Software floating-point is supplied with the cross-compiler and the target Forth. The target floating point wordset is not ANS or Forth-2012 compliant, but satisfies the needs of embedded systems without undue complexity. The Forth data stack and the floating point stack are the same (combined) in Common/SoftFP32CX.fth and separate in Common/SoftFP32SX.fth. The floating point data storage format is not IEEE, but is optimised for performance on small controllers. If you need a separate floating point stack or IEEE double format storage, please contact MPE. Any variations in the implementation will be documented in the target specific section of the manual.
The cross-compiler has a more limited floating-point support than the target. Some words are available during compilation of colon definitions, but not while interpreting.
The source code is in two sets of files, one for 32 bit Forth targets, the other for 16 bit targets. The files are:
Common\SoftFP32SX current 32 bit code for separate stacks
Common\SoftFP32CX current 32 bit code for a combined stack
Common\sfp32hi old 32 bit primitives
Common\sfp32com old 32 bit high level code
Common\sfp16hi 16 bit primitives
Common\sfp16com 16 bit high level code
The SoftFP32xx files use no assembler definitions. Some targets have code versions of the primitives, and these will be found in the CPU-specific code directory. An increase in performance can be obtained by using the code files.
Floating point number entry is enabled by REALS
and
disabled by INTEGERS
.
Floating-point numbers of the form 0.1234e1 are required
(see FNUMBER?
) during interpretation and compilation
of source code. Floating-point numbers are compiled as
literal numbers when in a colon definition (compiling) and
placed on the stack when outside a definition (interpreting).
Inside a colon definition, a floating point literal number
must be preceded by F#
.
: foo ... f# 1.234e0 ... ;
The more flexible word >FLOAT
accepts numbers in two
forms, 1.234 and 0.1234e1. Both words are documented later
in this chapter. See also the section on Gotchas later
in this chapter.
Note also that by default, MPE Forths use ',' as the double number indicator - it makes life much easier for Europeans.
A floating-point number is placed on the Forth data stack. In
the Forth literature, this is referred to as a combined floating
point and data stack.
For 32 bit targets, a floating point number consists of two
32-bit numbers, one for the mantissa and one for the exponent.
For 16 bit targets, it consists of a 32-bit double mantissa
and a single 16-bit exponent. The mantissa is normalised.
The exponent is on the top of the stack.
Note that for 16 bit targets, number conversion is affected
by the cross-compiler directives HOST-MATH
and TARGET-MATH
.
HOST-MATH
leaves double numbers and floats in 32-bit form,
whereas TARGET-MATH
leaves them in 16-bit form.
To create a variable, use FVARIABLE
. FVARIABLE
works in the
same way as VARIABLE
. For example, to create a floating-point
variable called VAR1
you code:
FVARIABLE VAR1
When VAR1
is used, it returns the address of the floating-point
number.
Two words are used to access floating-point variables,
F@
and F!
. These are analogous to @
and
!
.
To create a floating-point constant, use FCONSTANT
, which
is analogous to CONSTANT
. For example, to generate a
floating-point constant called CON1
with a value of 1.234,
you enter:
1.234e0 FCONSTANT CON1
When CON1
is executed, it returns 1.234 on the Forth
stack.
The supplied words split into several groups:
The following functions only exist as target words so you cannot use them in calculations in your source code when outside a colon definition.
To calculate sine, cosine and tangent, use FSIN
, FCOS
and
FTAN
respectively. Angles are expressed in radians.
To calculate arc sine, cosine and tangent, use FASIN
, FACOS
and FATAN
respectively. They return an angle in radians.
Two words are supplied to calculate logarithms, FLOG
and FLN
.
FLOG
calculates a logarithm to base 10 (decimal).
FLN
calculates a logarithm to base e. Both take a
floating-point number in the range from 0 to Einf.
Three power functions are supplied:
FE^X F10^X X^Y
The angular measurement used in the trigonometric functions
are in radians. To convert between degrees and radians use
RAD>DEG
or DEG>RAD
. RAD>DEG
converts an angle from radians
to degrees. DEG>RAD
converts an angle from degrees to radians.
Two words are available for displaying floating-point numbers,
F.
and E.
. The word F.
takes a floating-point
number from the stack and displays it in the form xxxx.xxxxx
or x.xxxxxEyy depending on the size of the number. The word
E.
displays the number in the latter form.
The ANS Forth standard specifies that floating point numbers must be entered in the form 1.234e5 and must contain a point '.' and 'e' or 'E', and that double integers are terminated by a point '.'.
This situation prevents the use of the standard conversion
words in international applications because of the
interchangable use of the '.' and ',' characters in numbers.
Because of this, the cross-compiler's host VFX Forth uses
two four-byte arrays, FP-CHAR
and DP-CHAR
, to
hold the characters used as the floating point and double
integer indicator characters. By default, FP-CHAR
is
initialised to '.' and DP-CHAR
is initialised to to
',' and '.'. For strict ANS compliance, you should set them
as follows before CROSS-COMPILE
is run.
\ ANS standard setting
char . dp-char !
char . fp-char !
: ans-floats \ -- ; for strict ANS compliance
[char] . dp-char !
[char] . fp-char !
;
\ MPE defaults
char , dp-char !
char . dp-char 1+ c!
char . fp-char !
: mpe-floats \ -- ; for existing and most legacy code
[char] , dp-char !
[char] . dp-char 1+ c!
[char] . fp-char !
;
\ Legacy defaults, including ProForth
char , dp-char !
char . fp-char !
: legacy-floats \ -- ; for legacy code
[char] , dp-char !
[char] . fp-char !
;
You can of course set these arrays to hold any values which
suit your application's language and locale. Note that integer
conversion is always attempted before floating point
conversion. This means that if the FP-CHAR
and DP-CHAR
arrays contain the same character, floating point numbers must
contain 'e' or 'E'. If the arrays are all different, a number
containing the FP-CHAR
will be successfully converted
as a floating point number, even if it does not contain 'e'
or 'E'.
Before July 2010, the floating point separator, '.', was fixed. To ease internationalisation, it is now variable.
variable fp-char \ -- addr
Holds up to four character(s) to be treated as floating
point indicators. Set to '.' for ANS compatibility. Note
that this should be accessed as a one to four byte array.
The first character is used as the point character for
output.
0 equ SepArray? \ -- flag
If the equate is non-zero, fp-char
is treated as
a four byte array, otherwise as a one byte array. This is
a flag for future expansion.
: isSep? \ char addr -- flag
Return true if char is one of the four bytes at addr.
If less than than four bytes are needed, a zero byte acts as
a terminator. Used when SepArray?
is true.
: isSep? c@ = ;
A compiler macro used when SepArray?
is false.
8 equ FPCELL \ -- u
Size of a floating point number.
: finit \ F: i*f -- ; resets FPU and FP stack
Reset the floating point stack.
Do not forget to use this in a task before using floating
point.
: fdepth \ -- #f
Floating point equivalent of DEPTH
. The result is returned
on the Forth data stack.
: fs@ \ -- f ; F: f -- f
Copy the top of the floating point stack to the data stack.
: fs> \ F: f -- ; -- f
Move the top of the floating point stack to the data stack.
: fs! \ f1 -- ; F: f2 -- f1
Move a float from the data stack and overwrite the top of the
float stack.
: >fs \ f -- ; F: -- f
Move a float from the data stack to a new position on the float
stack.
: exp@ \ F: f -- f ; -- exp
Copy the exponent of the top float to the data stack.
: exp! \ exp -- ; F: f -- f'
Change/Set the exponent of the top float.
: mant@ \ F: f -- f ; -- mant
Copy the mantissa of the top float to the data stack.
: mant! \ F: f -- f ; -- mant
Change/Set the mantissa of the top float.
The software floating point pack requires several support primitives. High level versions are provided in SFP16HI.FTH and SFP32HI.FTH for 16 and 32 bit targets. Some targets have coded versions in the CPU directory and these will provide much better performance. The support file should be compiled before the common file.
In SoftFP32CX.fth and SoftFP32SX.fth a set of high-level primitives are compiled if the primitives have not yet been supplied.
: S-> \ n1 carry-in-flag --- n2 carry-out-flag
Perform a right shift, applying the carry in to the m.s. bit and
returning the carry out as 1 or 0.
: <-S \ n1 carry-in-flag --- n2 carry-out-flag
Perform a left shift, applying the carry in to the l.s. bit and
returning the carry out as 1 or 0.
: d<<1 \ xd -- xd<<1
One bit double left shift.
: d>>1 \ xd -- xd>>1
One bit double right logical shift.
: D>>N \ d n -- d>>n
N bit double right logical shift.
: rshiftx \ x u -- x' ; right shift
Used for right shifts that may exceed 31 bits. In the ANS and
Forth 2012 standards, this is an ambiguous condition. We need
shifts over 31 bits to return 0. On x86 targets, a check is made
for shifts over 31 bits.
: F! \ F: r -- ; addr --
Stores r at addr
: F@ \ addr -- ; F: -- r
Fetches r from addr.
: F, \ F: r --
Lays a real number into the dictionary, reserving 8 bytes.
: FDUP \ F: r -- r r
Floating point equivalent of DUP
.
: FOVER \ F: r1 r2 -- r1 r2 r1
Floating point equivalent of OVER
.
: FSWAP \ F: r1 r2 -- r2 r1
Floating point equivalent of SWAP
.
: FPICK \ F: fu..f0 u -- fu..f0 fu
Floating point equivalent of PICK
.
: FROT \ F: r1 r2 r3 -- r2 r3 r1
Floating point equivalent of ROT
.
: F-ROT \ F: r1 r2 r3 -- r3 r1 r2
Floating point equivalent of -ROT
.
: FROLL \ F: f1 f2 f3 -- f2 f3 f1
Floating point equivalent of ROLL
.
: FDROP \ F: r --
Floating point equivalent of DROP
.
: FNIP \ F: r1 r2 -- r2
Floating point equivalent of NIP
.
: FVARIABLE \ "<spaces>name" -- ; Run: -- f-addr
Use in the form: FVARIABLE <name>
to create a variable
that will hold a floating point number.
: FCONSTANT \ r "<spaces>name" -- ; Run: -- r
Use in the form: <float> FCONSTANT <name>
to create a
constant that returns a floating point number.
: FARRAY \ "<spaces>name" fn-1..f0 n -- ; Run: i -- ri
Create an initialised array of floating point numbers. Use
in the form:
fn-1 .. f1 f0 n FARRAY <name>
to create an array of n floating point numbers. When the
array name
is executed, the index i is used to return
the address of the i'th 0 zero-based element in the array.
For example:
4e0 3e0 2e0 1e0 0e0 5 FARRAY TEST
will set up an array of five elements.
Note that the rightmost float (0e0) is element 0.
Then i TEST
will return the *\{i}th element.
If you create this array in IDATA
, restore CDATA
afterwards.
: FBUFF \ u "name" -- ; i -- addr
Creates a buffer for u floats in the current memory
section. The child action is to return the address of the
ith element (zero-based).
10 fbuff foo
Creates an buffer for ten float elements in the current memory section.
3 foo
Returns the address of element 3 in the buffer.
The default section is CDATA
, and we recommend that
you leave it that way! To create a ten element array in
UDATA
space, you can use:
udata
10 fbuff MyFloats
cdata
: NORM \ n exp -- f
Normalise a single integer and a single exponent to produce a
floating point number on the data stack. INTERNAL.
: DNORM \ d exp -- fn ; normalise a 64 bit double
Normalise a double integer and a single exponent to produce a
floating point number on the data stack. INTERNAL.
: (FSIGN) \ fn -- |fn| flag ; true if negative
Return the absolute value of fn and a flag which is true
if fn is negative. Data stack operation.
: FSIGN \ F: fn -- |fn| ; -- flag ; true if negative
Return the absolute value of fn and a flag which is true
if fn is negative. F.P. stack operation.
: S>F \ n -- ; F: -- fn
Converts a single integer to a float.
: D>F \ d -- ; F: -- fn
Converts a double integer to a float.
: F>S \ F: fn -- ; -- n
Converts a float to a single integer.
Note that F>S
truncates the number towards zero
according to the ANS specification. If |fn| is greater
than maxint, +/-maxint is returned.
: F>D \ F: fn -- ; -- d
Converts a float to a double integer.
Note that F>D
truncates the number towards zero
according to the ANS specification. If |fn| is greater
than dmaxint, +/-dmaxint is returned.
: FINT \ F: f1 -- f2
Chop the number towards zero to produce a floating point
representation of an integer.
: FNEGATE \ F: r1 -- r2
Floating point negate.
: ?FNEGATE \ n -- ; F: fn -- fn|-fn
If n is negative, negate fn.
: FABS \ F: fn -- |fn|
Floating point absolute.
: F* \ F: r1 r2 -- r3
Floating point multiply.
: F/ \ F: r1 r2 -- r3
Floating point divide.
: F+ \ F: r1 r2 -- r3
Floating point addition.
: F- \ F: r1 r2 -- r3
Floating point subtraction, r3=r1-r2
: FSEPARATE \ F: f1 f2 -- f3 f4
Leave the signed integer quotient f4 and remainder f3 when
f1 is divided by f2. The remainder has the same sign as the
dividend.
: FFRAC \ f1 f2 -- f3
Leave the fractional remainder from the division f1/f2. The
remainder takes the sign of the dividend.
: F0< \ F: f1 -- ; -- flag
Floating point 0<
.
: F0> \ F: f1 -- ; -- flag
Floating point 0>
.
: F0= \ F: f1 -- ; -- flag
Floating point 0=
.
: F0<> \ F: f1 -- ; -- flag
Floating point 0<>
.
: F= \ F: f1 f2 -- ; -- flag
Floating point =
.
: F< \ F: r1 r2 -- ; -- flag
Floating point <
.
: F> \ F: f1 f2 -- ; -- flag
Floating point >
.
: FMAX \ F: r1 r2 -- r1|r2
Floating point MAX
.
: FMIN \ F: r1 r2 -- r1|r2
Floating point MIN
.
: FALIGNED \ addr -- f-addr
Aligns the address to accept an 8-byte float.
: FALIGN \ --
Aligns the dictionary to accept an 8-byte float.
: FLOAT+ \ f-addr1 -- f-addr2
Increments addr by 8, the size of a float.
: FLOATS \ n1 -- n2
Returns n2, the size of n1 floats.
1 s>f 10 s>f f/ fconstant %.1
Floating point 0.1.
1 s>f fconstant %1
Floating point 1.0.
10 s>f fconstant %10
Floating point 10.0.
1250000000 34 fconstant %10^10
Floating point 10^10.
1844674407 -33 fconstant %10^-10
Floating point 10^-10.
F# 1.0E256 FCONSTANT %10^256
Floating point 10^256.
F# 1.0E-1 FCONSTANT %10E-1
Floating point 10^-1.
F# 1.0E-10 FCONSTANT %10E-10
Floating point 10^-10.
F# 1.0E-256 FCONSTANT %10^-256
Floating point 10^-256.
16 FARRAY POWERS-OF-10E1
An array of 16 powers of ten starting at 10^0
in steps of 1.
17 FARRAY POWERS-OF-10E16
An array of 17 powers of ten starting at 10^0
in steps of 16.
16 FARRAY POWERS-OF-10E-1
An array of 16 powers of ten starting at 10^0
in steps of -1.
17 FARRAY POWERS-OF-10E-16
An array of 17 powers of ten starting at 10^0
in steps of -16.
: RAISE_POWER \ exp(10) -- ; F: f -- f'
Raise the power in preparation for number formatting.
: SINK_FRACTION \ exp(10) -- ; F: f -- f'
Reduce the power in preparation for number formatting.
: *10^X \ exp(10) -- ; F: f -- f'
Generate float' = float *10^dec_exp. INTERNAL.
Note that number conversion takes place in PAD
.
: FLITERAL \ Comp: F: r -- ; Run: F: -- r
Compiles a float as a literal into the current definition.
At execution time, a float is returned. For example,
[ %PI F2* ] FLITERAL
will compile 2PI as a floating point
literal. Note that FLITERAL
is immediate.
: CONVERT-EXP \ c-addr --
If the character at c-addr is 'D' convert it to 'E'. INTERNAL.
: CONVERT-FPCHAR \ c-addr --
Convert the f.p. char '.' to the double char ',' for
conversion. INTERNAL.
: ALL-BLANKS? \ c-addr len -- flag
Return true if string is all blanks (spaces). INTERNAL.
: FCHECK \ -- am lm ae le e-flag .-flag
Check the input string at PAD
, returning the separated
mantissa and exponent flags. The e-flag is returned true
if the string contained an exponent indicator 'E' and
the .-flag is returned true if a '.' was found. INTERNAL.
: doMNUM \ c-addr u -- d 2 | 0
Convert the mantissa string to a double number and 2. If
conversion fails, just return 0. INTERNAL.
: doENUM \ c-addr u -- n 1 | 0 ; str as above
Convert the exponent string to a single number and 1. If
conversion fails, just return 0. INTERNAL.
: FIXEXP \ dmant exp(10) -- ; F: -- f
Convert a double integer mantissa and a single integer
exponent into a floating point number. INTERNAL.
: FNUMBER? \ addr -- 0/.../mant exp 2
Behaves like the integer version of NUMBER?
except that if
the number is in F.P. format and BASE
is decimal, a floating
point conversion is attempted. If conversion is successful,
the floating point number is left on the float stack and
the result code is 2. This word only accepts text with an
'E' as a floating point indicator, e.g, 1.2345e0.
If *\fo{BASE is not decimal all numbers are treated as integers.
The integer prefixes '#','$','0x' etc. are recognised and
cause integer conversion to be used.
: >FLOAT \ c-addr u -- true|false ; F: -- [f]
Try to convert the string at c-addr/u to a floating point number.
If conversion is successful, flag is returned true, and a floating
number is returned on the float stack, otherwise just flag=0 is
returned. This word accepts several forms, e.g. 1.2345e0, 1.2345,
12345 and converts them to a float. Note that double numbers
(containing a ',') cannot be converted.
Number conversion is decimal only, regardless of the current BASE
.
: (F#) \ addr -- 2|0 ; F: -- [f]
The primitive for F#
and F#IN
below.
: F#IN \ -- 2|0 ; F: -- [f]
Attempts to convert a token from the input stream to a
floating-point number. Numbers in integer format will be
converted to floating-point. An indicator (0 or 2/3) is
returned in the same way as an indicator is returned by
FNUMBER?
.
: F# \ F: -- [f] ; or compiles it [ state smart ]
If interpreting, takes text from the input stream and,
if possible converts it to a f.p. number on the stack.
Numbers in integer format will be converted to floating-point.
If compiling, the converted number is compiled.
: REALS \ -- ; allow f.p input
Switch NUMBER?
to permit floating point input using
FNUMBER?
. This action can be reversed by INTEGERS
.
Both REALS
and INTEGERS
are in the FORTH
vocabulary.
: INTEGERS \ -- ; no f.p input
Switch NUMBER?
to restore integer only input.
variable places 8 places ! \ -- addr
Number of digits output after the decimal point.
: ROUND \ F: f1 -- f2
Rounds least significant eight bits to 0 if higher 2 bits
are all 0s or all 1s.
: ?10PWR \ F: f -- f ; -- exp[10]
Generate the power of ten corresponding to the float's power of two.
: SIGFIGS \ F: f -- f' ; n -- dec_exponent
Scale f and generate a decimal exponent corresponding to n
significant digits.
: op-prepare \ F: fn -- ; -- d exp(10) sign
From fn, generate a double number corresponding to 8 significant digits,
a decimal exponent and a sign indicator (nz=negative). INTERNAL.
: .EXP \ exp --
Display the exponent. INTERNAL.
: N# \ d n -- d'
Convert n digits. INTERNAL.
: .FPsign \ flag --
If flag is non-zero, generate a '-' otherwise a space.
: .FPsep \ --
Issue the FP separator, usually '.'.
: E. \ F: f --
Print the f.p. number on the stack in exponential form,
x.xxxxxEyy.
: REPRESENT \ F: r -- ; c-addr u -- n flag1 flag2
Assume that the floating number is of the form +/-0.xxxxEyy.
Place the significand xxxxx at c-addr with a maximum of u digits.
Return n the signed integer version of yy. Return flag1 true
if f is negative, and return flag2 true if the results are
valid. In this implementation all errors are handled by
exceptions, and so flag2 is always true.
: F. \ F: f --
Print the f.p. number in free format, xxxx.yyyy, if
possible. Otherwise display using the x.xxxxEyy format.
f# 1.0 fconstant %ONE
Floating point 1.0.
: FLOOR \ r1 -- r2
Floored round towards -infinity.
: FROUND \ r1 -- r2
Round the number to nearest or even.
N.B. All angles are in radians.
: DEG>RAD \ F: n1 -- n2
Convert degrees to radians.
: RAD>DEG \ F: n1 -- n2
convert radians to degrees.
: FSQR \ F: f1 -- f2 ; FSQR by Heron's formula
F2=sqrt(f1) by Heron's formula.
: FSIN \ F: f1 -- f2
f2=sin(f1).
: FCOS \ F: f1 -- f2
f2=cos(f1).
: FTAN \ f1 -- f2
f2=tan(f1).
: FASIN \ F: f1 -- f2
f2=arcsin(f1).
: FACOS \ F: f1 -- f2
f2=arccos(f1).
: FATAN \ F: f1 -- f2
f2=arctan(f1).
: FLN \ F: f1 -- f2
Take the logarithm of f1 to base e and return the result.
: FLOG \ F: f1 -- f2
Take the logarithm of f1 to base 10 and return the result.
: FE^X \ F: f1 -- f2
f2=e^f1.
: F10^X \ F: f1 -- f2
f2=10^f1
: FX^N \ x-real n-integer -- fx^n
fx^n=x^n where x is a float and n is an integer.
: FX^Y \ F: fx fy -- fx^fy
fn=X^Y where Y and Y are both floats.
: FP>IEEE \ F: fp -- ; -- ieee32
Convert native FP value to IEEE 32 bit format.
: IEEE>FP \ ieee32 -- ; F: -- fp
Convert IEEE 32 bit float to native format.
The ANS and Forth200x specifications define the format of floating point numbers during text interpretation as:
Convertible string := <significand><exponent>
<significand> := [<sign>]<digits>[.<digits0>]
<exponent> := E[<sign>]<digits0>
<sign> := { + | - }
<digits> := <digit><digits0>
<digits0> := <digit>*
<digit> := { 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 }
This format is handled by the word FNUMBER?
. The word
>FLOAT
accepts a more relaxed format.
Convertible string := <significand>[<exponent>]
<significand> := [<sign>]{<digits>[.<digits0>] | .<digits> }
<exponent> := <marker><digits0>
<marker> := {<e-form> | <sign-form>}
<e-form> := <e-char>[<sign-form>]
<sign-form> := { + | - }
<e-char> := { D | d | E | e }
This restriction makes it difficult to use the text interpreter
during program execution as it requires floating point numbers
to contain 'D' or 'E' indicators, which is not profane practice.
A quick kluge to fix this is to change FNUMBER?
as below.
Replace:
fcheck drop if \ valid f.p. number?
with:
fcheck or if \ valid f.p. number?
Note that this change can/will cause problems if number base
is not DECIMAL
.