this result is that it is wrong.
Not just a little bit, but completely wrong when you compare it
to the exact result, which is 1.0. Even worse, the software
doesn’t give you any clue. Rest assured that it is
not AWK’s fault. AWK relies on the arithmetic implemented in the
underlying operating system (which will produce the same result
no matter which programming language you use for the calculation).
So, what can MPFR do better about this ? First, the problem has to be recast in a different syntax. The usual arithmetic operators of AWK have to be replaced by the equivalent MPFR operators, making it a bit harder to read the program. The following example uses the MPFR extension to evaluate the same polynomial.
gawk -l mpfr 'BEGIN {x=665857; y=470832; \
print mpfr_sub(mpfr_sub(mpfr_pow(x, 4), mpfr_mul(4, mpfr_pow(y, 4))), 4 * y^2)}'
1.1885568000000000E7
By default, the MPFR extension calculates with the same precision (53 bits in the mantissa) as the usual IEEE 754 compatible operators implemented in the operating system. Thus, the result is the same as above. We see no advantage up to now. So, how can we eventually exploit the arbitrary precision capabilities of MPFR ? We have to tell MPFR to use some more bits in the mantissa of the numbers. In this case, 80 bits are enough.
gawk -l mpfr 'BEGIN {MPFR_PRECISION = 80; x=665857; y=470832; \
print mpfr_sub(mpfr_sub(mpfr_pow(x, 4), mpfr_mul(4, mpfr_pow(y, 4))), 4 * y^2)}'
1.0000000000000000000000000
You can see, when calculating with numbers that have 80 bits
in their mantissa, the result of the whole evaluation is correct (1.0).
Now, look at the program again. Notice the end of the polynomial.
The last term of the polynomial has not been recast in terms of
MPFR operators — the usual operators are still used. This example
demonstrates that you can mix ordinary numbers with the long numbers
returned by MPFR. Mixing ordinary numbers with long numbers is quite
convenient and improves readability of the program. But (from an analytic
point of view), this is bad practice. Ordinary numbers are potentially
less precise and one such term in a polynomial might spoil the complete
evaluation. In the case of the polynomial evaluation above it doesn’t
matter (because the term is only quadratic in y, requiring not
as long a mantissa as the quadric terms in x and y).
But in the more general cases (where the variables actually vary
and are not constant), you should do the complete evaluation in terms
of MPFR functions.
Let’s summarize: MPFR is a portable library for arbitrary precision arithmetic on floating-point numbers. The precision in bits can be set exactly to any valid value for each variable. The semantics of a calculation in MPFR is specified as follows: Compute the requested operation exactly (with infinite accuracy), and round the result to the precision of the destination variable, with the given rounding mode. The MPFR floating-point functions are intended to be a smooth extension of the IEEE 754-1985 arithmetic.
The internal representation of the numbers is not visible to the user of the MPFR extension. To the user, the numbers appear as strings of varying length. As a general rule, all MPFR functions return the result of the numerical calculation as a string containing a number. Each initialization and each calculation of a variable is controlled by the following global variables:
The remaining sections of this appendix contain a list of all
functions provided by the MPFR extension. Notice that only the
functions listed here can actually be used — some obsolete
legacy functions of old MPFR versions are not supported.
Supported functions can only be used after either invoking
gawk with a command-line argument of -l mpfr
or by inserting @load "mpfr" in your script. Optional
parameters are enclosed in square brackets ([ ]).
In the following sections, the functions are presented in groups.
The distinction between groups is based on the
arity (the
signature) of the function’s parameters and their return
values.
Next: Unary Functions, Previous: Who Needs Arbitrary Precision Arithmetic ?, Up: MPFR Extension Reference [Contents][Index]
Nullary functions take no (null) arguments, but they return some useful number. These functions are meant to provide you with the best approximation of a specific constant that is possible under the given circumstance (chosen precision, number base and rounding).
The following functions return the base-e logarithm of 2, the value of Pi, of Euler’s constant 0.577... respectively, rounded in the currently set direction MPFR_ROUND.
The following examples will not only show you how to use nullary functions, they will also demonstrate the limitations that are inherent to any implementation of arithmetical operators. It will not surprise you that it is easy to print the famous constant Pi to a thousand binary digits.
gawk -l mpfr 'BEGIN {MPFR_PRECISION=1000; print "pi = " mpfr_const_pi() }'
pi = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127360
You could easily change the example to let it print Pi to a million binary digits (the calculation would take just a few seconds more). But what about working with a precision of only four bits ?
gawk -l mpfr 'BEGIN {MPFR_PRECISION=4; print "pi = " mpfr_const_pi() }'
pi = 3.25
You know that the result 3.25 is wrong, but is it really that wrong ?
What actually is the right value of Pi ?
Is it the one in the previous example ? No, none of them
is really exact. Like many other numbers, Pi has
infinitely many places. Every representation of such a
number in floating-point arithmetic can only be an
approximation (3.25 if you have only four bits in the mantissa
and rounding is done to the nearest number). If any calculation
with floating-point numbers returns an exact result
to you, then you were just in luck. Exact results are
an exception, not the rule.
Next: Binary Functions, Previous: Nullary Functions, Up: MPFR Extension Reference [Contents][Index]
Unary functions take one argument and return some useful number. These functions are meant to provide you with the best approximation of a specific function that is possible under the given circumstance (chosen precision, number base and rounding). The names of the functions in the following list should explain what is meant. In case of doubt, refer to the documentation of the MPFR library.
Next: Nullary Predicates, Previous: Unary Functions, Up: MPFR Extension Reference [Contents][Index]
Binary functions take two arguments and return a result. To many users, these functions are the most commonly needed functions. Among others, they provide the four elementary arithmetic operations: addition, subtraction, multiplication and division. These functions are meant to provide you with the best approximation of a specific function that is possible under the given circumstance (chosen precision, number base and rounding). The names of the functions in the following list should explain what is meant. In case of doubt, refer to the documentation of the MPFR library.
Next: Unary Predicates, Previous: Binary Functions, Up: MPFR Extension Reference [Contents][Index]
Predicates are boolean-valued functions. They are indicator-functions, testing for some condition and revealing the presence or absence of the condition. Most importantly, error conditions can be checked by using the following functions. Notice that nullary predicates take no argument. They check for a global condition that is unrelated to any specific number.
Next: Binary Predicates, Previous: Nullary Predicates, Up: MPFR Extension Reference [Contents][Index]
Unary predicates are similar to nullary predicates in that they detect the presence or absence of a specific condition. But in unary predicates, this condition is bound to a specific number. The importance of these predicates is often underestimated by beginners. For example, detecting a result that is NaN (not a number) may be important. Another subtle question is the equality of numbers, especially equality to zero. In case of doubt, look up the documentation of the MPFR library.
Next: Input and Output of Numbers, Previous: Unary Predicates, Up: MPFR Extension Reference [Contents][Index]
Binary predicates are the most common indicator functions.
They allow you to detect equality of two arguments.
Notice the slight difference between testing for equality
and comparing two arguments (with mpfr_cmp()).
Also notice that any number can be NaN (not a number)
and comparing to NaN or Inf is dubious.
Previous: Binary Predicates, Up: MPFR Extension Reference [Contents][Index]
Conversion between internal and external representation.
Next: Index, Previous: MPFR Extension Reference, Up: Top [Contents][Index]
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