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<?xml version="1.0" encoding="ISO8859-1"?>
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$Id: math.xml,v 1.4 2005/04/29 07:57:29 michael Exp $
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This file is part of the FPC documentation.
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Copyright (C) 1997, by Michael Van Canneyt
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The FPC documentation is free text; you can redistribute it and/or
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modify it under the terms of the GNU Library General Public License as
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published by the Free Software Foundation; either version 2 of the
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License, or (at your option) any later version.
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The FPC Documentation is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Library General Public License for more details.
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You should have received a copy of the GNU Library General Public
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License along with the FPC documentation; see the file COPYING.LIB. If not,
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write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330,
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Boston, MA 02111-1307, USA.
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<short>Additional mathematical routines.</short>
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<!-- \FPCexampledir{mathex} -->
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This document describes the <file>math</file> unit. The <var>math</var> unit
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was initially written by Florian Klaempfl. It provides mathematical
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functions which aren't covered by the system unit.
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This chapter starts out with a definition of all types and constants
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that are defined, after which an overview is presented of the available
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functions, grouped by category, and the last part contains a
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complete explanation of each function.
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The following things must be taken into account when using this unit:
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<li>This unit is compiled in Object Pascal mode so all <var>integers</var> are 32 bit.</li>
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<li> Some overloaded functions exist for data arrays of integers and
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floats. When using the address operator (<var>@</var>) to pass an array of
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data to such a function, make sure the address is typecasted to the
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right type, or turn on the 'typed address operator' feature. failing to
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do so, will cause the compiler not be able to decide which function you
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<topic name="MinMaxRoutines">
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<short>Min/max determination</short>
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Functions to determine the minimum or maximum of numbers:
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<th><td>Name</td><td>Description</td></th>
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<tr><td><link id="max"/></td><td>Maximum of 2 values</td></tr>
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<tr><td><link id="maxIntValue"/></td><td>Maximum of an array of integer values</td></tr>
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<tr><td><link id="maxvalue"/></td><td>Maximum of an array of values</td></tr>
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<tr><td><link id="min"/></td><td>Minimum of 2 values</td></tr>
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<tr><td><link id="minIntValue"/></td><td>Minimum of an array of integer values</td></tr>
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<tr><td><link id="minvalue"/></td><td>Minimum of an array of values</td></tr>
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<topic name="AngleConversionRoutines">
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<short>Angle unit conversion</short>
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Routines to convert angles between different angle units.
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<th><td>Name</td><td>Description</td></th>
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<tr><td><link id="cycletorad"/></td><td>convert cycles to radians</td></tr>
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<tr><td><link id="degtograd"/></td><td>convert degrees to grads</td></tr>
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<tr><td><link id="degtorad"/></td><td>convert degrees to radians</td></tr>
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<tr><td><link id="gradtodeg"/></td><td>convert grads to degrees</td></tr>
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<tr><td><link id="gradtorad"/></td><td>convert grads to radians</td></tr>
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<tr><td><link id="radtocycle"/></td><td>convert radians to cycles</td></tr>
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<tr><td><link id="radtodeg"/></td><td>convert radians to degrees</td></tr>
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<tr><td><link id="radtograd"/></td><td>convert radians to grads</td></tr>
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<topic name="TrigoniometricRoutines">
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<short>Trigoniometric functions</short>
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<th><td>Name</td><td>Description</td></th>
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<tr><td><link id="arccos"/></td><td>calculate reverse cosine</td></tr>
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<tr><td><link id="arcsin"/></td><td>calculate reverse sine</td></tr>
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<tr><td><link id="arctan2"/></td><td>calculate reverse tangent</td></tr>
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<tr><td><link id="cotan"/></td><td>calculate cotangent</td></tr>
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<tr><td><link id="sincos"/></td><td>calculate sine and cosine</td></tr>
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<tr><td><link id="tan"/></td><td>calculate tangent</td></tr>
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<topic name="HyperbolicRoutines">
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<short>Hyperbolic functions</short>
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<th><td>Name</td><td>Description</td></th>
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<tr><td><link id="arcosh"/></td><td>caculate reverse hyperbolic cosine</td></tr>
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<tr><td><link id="arsinh"/></td><td>caculate reverse hyperbolic sine</td></tr>
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<tr><td><link id="artanh"/></td><td>caculate reverse hyperbolic tangent</td></tr>
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<tr><td><link id="cosh"/></td><td>calculate hyperbolic cosine</td></tr>
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<tr><td><link id="sinh"/></td><td>calculate hyperbolic sine</td></tr>
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<tr><td><link id="tanh"/></td><td>calculate hyperbolic tangent</td></tr>
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<topic name="ExpLogRoutines">
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<short>Exponential and logarithmic functions</short>
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<th><td>Name</td><td>Description</td></th>
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<tr><td><link id="intpower"/></td><td>Raise float to integer power</td></tr>
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<tr><td><link id="ldexp"/></td><td>Calculate $2^p x$</td></tr>
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<tr><td><link id="lnxp1"/></td><td>calculate <var>log(x+1)</var></td></tr>
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<tr><td><link id="log10"/></td><td>calculate 10-base log</td></tr>
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<tr><td><link id="log2"/></td><td>calculate 2-base log</td></tr>
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<tr><td><link id="logn"/></td><td>calculate N-base log</td></tr>
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<tr><td><link id="power"/></td><td>raise float to arbitrary power</td></tr>
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<topic name="NumberConversionRoutines">
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<short>Number converting</short>
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<th><td>Name</td><td>Description</td></th>
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<tr><td><link id="ceil"/></td><td>Round to infinity</td></tr>
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<tr><td><link id="floor"/></td><td>Round to minus infinity</td></tr>
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<tr><td><link id="frexp"/></td><td>Return mantissa and exponent</td></tr>
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<topic name="StatisticalRoutines">
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<short>Statistical functions</short>
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<th><td>Name</td><td>Description</td></th>
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<tr><td><link id="mean"/></td><td>Mean of values</td></tr>
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<tr><td><link id="meanandstddev"/></td><td>Mean and standard deviation of values</td></tr>
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<tr><td><link id="momentskewkurtosis"/></td><td>Moments, skew and kurtosis</td></tr>
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<tr><td><link id="popnstddev"/></td><td>Population standarddeviation </td></tr>
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<tr><td><link id="popnvariance"/></td><td>Population variance</td></tr>
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<tr><td><link id="randg"/></td><td>Gaussian distributed randum value</td></tr>
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<tr><td><link id="stddev"/></td><td>Standard deviation</td></tr>
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<tr><td><link id="sum"/></td><td>Sum of values</td></tr>
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<tr><td><link id="sumofsquares"/></td><td>Sum of squared values</td></tr>
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<tr><td><link id="sumsandsquares"/></td><td>Sum of values and squared values</td></tr>
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<tr><td><link id="totalvariance"/></td><td>Total variance of values</td></tr>
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<tr><td><link id="variance"/></td><td>variance of values</td></tr>
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<topic name="GeometricalRoutines">
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<short>Geometrical functions</short>
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<th><td>Name</td><td>Description</td></th>
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<tr><td><link id="hypot"/></td><td>Hypotenuse of triangle</td></tr>
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<tr><td><link id="norm"/></td><td>Euclidian norm</td></tr>
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<element name="Float">
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<short>Float type used in all calls</short>
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All calculations are done with the Float type. This allows to
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recompile the unit with a different float type to obtain a
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desired precision. The pointer type <link id="PFloat"/>
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is used in functions that accept an array of values of arbitrary length.
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<element name="PFloat">
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<short>Pointer to <link id="Float"/> type.</short>
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<element name="TPaymentTime">
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<short>Type used in financial (interest) calculations.</short>
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<element name="TPaymentTime.PTEndOfPeriod">
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<short>End of period.</short>
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<element name="TPaymentTime.PTStartOfPeriod">
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<short>Start of period.</short>
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<element name="EInvalidArgument">
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<short>Exception raised when invalid arguments are passed to a function.</short>
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<element name="arccos">
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<short>Return inverse cosine</short>
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<var>Arccos</var> returns the inverse cosine of its argument <var>x</var>. The
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argument <var>x</var> should lie between -1 and 1 (borders included).
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If the argument <var>x</var> is not in the allowed range, an
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<var>EInvalidArgument</var> exception is raised.
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<example file="mathex/ex1"/>
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<element name="arcosh">
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<short>Return inverse hyperbolic cosine</short>
233
<var>Arcosh</var> returns the inverse hyperbolic cosine of its argument <var>x</var>.
234
The argument <var>x</var> should be larger than 1.
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The <var>arccosh</var> variant of this function is supplied for Delphi
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If the argument <var>x</var> is not in the allowed range, an <var>EInvalidArgument</var>
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<example file="mathex/ex3"/>
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<element name="arcsin">
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<short>Return inverse sine</short>
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<var>Arcsin</var> returns the inverse sine of its argument <var>x</var>. The
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argument <var>x</var> should lie between -1 and 1.
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If the argument <var>x</var> is not in the allowed range, an <var>EInvalidArgument</var>
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<example file="mathex/ex2"/>
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<element name="arctan2">
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<short>Return arctangent of (y/x)</short>
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<var>arctan2</var> calculates <var>arctan(y/x)</var>, and returns an angle in the
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correct quadrant. The returned angle will be in the range $-\pi$ to
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The values of <var>x</var> and <var>y</var> must be between -2\^{}64 and 2\^{}64,
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moreover <var>x</var> should be different from zero.
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On Intel systems this function is implemented with the native intel
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<var>fpatan</var> instruction.
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If <var>x</var> is zero, an overflow error will occur.
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<example file="mathex/ex6"/>
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<element name="arsinh">
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<short>Return inverse hyperbolic sine</short>
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<var>arsinh</var> returns the inverse hyperbolic sine of its argument <var>x</var>.
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The <var>arscsinh</var> variant of this function is supplied for Delphi
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<example file="mathex/ex4"/>
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<element name="artanh">
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<short>Return inverse hyperbolic tangent</short>
322
<var>artanh</var> returns the inverse hyperbolic tangent of its argument <var>x</var>,
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where <var>x</var> should lie in the interval [-1,1], borders included.
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The <var>arctanh</var> variant of this function is supplied for Delphi compatibility.
328
In case <var>x</var> is not in the interval [-1,1], an <var>EInvalidArgument</var>
337
<example file="mathex/ex5"/>
341
<element name="ceil">
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<short>Return the lowest integer number greater than or equal to argument</short>
344
<var>Ceil</var> returns the lowest integer number greater than or equal to <var>x</var>.
345
The absolute value of <var>x</var> should be less than <var>maxint</var>.
348
If the asolute value of <var>x</var> is larger than maxint, an overflow error will
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<example file="mathex/ex7"/>
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<element name="cosh">
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<short>Return hyperbolic cosine</short>
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<var>Cosh</var> returns the hyperbolic cosine of it's argument {x}.
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<example file="mathex/ex8"/>
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<element name="cotan">
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<short>Return cotangent</short>
377
<var>Cotan</var> returns the cotangent of it's argument <var>x</var>. <var>x</var> should
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be different from zero.
381
If <var>x</var> is zero then a overflow error will occur.
386
<example file="mathex/ex9"/>
390
<element name="cycletorad">
391
<short>Convert cycle angle to radians angle</short>
393
<var>Cycletorad</var> transforms it's argument <var>cycle</var>
394
(an angle expressed in cycles) to radians.
395
(1 cycle is $2 \pi$ radians).
401
<link id="degtograd"/>
402
<link id="degtorad"/>
403
<link id="radtodeg"/>,
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<link id="radtograd"/>
405
<link id="radtocycle"/>
407
<example file="mathex/ex10"/>
411
<element name="degtograd">
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<short>Convert degree angle to grads angle</short>
414
<var>Degtograd</var> transforms it's argument <var>deg</var> (an angle in degrees)
417
(90 degrees is 100 grad.)
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<link id="cycletorad"/>
424
<link id="degtorad"/>
425
<link id="radtodeg"/>,
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<link id="radtograd"/>
427
<link id="radtocycle"/>
429
<example file="mathex/ex11"/>
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<element name="degtorad">
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<short>Convert degree angle to radians angle.</short>
436
<var>Degtorad</var> converts it's argument <var>deg</var> (an angle in degrees) to
439
(pi radians is 180 degrees)
445
<link id="cycletorad"/>
446
<link id="degtograd"/>
447
<link id="radtodeg"/>,
448
<link id="radtograd"/>
449
<link id="radtocycle"/>
451
<example file="mathex/ex12"/>
455
<element name="floor">
456
<short>Return the largest integer smaller than or equal to argument</short>
458
<var>Floor</var> returns the largest integer smaller than or equal to <var>x</var>.
459
The absolute value of <var>x</var> should be less than <var>maxint</var>.
462
If <var>x</var> is larger than <var>maxint</var>, an overflow will occur.
467
<example file="mathex/ex13"/>
471
<element name="frexp">
472
<short>Return mantissa and exponent.</short>
474
<var>Frexp</var> returns the mantissa and exponent of it's argument
475
<var>x</var> in <var>mantissa</var> and <var>exponent</var>.
482
<example file="mathex/ex14"/>
486
<element name="gradtodeg">
487
<short>Convert grads angle to degrees angle</short>
489
<var>Gradtodeg</var> converts its argument <var>grad</var> (an angle in grads)
492
(100 grad is 90 degrees)
498
<link id="cycletorad"/>
499
<link id="degtograd"/>
500
<link id="radtodeg"/>,
501
<link id="radtograd"/>
502
<link id="radtocycle"/>
503
<link id="gradtorad"/>
505
<example file="mathex/ex15"/>
509
<element name="gradtorad">
510
<short>Convert grads angle to radians angle</short>
512
<var>Gradtorad</var> converts its argument <var>grad</var> (an angle in grads)
515
(200 grad is pi degrees).
521
<link id="cycletorad"/>
522
<link id="degtograd"/>
523
<link id="radtodeg"/>,
524
<link id="radtograd"/>
525
<link id="radtocycle"/>
526
<link id="gradtodeg"/>
528
<example file="mathex/ex16"/>
532
<element name="hypot">
533
<short>Return hypotenuse of triangle</short>
535
<var>Hypot</var> returns the hypotenuse of the triangle where the sides
536
adjacent to the square angle have lengths <var>x</var> and <var>y</var>.
538
The function uses Pythagoras' rule for this.
545
<example file="mathex/ex17"/>
549
<element name="intpower">
550
<short>Return integer power.</short>
552
<var>Intpower</var> returns <var>base</var> to the power <var>exponent</var>,
553
where exponent is an integer value.
556
If <var>base</var> is zero and the exponent is negative, then an
557
overflow error will occur.
562
<example file="mathex/ex18"/>
566
<element name="ldexp">
567
<short>Return (2 to the power p) times x</short>
569
<var>Ldexp</var> returns (2 to the power <var>p</var>) times <var>x</var>.
580
<example file="mathex/ex19"/>
584
<element name="lnxp1">
585
<short>Return natural logarithm of 1+X</short>
587
<var>Lnxp1</var> returns the natural logarithm of <var>1+X</var>. The result
588
is more precise for small values of <var>x</var>. <var>x</var> should be larger
592
If $x\leq -1$ then an <var>EInvalidArgument</var> exception will be raised.
600
<example file="mathex/ex20"/>
603
<element name="log10">
604
<short>Return 10-Based logarithm.</short>
606
<var>Log10</var> returns the 10-base logarithm of <var>X</var>.
609
If <var>x</var> is less than or equal to 0 an 'invalid fpu operation' error
618
<example file="mathex/ex21"/>
622
<element name="log2">
623
<short>Return 2-based logarithm</short>
625
<var>Log2</var> returns the 2-base logarithm of <var>X</var>.
628
If <var>x</var> is less than or equal to 0 an 'invalid fpu operation' error
637
<example file="mathex/ex22"/>
641
<element name="logn">
642
<short>Return N-based logarithm.</short>
644
<var>Logn</var> returns the n-base logarithm of <var>X</var>.
647
If <var>x</var> is less than or equal to 0 an 'invalid fpu operation' error
656
<example file="mathex/ex23"/>
660
<short>Return largest of 2 values</short>
662
<var>Max</var> returns the maximum of <var>Int1</var> and <var>Int2</var>.
669
<link id="maxIntValue"/>
670
<link id="maxvalue"/>
672
<example file="mathex/ex24"/>
675
<element name="maxIntValue">
676
<short>Return largest element in integer array</short>
679
<var>MaxIntValue</var> returns the largest integer out of the <var>Data</var>
683
This function is provided for Delphi compatibility, use the <link id="maxvalue"/>
691
<link id="maxvalue"/>
692
<link id="minvalue"/>
693
<link id="minIntValue"/>
695
<example file="mathex/ex25"/>
699
<element name="maxvalue">
700
<short>Return largest value in array</short>
703
<var>Maxvalue</var> returns the largest value in the <var>data</var>
704
array with integer or float values. The return value has
705
the same type as the elements of the array.
708
The third and fourth forms accept a pointer to an array of <var>N</var>
709
integer or float values.
716
<link id="maxIntValue"/>
717
<link id="minvalue"/>
718
<link id="minIntValue"/>
720
<example file="mathex/ex26"/>
723
<element name="mean">
724
<short>Return mean value of array</short>
726
<var>Mean</var> returns the average value of <var>data</var>.
728
The second form accepts a pointer to an array of <var>N</var> values.
734
<link id="meanandstddev"/>
735
<link id="momentskewkurtosis"/>
738
<example file="mathex/ex27"/>
741
<element name="meanandstddev">
742
<short>Return mean and standard deviation of array</short>
744
<var>meanandstddev</var> calculates the mean and standard deviation of <var>data</var>
745
and returns the result in <var>mean</var> and <var>stddev</var>, respectively.
746
Stddev is zero if there is only one value.
748
The second form accepts a pointer to an array of <var>N</var> values.
756
<link id="sumofsquares"/>
757
<link id="momentskewkurtosis"/>
759
<example file="mathex/ex28"/>
764
<short>Return smallest of two values.</short>
766
<var>min</var> returns the smallest value of <var>Int1</var> and <var>Int2</var>;
774
<example file="mathex/ex29"/>
777
<element name="minIntValue">
778
<short>Return smallest value in integer array</short>
781
<var>MinIntvalue</var> returns the smallest value in the <var>Data</var> array.
784
This function is provided for Delphi compatibility, use <var>minvalue</var>
792
<link id="minvalue"/>
793
<link id="maxIntValue"/>
794
<link id="maxvalue"/>
796
<example file="mathex/ex30"/>
800
<element name="minvalue">
801
<short>Return smallest value in array</short>
804
<var>Minvalue</var> returns the smallest value in the <var>data</var>
805
array with integer or float values. The return value has
806
the same type as the elements of the array.
809
The third and fourth forms accept a pointer to an array of <var>N</var>
810
integer or float values.
817
<link id="maxIntValue"/>
818
<link id="maxvalue"/>
819
<link id="minIntValue"/>
821
<example file="mathex/ex31"/>
825
<element name="momentskewkurtosis">
826
<short>Return 4 first moments of distribution</short>
828
<var>momentskewkurtosis</var> calculates the 4 first moments of the distribution
829
of valuesin <var>data</var> and returns them in <var>m1</var>,<var>m2</var>,<var>m3</var> and
830
<var>m4</var>, as well as the <var>skew</var> and <var>kurtosis</var>.
837
<link id="meanandstddev"/>
839
<example file="mathex/ex32"/>
842
<element name="norm">
843
<short>Return Euclidian norm</short>
846
<var>Norm</var> calculates the Euclidian norm of the array of data.
847
This equals <var>sqrt(sumofsquares(data))</var>.
850
The second form accepts a pointer to an array of <var>N</var> values.
857
<link id="sumofsquares"/>
859
<example file="mathex/ex33"/>
863
<element name="popnstddev">
864
<short>Return population variance</short>
867
<var>Popnstddev</var> returns the square root of the population variance of
868
the values in the <var>Data</var> array. It returns zero if there is only one value.
871
The second form of this function accepts a pointer to an array of <var>N</var>
879
<link id="popnvariance"/>
881
<link id="meanandstddev"/>
883
<link id="momentskewkurtosis"/>
885
<example file="mathex/ex35"/>
889
<element name="popnvariance">
890
<short>Return population variance</short>
893
<var>Popnvariance</var> returns the square root of the population variance of
894
the values in the <var>Data</var> array. It returns zero if there is only one value.
897
The second form of this function accepts a pointer to an array of <var>N</var>
905
<link id="popnstddev"/>
907
<link id="meanandstddev"/>
909
<link id="momentskewkurtosis"/>
911
<example file="mathex/ex36"/>
915
<element name="power">
916
<short>Return real power.</short>
918
<var>power</var> raises <var>base</var> to the power <var>power</var>. This is equivalent
919
to <var>exp(power*ln(base))</var>. Therefore <var>base</var> should be non-negative.
925
<link id="intpower"/>
927
<example file="mathex/ex34"/>
931
<element name="radtocycle">
932
<short>Convert radians angle to cycle angle</short>
935
<var>Radtocycle</var> converts its argument <var>rad</var> (an angle expressed in
936
radians) to an angle in cycles.
939
(1 cycle equals 2 <var>pi</var> radians)
946
<link id="degtograd"/>
947
<link id="degtorad"/>
948
<link id="radtodeg"/>,
949
<link id="radtograd"/>
950
<link id="cycletorad"/>
952
<example file="mathex/ex37"/>
956
<element name="radtodeg">
957
<short>Convert radians angle to degrees angle</short>
959
<var>Radtodeg</var> converts its argument <var>rad</var> (an angle expressed in
960
radians) to an angle in degrees.
962
(180 degrees equals pi radians)
968
<link id="degtograd"/>
969
<link id="degtorad"/>
970
<link id="radtocycle"/>,
971
<link id="radtograd"/>
972
<link id="cycletorad"/>
974
<example file="mathex/ex38"/>
978
<element name="radtograd">
979
<short>Convert radians angle to grads angle</short>
982
<var>Radtodeg</var> converts its argument <var>rad</var> (an angle expressed in
983
radians) to an angle in grads.
986
(200 grads equals pi radians)
993
<link id="degtograd"/>
994
<link id="degtorad"/>
995
<link id="radtocycle"/>,
996
<link id="radtodeg"/>
997
<link id="cycletorad"/>
999
<example file="mathex/ex39"/>
1003
<element name="randg">
1004
<short>Return gaussian distributed random number.</short>
1006
<var>randg</var> returns a random number which - when produced in large
1007
quantities - has a Gaussian distribution with mean <var>mean</var> and
1008
standarddeviation <var>stddev</var>.
1016
<link id="meanandstddev"/>
1018
<example file="mathex/ex40"/>
1022
<element name="sincos">
1023
<short>Return sine and cosine of argument</short>
1026
<var>Sincos</var> calculates the sine and cosine of the angle <var>theta</var>,
1027
and returns the result in <var>sinus</var> and <var>cosinus</var>.
1030
On Intel hardware, This calculation will be faster than making 2 calls
1031
to calculate the sine and cosine separately.
1039
<link id="arccos"/>.
1041
<example file="mathex/ex41"/>
1045
<element name="sinh">
1046
<short>Return hyperbolic sine</short>
1048
<var>Sinh</var> returns the hyperbolic sine of its argument <var>x</var>.
1058
<example file="mathex/ex42"/>
1062
<element name="stddev">
1063
<short>Return standard deviation of data</short>
1066
<var>Stddev</var> returns the standard deviation of the values in <var>Data</var>.
1067
It returns zero if there is only one value.
1070
The second form of the function accepts a pointer to an array of <var>N</var>
1079
<link id="meanandstddev"/>
1080
<link id="variance"/>
1081
<link id="totalvariance"/>
1083
<example file="mathex/ex43"/>
1087
<element name="sum">
1088
<short>Return sum of values</short>
1091
<var>Sum</var> returns the sum of the values in the <var>data</var> array.
1094
The second form of the function accepts a pointer to an array of <var>N</var>
1102
<link id="sumofsquares"/>
1103
<link id="sumsandsquares"/>
1104
<link id="totalvariance"/>
1105
<link id="variance"/>
1107
<example file="mathex/ex44"/>
1111
<element name="sumofsquares">
1112
<short>Return sum of squares of values</short>
1115
<var>Sumofsquares</var> returns the sum of the squares of the values in the <var>data</var>
1119
The second form of the function accepts a pointer to an array of <var>N</var>
1128
<link id="sumsandsquares"/>
1129
<link id="totalvariance"/>
1130
<link id="variance"/>
1132
<example file="mathex/ex45"/>
1136
<element name="sumsandsquares">
1137
<short>Return sum and sum of squares of values.</short>
1140
<var>sumsandsquares</var> calculates the sum of the values and the sum of
1141
the squares of the values in the <var>data</var> array and returns the
1142
results in <var>sum</var> and <var>sumofsquares</var>.
1145
The second form of the function accepts a pointer to an array of <var>N</var>
1154
<link id="sumofsquares"/>
1155
<link id="totalvariance"/>
1156
<link id="variance"/>
1158
<example file="mathex/ex46"/>
1162
<element name="tan">
1163
<short>Return tangent</short>
1165
<var>Tan</var> returns the tangent of <var>x</var>.
1168
If <var>x</var> (normalized) is pi/2 or 3pi/2 then an overflow will occur.
1176
<example file="mathex/ex47"/>
1180
<element name="tanh">
1181
<short>Return hyperbolic tangent</short>
1183
<var>Tanh</var> returns the hyperbolic tangent of <var>x</var>.
1193
<example file="mathex/ex48"/>
1197
<element name="totalvariance">
1198
<short>Return total varians of values</short>
1201
<var>TotalVariance</var> returns the total variance of the values in the
1202
<var>data</var> array. It returns zero if there is only one value.
1205
The second form of the function accepts a pointer to an array of <var>N</var>
1213
<link id="variance"/>
1217
<example file="mathex/ex49"/>
1221
<element name="variance">
1222
<short>Return variance of values</short>
1225
<var>Variance</var> returns the variance of the values in the
1226
<var>data</var> array. It returns zero if there is only one value.
1229
The second form of the function accepts a pointer to an array of <var>N</var>
1237
<link id="totalvariance"/>
1241
<example file="mathex/ex50"/>
1244
<!-- unresolved type reference Visibility: default -->
1245
<element name="sysutils">
1246
<short>Used for exception definitions.</short>
1249
<!-- constant Visibility: default -->
1250
<element name="MinExtended">
1251
<short>Minimum value (closest to zero) of extended type</short>
1254
<!-- constant Visibility: default -->
1255
<element name="MaxExtended">
1256
<short>Maximum value of extended type</short>
1259
<!-- constant Visibility: default -->
1260
<element name="MinFloat">
1261
<short>Minimum value (closest to zero) of float type</short>
1264
<!-- constant Visibility: default -->
1265
<element name="MaxFloat">
1266
<short>Maximum value of float type</short>
1269
<!-- pointer type Visibility: default -->
1270
<element name="PInteger">
1271
<short>Pointer to integer type</short>
1274
<!-- range type Visibility: default -->
1275
<element name="TValueRelationship">
1276
<short>Type to describe relational order between values</short>
1279
<!-- constant Visibility: default -->
1280
<element name="EqualsValue">
1281
<short>Values are the same</short>
1284
<!-- constant Visibility: default -->
1285
<element name="LessThanValue">
1286
<short>First value is less than second value</short>
1289
<!-- constant Visibility: default -->
1290
<element name="GreaterThanValue">
1291
<short>First values is greater than second value</short>
1294
<!-- constant Visibility: default -->
1295
<element name="NaN">
1296
<short>Value is Not a Number</short>
1299
<!-- constant Visibility: default -->
1300
<element name="Infinity">
1301
<short>Value is infinity</short>
1304
<!-- function Visibility: default -->
1305
<element name="InRange">
1306
<short>Check whether value is in range.</short>
1308
<var>InRange</var> returns <var>True</var> if <var>AValue</var> is in the
1309
range <var>AMin</var>..<var>AMax</var>. It returns <var>False</var> if
1310
<var>Value</var> lies outside the specified range.
1313
<link id="EnsureRange"/>
1317
<!-- function Visibility: default -->
1318
<element name="EnsureRange">
1319
<short>Change value to it falls in specified range.</short>
1321
<var>EnsureRange</var> returns <var>Value</var> if <var>AValue</var> is in
1322
the range <var>AMin</var>..<var>AMax</var>. It returns <var>AMin</var> if
1323
the value is less than <var>AMin</var>, or <var>AMax</var> if the value is
1324
larger than <var>AMax</var>.
1327
<link id="InRange"/>
1331
<!-- procedure Visibility: default -->
1332
<element name="DivMod">
1333
<short>Return DIV and MOD of arguments </short>
1335
<var>DivMod</var> returns <var>Dividend</var> DIV <var>Divisor</var> in
1336
<var>Result</var>, and <var>Dividend</var> MOD <var>Divisor</var> in
1337
<var>Remainder</var>
1341
<!-- range type Visibility: default -->
1342
<element name="TValueSign">
1343
<short>Type indicating sign of a valuea</short>
1346
<!-- constant Visibility: default -->
1347
<element name="NegativeValue">
1348
<short>Value is negative</short>
1351
<!-- constant Visibility: default -->
1352
<element name="ZeroValue">
1353
<short>Value is zero</short>
1356
<!-- constant Visibility: default -->
1357
<element name="PositiveValue">
1358
<short>Value is positive</short>
1361
<!-- function Visibility: default -->
1362
<element name="Sign">
1363
<short>Return sign of argument</short>
1365
<var>Sign</var> returns the sign of it's argument, which can be an Integer,
1366
64 bit integer, or a double. The returned value is an integer which is -1, 0
1367
or 1, and can be used to do further calculations with.
1371
<!-- function Visibility: default -->
1372
<element name="IsZero">
1373
<short>Check whether value is zero</short>
1376
<var>IsZero</var> checks whether the float value <var>A</var> is zero, up to a
1377
precision of <var>Epsilon</var>. It returns <var>True</var> if Abs(<var>A</var>) is
1378
less than <var>Epsilon</var>.
1381
The default value for <var>Epsilon</var> is dependent on the type of the
1382
arguments, but is <link id="MinFloat"/> for the float type.
1387
<link id="IsInfinite"/>
1388
<link id="SameValue"/>
1392
<!-- function Visibility: default -->
1393
<element name="IsNan">
1394
<short>Check whether value is Not a Number</short>
1396
<var>IsNan</var> returns <var>True</var> if the double <var>d</var>
1397
contains Not A Number (a value which cannot be represented correctly
1402
<link id="IsInfinite"/>
1406
<!-- function Visibility: default -->
1407
<element name="IsInfinite">
1408
<short>Check whether value is infinite</short>
1410
<var>IsInfinite</var> returns <var>True</var> if the double <var>d</var>
1411
contains the infinite value.
1415
<link id="IsInfinite"/>
1419
<!-- function Visibility: default -->
1420
<element name="SameValue">
1421
<short>Check whether 2 float values are the same</short>
1424
<var>SameValue</var> returns <var>True</var> if the floating-point values
1425
<var>A</var> and <var>B</var> are the same, i.e. whether the absolute value
1426
of their their difference is smaller than <var>Epsilon</var>. If their
1427
difference is larger, then <var>False</var> is returned.
1430
The default value for <var>Epsilon</var> is dependent on the type of the
1431
arguments, but is <link id="MinFloat"/> for the float type.
1435
<link id="MinFloat"/>
1440
<!-- function Visibility: default -->
1441
<element name="arccosh">
1442
<short>Return inverse hyperbolic cosine</short>
1445
<var>arccosh</var> returns the inverse hyperbolic cosine of it's argument
1449
This function is an alias for <link id="arcosh"/>, provided for Delphi
1458
<!-- function Visibility: default -->
1459
<element name="arcsinh">
1460
<short>Return inverse hyperbolic sine</short>
1463
<var>arcsinh</var> returns the inverse hyperbolic sine of it's argument
1467
This function is an alias for <link id="arsinh"/>, provided for Delphi
1476
<!-- function Visibility: default -->
1477
<element name="arctanh">
1478
<short>Return inverse hyperbolic tangent</short>
1481
<var>arcsinh</var> returns the inverse hyperbolic tangent of it's argument
1485
This function is an alias for <link id="artanh"/>, provided for Delphi
1494
<!-- enumeration type Visibility: default -->
1495
<element name="TFPURoundingMode">
1496
<short>Type describing the rounding mode for the Floating Point processor.</short>
1499
<!-- enumeration value Visibility: default -->
1500
<element name="TFPURoundingMode.rmNearest">
1501
<short>Round to nearest integer value</short>
1504
<!-- enumeration value Visibility: default -->
1505
<element name="TFPURoundingMode.rmDown">
1506
<short>Round to biggest integer smaller than value.</short>
1509
<!-- enumeration value Visibility: default -->
1510
<element name="TFPURoundingMode.rmUp">
1511
<short>Round to smallest integer larger than value.</short>
1514
<!-- enumeration value Visibility: default -->
1515
<element name="TFPURoundingMode.rmTruncate">
1516
<short>Cut off fractional part.</short>
1519
<!-- enumeration type Visibility: default -->
1520
<element name="TFPUPrecisionMode">
1521
<short>Type describing the default precision for the Floating Point processor.</short>
1524
<!-- enumeration value Visibility: default -->
1525
<element name="TFPUPrecisionMode.pmSingle">
1526
<short>Single-type precision</short>
1529
<!-- enumeration value Visibility: default -->
1530
<element name="TFPUPrecisionMode.pmReserved">
1534
<!-- enumeration value Visibility: default -->
1535
<element name="TFPUPrecisionMode.pmDouble">
1536
<short>Double-type precision</short>
1539
<!-- enumeration value Visibility: default -->
1540
<element name="TFPUPrecisionMode.pmExtended">
1541
<short>Extended-type precision</short>
1544
<!-- enumeration type Visibility: default -->
1545
<element name="TFPUException">
1546
<short>Type describing Floating Point processor exceptions.</short>
1549
<!-- enumeration value Visibility: default -->
1550
<element name="TFPUException.exInvalidOp">
1551
<short>Invalid operation error</short>
1554
<!-- enumeration value Visibility: default -->
1555
<element name="TFPUException.exDenormalized">
1559
<!-- enumeration value Visibility: default -->
1560
<element name="TFPUException.exZeroDivide">
1561
<short>Division by zero error.</short>
1564
<!-- enumeration value Visibility: default -->
1565
<element name="TFPUException.exOverflow">
1566
<short>Float overflow error</short>
1569
<!-- enumeration value Visibility: default -->
1570
<element name="TFPUException.exUnderflow">
1571
<short>Float underflow error</short>
1574
<!-- enumeration value Visibility: default -->
1575
<element name="TFPUException.exPrecision">
1576
<short>Precision error</short>
1579
<!-- set type Visibility: default -->
1580
<element name="TFPUExceptionMask">
1581
<short>Type to set the Floating Point Unit exception mask.</short>
1584
<!-- function Visibility: default -->
1585
<element name="GetRoundMode">
1586
<short>Return the Floating Point Unit rounding mode.</short>
1591
<!-- function Visibility: default -->
1592
<element name="SetRoundMode">
1593
<short>Set the Floating Point Unit rounding mode.</short>
1598
<!-- function Visibility: default -->
1599
<element name="GetPrecisionMode">
1600
<short>Return the Floating Point Unit precision mode.</short>
1605
<!-- function Visibility: default -->
1606
<element name="SetPrecisionMode">
1607
<short>Set the Floating Point Unit precision mode.</short>
1612
<!-- function Visibility: default -->
1613
<element name="GetExceptionMask">
1614
<short>Get the Floating Point Unit exception mask.</short>
1619
<!-- function Visibility: default -->
1620
<element name="SetExceptionMask">
1621
<short>Set the Floating Point Unit exception mask.</short>
1626
<!-- procedure Visibility: default -->
1627
<element name="ClearExceptions">
1628
<short>Clear Floating Point Unit exceptions</short>
1633
<element name="GetSSECSR">
1634
<short>Get MXCSR control word (Intel only)</short>
1636
<var>GetSSECSR</var> can be used to get the SSE/SSE2 control DWord.
1637
It is equivalent to the <var>LDMXCSR</var> assembler instruction, and
1638
returns the control dword.
1642
<element name="ifthen">
1643
<short>Return one of two values, depending on a boolean condition</short>
1646
<var>ifthen</var> returns <var>iftrue</var> if <var>val</var> is
1647
<var>True</var>, and <var>False</var> if <var>val</var> is <var>False</var>.
1650
This function can be used in expressions.
1655
<element name="SetSSECSR">
1656
<short>Set MXCSR control word (Intel only)</short>
1658
<var>SetSSECSR</var> can be used to set the SSE/SSE2 control DWord.
1659
It is equivalent to the <var>STMXCSR</var> assembler instruction, and stores
1660
<var>w</var> in the control dword.
1665
<!-- unresolved type reference Visibility: default -->
1666
<element name="sysutils">
1667
<short>Exception support</short>
1670
<!-- function Visibility: default -->
1671
<element name="operator **">
1672
<short>Exponent operator</short>
1674
<var>**</var> raises <var>base</var> to the power <var>expt</var>. Both numbers can be floats or 64-bit integers.
1677
Overflow exceptions can occur.
1683
<!-- function Visibility: default -->
1684
<element name="sumInt">
1685
<short>Return the sum of an array of integers</short>
1687
<var>SumInt</var> returns the sum of the <var>N</var> integers in the <var>Data</var>
1688
array, where this can be an open array or a pointer to an array.
1691
An overflow may occur.
1698
</fpdoc-descriptions>
added
removed
fpcdocs/math.xml
