"{ Encoding: utf8 }"

"

 COPYRIGHT (c) 1994 by Claus Gittinger

	      All Rights Reserved

 This software is furnished under a license and may be used

 only in accordance with the terms of that license and with the

 inclusion of the above copyright notice.   This software may not

 be provided or otherwise made available to, or used by, any

 other person.  No title to or ownership of the software is

 hereby transferred.

"

"{ Package: 'stx:libbasic' }"

"{ NameSpace: Smalltalk }"

Number subclass:#LimitedPrecisionReal

	instanceVariableNames:''

	classVariableNames:''

	poolDictionaries:''

	category:'Magnitude-Numbers'

!

!LimitedPrecisionReal class methodsFor:'documentation'!

copyright

"

 COPYRIGHT (c) 1994 by Claus Gittinger

	      All Rights Reserved

 This software is furnished under a license and may be used

 only in accordance with the terms of that license and with the

 inclusion of the above copyright notice.   This software may not

 be provided or otherwise made available to, or used by, any

 other person.  No title to or ownership of the software is

 hereby transferred.

"

!

documentation

"

    Abstract superclass for any-precision floating point numbers (i.e. IEE floats and doubles).

    Due to historic reasons, ST/X's Floats are what Doubles are in ST-80.

    The reason is that in some Smalltalk's, double floats are called Float, and no single float exists (VSE, V'Age),

    whereas in others, there are both Float and Double classes (VisualWorks).

    In order to allow code from both families to be loaded into ST/X without a missing class error,

    we decided to use IEE doubles as the internal representation of Float and make Double an alias to it.

    This should work for either family.

    If you really only want single precision floating point numbers, use ShortFloat instances.

    But be aware, that there is usually no advantage (neither in memory usage, due to memory alignment restrictions),

    nor in speed, as these days, the CPUs are just as fast doing double precision operations.

    (There is a difference when doing bulk operations, and you should consider using FloatArray for those).

    Range and Precision of Storage Formats:

    =======================================

      Format |   Class    |   Array Class   | Bits / Significant  | Smallest Pos Number | Largest Pos Number | Significant Digits 

             |            |                 |      (Binary)       |                     |                    |     (Decimal)

      -------+------------+-----------------+---------------------+---------------------+--------------------+--------------------    

      half   |     --     | HalfFloatArray  |    16 / 11          |  6.10... x 10−5     |  6.55...  x 10+5   |      3.3

      -------+------------+-----------------+---------------------+---------------------+--------------------+--------------------    

      single | ShortFloat | FloatArray      |    32 / 24          |  1.175...x 10-38    |  3.402... 10+38    |      6-9

      -------+------------+-----------------+---------------------+---------------------+--------------------+--------------------    

      double | Float      | DoubleArray     |    64 / 53          |  2.225...x 10-308   |  1.797... 10+308   |     15-17

      -------+------------+-----------------+---------------------+---------------------+--------------------+--------------------    

      double | LongFloat  |     --          |   128 / 113         |  3.362...x 10-4932  |  1.189... 10+4932  |     33-36

      ext    |            |                 |                     |                     |                    |

      (SPARC)|            |                 |                     |                     |                    |

      -------+            |                 |---------------------+---------------------+--------------------+--------------------    

      double |            |                 |    96 / 64          |  3.362...x 10-4932  |  1.189... 10+4932  |     18-21

      ext    |            |                 |                     |                     |                    |

      (x86)  |            |                 |                     |                     |                    |

      -------+------------+-----------------+---------------------+---------------------+--------------------+--------------------    

        --   | LargeFloat |     --          |     arbitrary       |  arbitrarily small  |  arbitrarily large |     arbitrary

      -------+------------+-----------------+---------------------+---------------------+--------------------+--------------------    

    HalfFloats are only supported in fixed array containers. This was added for OpenGL and other graphic libraries which allow

    for texture, and vertex data to be passed quickly in that format (see http://www.opengl.org/wiki/Small_Float_Formats).

    Long- and LargeFloat are not supported as array containers. 

    These formats are seldom used for bulk data.

    Bulk Containers:

    ================

    If you have a vector or matrix (and especially: large one's) of floating point numbers, the well known

    Array is a very inperformant choice. The reason is that it keeps pointers to each of its elements, and each element

    (if it is a float) is itself stored somewhere in the object memory.

    Thus, there is both a space overhead (every float object has an object header, for class and other information), and

    also a performance overhead (extra indirection, cache misses and alignment inefficiencies).

    For this, the bulk numeric containers are provided, which keep the elements unboxed and properly aligned.

    Use them for matrices and large numeric vectors. They also provide some optimized bulk operation methods,

    such as adding, multiplying etc.

    Comparing Floats:

    =================

    Due to rounding errors (usually on the last bit), you shalt not compare two floating point numbers

    using the #= operator. For example, the value 0.1 cannot be represented as a sum of powers of two fractions,

    and will therefore be always an approximation with a half bit error in the last bit of the mantissa.

    Usually, the print functions take this into consideration and return a (faked) '0.1'.

    However, this half bit error may accumulate, for example, when multiplying that by 10, the error may get large

    enough to be no longer pushed under the rug by the print function, and you will get '0.9999999999999' from it.

    Also, comparing against a proper 1.0 (which is representable as an exact power of 2), you will get a false result.

    (i.e. 0.1 * 10 ~= 1.0)

    This often confuses non-computer scientists (and even some of those are ocasionally).

    For this, you should always provide an epsilon value, when comparing two numbers. The epsilon value is

    the distance you accept two number to be apart to be still considered equal. Effectively the epsilon says

    are those nearer than this epsilon?.

    Now we could say is the delta between two numbers smaller than 0.00001,

    and get a reasonable answer for big numbers. But what if we compare two tiny numbers?

    Then a reasonable epsilon must also be much smaller. 

    Actually, the epsilon should always be computed dynamically depending on the two values compared.

    That is what the #isAlmostEqualTo:nEpsilon: method does for you. It does not take an absolute epsilon,

    but instead the number of distinct floating point numbers that the two compared floats may be apart.

    That is: the number of actually representable numbers between those two. Effectively, that is the difference between

    the two mantissas, when the numbers are scaled to the same exponent, taking the number of mantissa bits into account.

    [author:]

        Claus Gittinger

    [see also:]

        Fraction FixedPoint

"

! !

!LimitedPrecisionReal class methodsFor:'instance creation'!

fromInteger:anInteger

    "return a float with anInteger's value.

     Since floats have a limited precision, you usually loose bits when doing this

     (see Float decimalPrecision, LongFloat decimalPrecision."

    |newFloat sign absVal completeAbsVal delta mask|

    newFloat := self zero.

    sign := anInteger sign.

    (sign ~~ 0) ifTrue:[

        "be careful to do round to nearest even floating point value -

         thanks to Nicolas Cellier for this code"

        absVal := anInteger abs.

        delta := absVal highBit - self precision.

        delta > 0 ifTrue: [

            completeAbsVal := absVal.

            "eliminate insignificant/trailing bits"

            absVal := absVal bitShift: delta negated.

            "inexact := trailingBits ~= 0.

             Round to nearest even"

            (completeAbsVal bitAt:delta) ~= 0 ifTrue:[

                mask := (1 bitShift:delta-1) - 1.

                ((completeAbsVal bitAnd:mask) ~= 0 or:[absVal odd]) ifTrue:[

                    absVal := absVal + 1

                ].

            ].

        ] ifFalse: [

            delta := 0

        ].

        absVal digitLength to:1 by:-1 do:[:i |

            newFloat := (newFloat * 256) + (absVal digitByteAt:i).

        ].

        delta ~~ 0 ifTrue: [

            newFloat := newFloat timesTwoPower:delta.

        ].

        (sign < 0) ifTrue:[

            newFloat := newFloat negated

        ]

    ].

    ^ newFloat

    "

     ShortFloat fromInteger:2

     12345678901234567890 asShortFloat            

     1234567890 asFloat                     

     1234567890 asFloat asInteger                    

     -1234567890 asFloat asInteger                    

     12345678901234567890 asFloat storeString            

     12345678901234567890 asFloat asInteger   

     -12345678901234567890 asFloat asInteger   

     12345678901234567890 asLongFloat           

     12345678901234567890 asLongFloat asInteger 

     -12345678901234567890 asLongFloat asInteger 

     123456789012345678901234567890 asLongFloat           

     123456789012345678901234567890 asLongFloat asInteger  

     -123456789012345678901234567890 asLongFloat asInteger  

     1234567890123456789012345678901234567890 asLongFloat           

     1234567890123456789012345678901234567890 asLongFloat asInteger  

     -1234567890123456789012345678901234567890 asLongFloat asInteger

     'this test is on 65 bits'.

     self assert: 16r1FFFFFFFFFFFF0801 asDouble ~= 16r1FFFFFFFFFFFF0800 asDouble.

     'this test is on 64 bits'.

     self assert: 16r1FFFFFFFFFFFF0802 asDouble ~= 16r1FFFFFFFFFFFF0800 asDouble.

     'nearest even is upper'.

     self assert: 16r1FFFFFFFFFFF1F800 asDouble = 16r1FFFFFFFFFFF20000 asDouble.

     'nearest even is lower'.

     self assert: 16r1FFFFFFFFFFFF0800 asDouble = 16r1FFFFFFFFFFFF0000 asDouble.

    "

!

fromLimitedPrecisionReal:anLPReal

    "return a float with anLPReals value.

     You might loose bits when doing this.

     Slow fallback."

    |fract|

    anLPReal isFinite ifFalse:[

        anLPReal isNaN ifTrue:[^ self NaN].

        anLPReal negative ifTrue:[^ self negativeInfinity].

        ^ self infinity

    ].

    fract := anLPReal asTrueFraction.

    ^ (self fromInteger:fract numerator) / (self fromInteger:fract denominator)

!

fromNumerator:numerator denominator:denominator

    "Create a limited precision real from a Rational.

     This version will answer the nearest flotaing point value,

     according to IEEE 754 round to nearest even default mode"

    |a b q r exponent floatExponent n ha hb hq q1 eMin precision nSubHq|

    a := numerator abs.

    b := denominator abs.

    ha := a highBit.

    hb := b highBit.    

    precision := self precision.

    "If both numerator and denominator are represented exactly in floating point number,

     then fastest thing to do is to use hardwired float division"

    (ha <= precision and:[hb <= precision]) ifTrue: [

        ^ (self coerce:numerator) / (self coerce:denominator)

    ].

    "Try and obtain a mantissa with 1 bit excess precision.

     First guess is rough, we might get one more bit or one less"

    exponent := ha - hb - precision - 1.

    exponent > 0

        ifTrue: [b := b bitShift: exponent]

        ifFalse: [a := a bitShift: exponent negated].

    q := a quo: b.

    r := a - (q * b).

    hq := q highBit.

    "check for gradual underflow, in which case we should use less bits"

    floatExponent := exponent + hq - 1.

    eMin := 2 - (1 bitShift: (self numBitsInExponent - 1)).

    n := floatExponent >= eMin

        ifTrue: [precision + 1]

        ifFalse: [precision + 1 + floatExponent - eMin].

    nSubHq := n - hq.

    nSubHq < 0 ifTrue: [

        exponent := exponent + hq - n.

        r := (q bitAnd: (1 bitShift:nSubHq) - 1) * b + r.

        q := q bitShift:nSubHq

    ] ifFalse:[nSubHq > 0 ifTrue: [

        exponent := exponent + hq - n.

        q1 := (r bitShift:nSubHq) quo: b.

        q := (q bitShift:nSubHq) bitAnd: q1.

        r := r - (q1 * b)

    ]].

    "check if we should round upward.

     The case of exact half (q bitAnd: 1) isZero not & (r isZero)

     will be handled by self fromInteger: conversion"

    ((q bitAnd:1) ~~ 0 and:[r ~~ 0]) ifTrue:[

        q := q + 1

    ].

    (numerator negative xor:denominator negative) ifTrue: [

        q := q negated.

    ].

    ^ (self coerce:q) timesTwoPower:exponent

    "

        Time millisecondsToRun:[

            1000000  timesRepeat:[

                Float fromNumerator:12345678901234567890 denominator:987654321

            ].

        ]

        |fraction|

        fraction := 12345678901234567890//987654321.

        Time millisecondsToRun:[

            1000000  timesRepeat:[

                fraction asFloat

            ].

        ]

    "

!

new:aNumber

    "catch this message - not allowed for floats/doubles"

    self error:'Floats/Doubles cannot be created with new:'

!

readFrom:aStringOrStream onError:exceptionBlock

    "read a float from a string"

    |num|

    num := super readFrom:aStringOrStream onError:exceptionBlock.

    ^ self coerce:num

    "

     Float readFrom:'.1'

     Float readFrom:'0.1'

     Float readFrom:'0'

     ShortFloat readFrom:'.1'

     ShortFloat readFrom:'0.1'

     ShortFloat readFrom:'0'

     LongFloat readFrom:'.1'

     LongFloat readFrom:'0.1'

     LongFloat readFrom:'0'

    "

    "Created: / 07-01-1998 / 16:17:19 / cg"

    "Modified: / 23-11-2010 / 14:35:41 / cg"

! !

!LimitedPrecisionReal class methodsFor:'class initialization'!

initialize

    "initialize ANSI compliant float globals"

    Smalltalk at:#FloatE put:ShortFloat.

    Smalltalk at:#FloatD put:Float.

    Smalltalk at:#FloatQ put:LongFloat.

    "

     self initialize

    "

    "Created: / 7.9.2001 / 14:02:45 / cg"

    "Modified: / 7.9.2001 / 14:02:54 / cg"

! !

!LimitedPrecisionReal class methodsFor:'constants & defaults'!

NaN

    "return the constant NaN (not a Number)."

    |zero|

    zero := self zero.

    ^ zero uncheckedDivide:zero.

    "

      ShortFloat NaN  

      Float NaN       

      LongFloat NaN   

      LargeFloat NaN   

    "

!

computeEpsilon

    "return the maximum relative spacing"

    ^ self radix asFloat raisedToInteger:(1 - self precision).

    "

      ShortFloat computeEpsilon  

      Float computeEpsilon       

      LongFloat computeEpsilon   

    "

!

e

    "return the closest approximation of the irrational number e"

    self subclassResponsibility

    "Modified: / 7.9.2001 / 14:05:02 / cg"

!

emax

    "return the largest exponent"

    ^ (1 << (self numBitsInExponent-1)) - 1

    "

     Float emax

     ShortFloat emax

    "

!

emin

    "return the smallest exponent"

    ^ self subclassResponsibility

    "

     Float emin  

     ShortFloat emin  

    "

!

epsilon

    "return the maximum relative spacing"

    ^ self radix asFloat raisedToInteger:(1 - self precision)

    "

      ShortFloat epsilon  

      Float epsilon       

      LongFloat epsilon   

    "

!

fmax

    "The largest value allowed by instances of this class."

    |radix|

    radix := self radix.

    ^(self fromNumber: 1) -

            ((self fromNumber: radix) timesTwoPower: self precision negated - 1) * radix

            * ((self fromNumber: radix) timesTwoPower: self emax - 1)

    "

     Float fmax    

     ShortFloat fmax 

     LongFloat fmax  

    "

!

fmin

    "The smallest value allowed by instances of this class."

    ^ self subclassResponsibility

!

infinity

    "return a float representing infinity"

    ^ self unity uncheckedDivide:self zero

    "

      ShortFloat infinity  

      Float infinity       

      LongFloat infinity   

      LargeFloat infinity   

    "

!

negativeInfinity

    "return a float representing negative infinity"

    ^ self unity negated uncheckedDivide:self zero

    "

      ShortFloat negativeInfinity   

      Float negativeInfinity       

      LongFloat negativeInfinity   

      LargeFloat negativeInfinity   

    "

!

pi

    "return the closest approximation of the irrational number pi"

    self subclassResponsibility

    "Created: / 7.9.2001 / 14:07:35 / cg"

! !

!LimitedPrecisionReal class methodsFor:'queries'!

decimalPrecision

    "return the number of valid decimal digits"

    ^ ((self precision + 1) / (10.0 log:self radix)) ceiling   "use ceiling, since the last decimal digit contains some precision"

    "

     ShortFloat decimalPrecision