Why Floating-Point?

STM AN4044: One alternative to floating-point is fixed-point, where the exponent field is fixed. But if fixed-point is giving better calculation speed on FPU-less processors, the range of numbers and their dynamic is low. As a consequence, a developer using the fixed-point technique will have to check carefully any scaling/saturation issues in the algorithm.

Floating-Point Unit

The STM32 ARM Cortex M4F MPUs (e.g. STM32WB, STM32F4, STM32L4) have a single precision floating-point unit. The STM32H7 MPUs have a double precision FPU (not supported yet).

Also from STM AN4044

Floating-point calculations require a lot of resources, as for any operation between two numbers. For example, we need to:

• Align the two numbers (have them with the same exponent)
• Perform the operation
• Round out the result
• Code the result

On an FPU-less processor, all these operations are done by software through the C compiler library (or Forth Words) and are not visible to the programmer; but the performances are very low. On a processor having an FPU, all of the operations are entirely done by hardware in a single cycle, for most of the instructions. The C (or Forth) compiler does not use its own floating-point library but directly generates FPU native instructions.

When implementing a mathematical algorithm on a microprocessor having an FPU, the programmer does not have to choose between performance and development time. The FPU brings reliability allowing to use directly any generated code through a high level tool, such as MATLAB or Scilab, with the highest level of performance.

Any integer with absolute value less than 2^24 can be exactly represented in the single-precision format, and any integer with absolute value less than 2^53 can be exactly represented in the double-precision format.

Normalized Numbers Range

IEEE.754 Single and Double Precision Floating-Point Coding

• Single-precision floating-point format
• Double-precision floating-point format

• Scientific notation
• Engineering notation
• Metric prefix

Some Hints for Using the FPU

It is better to be approximately (vaguely) right than exactly wrong. Carveth Read

• Do not use FPU in interrupt service routines.
• Tasks/Threads with FPU operations need much more return stack depth.
• Rounding is not always working properly. useful for precision more than 3.

0.1005e fs. 1.01E-1  ok.
0.1005e fm. 101m ok.
4 set-precision
0.100005e fs. 1.0000E-1  ok.
0.100005e fm. 100.00m ok.
1.00005e f>x x. 1,00004994869232177734375000000000  ok.
1,00005 x. 1,00004999991506338119506835937500  ok.

Some Links

• https://forth-standard.org/standard/float
• https://en.wikipedia.org/wiki/IEEE_754
• https://en.wikipedia.org/wiki/Single-precision_floating-point_format
• What Every Computer Scientist Should Know About Floating-Point Arithmetic
• Fixed Point, https://en.wikipedia.org/wiki/Q_(number_format)

• https://www.complang.tuwien.ac.at/forth/gforth/Docs-html/Number-Conversion.html#Number-Conversion
• https://www.complang.tuwien.ac.at/forth/gforth/Docs-html/Floating-Point.html#Floating-Point
• https://forth-standard.org/proposals/recognizer#contribution-142
• https://interrupt.memfault.com/blog/cortex-m-rtos-context-switching ARM Cortex-M RTOS Context Switching
• https://mcuoneclipse.com/2019/03/29/be-aware-floating-point-operations-on-arm-cortex-m4f/
• http://support.raisonance.com/content/how-should-i-use-floating-point-unit-fpu-cortex-m4

Floating-Point Words

No separate floating-point stack. A single precision floating-point number is one cell. The 32-bit base-2 format is officially referred to as binary32 IEEE 754-2008.

Bare FPU Words (Without C Math Library)

f+      ( r1 r2 -- r3 )     Add r1 to r2 giving the sum r3
f-      ( r1 r2 -- r3 )     Subtract r2 from r1, giving r3
f*      ( r1 r2 -- r3 )     Multiply r1 by r2 giving r3
f/      ( r1 r2 -- r3 )     Divide r1 by r2, giving the quotient r3
fsqrt   ( r1 -- r2 )        r2 is the square root of r1

fabs    ( r1 -- r2 )        r2 is the absolute value of r1
fnegate ( r1 -- r2 )        r2 is the negation of r1
fround  ( r1 -- r2 )        round r1 to an integral value using the "round to nearest" rule, giving r2

fflags@ ( -- u )            get the current value of the Floating Point Status/Control register FPSCR
fflags! ( u -- )            assign the given value to the Floating Point Status/Control register FPSCR

f0=     ( r -- ? )          flag is true if r is equal to zero
f0<     ( r -- ? )          flag is true if r is less than zero
f<      ( r1 r2 -- ? )      flag is true if r1 is less than r2
f~      ( r1 r2 r3 -- ? )   If r3 is positive, flag is true if the absolute value of (r1 minus r2) is less than r3
                            If r3 is zero, flag is true if the implementation-dependent encoding of r1 and r2 are exactly identical 
                             (positive and negative zero are unequal if they have distinct encodings).
                            If r3 is negative, flag is true if the absolute value of (r1 minus r2) is less than the absolute value 
                            of r3 times the sum of the absolute values of r1 and r2. 

f>s     ( r -- n )          n is the single-cell signed-integer equivalent of the integer portion of r
s>f     ( n -- r )          r is the floating-point equivalent of the single-cell value n
f>x     ( r -- x )          x is the fixed-point equivalent of the floating-point r
x>f     ( x -- r )          r is the floating-point equivalent of the fixed-point x

pi      (  -- r )           r is pi, approx. 3.14159274101257324
e       (  -- r )           r is e, approx. 2.7182818

f.            ( r --  )     display, with a trailing space, the floating-point number r in fixed-point notation
precision     ( -- u )      return the number of significant digits currently used by F., FE., or FS. as u
set-precision ( u -- )      set the number of significant digits currently used by F., FE., or FS. to u

Words Using the C Math Library

fnumber (a # -- r u )      convert the specified string by a and # to float r, on success u is 1, otherwise 0
>float  (a # -- r ? )      convert the specified string by a and # to float r, on success flag is true (more robust)

fs.     ( r --  )          display, with a trailing space, the floating-point number r in scientific notation
fe.     ( r --  )          display, with a trailing space, the floating-point number r in engineering notation
fm.     ( r --  )          display, with a trailing space, the floating-point number r in metric unit prefix notation

fsin    ( r1 -- r2 )       r2 is the sine of the radian angle r1
fcos    ( r1 -- r2 )       r2 is the cosine of the radian angle r1
ftan    ( r1 -- r2 )       r2 is the principal radian angle whose tangent is r1
fasin   ( r1 -- r2 )       r2 is the principal radian angle whose sine is r1
facos   ( r1 -- r2 )       r2 is the principal radian angle whose cosine is r1
fatan   ( r1 -- r2 )       r2 is the principal radian angle whose tangent is r1

fsinh   ( r1 -- r2 )       r2 is the hyperbolic sine of r1
fcosh   ( r1 -- r2 )       r2 is the hyperbolic cosine of r1
ftanh   ( r1 -- r2 )       r2 is the hyperbolic tangent of r1
fasinh  ( r1 -- r2 )       r2 is the floating-point value whose hyperbolic sine is r1
facosh  ( r1 -- r2 )       r2 is the floating-point value whose hyperbolic cosine is r1
fatanh  ( r1 -- r2 )       r2 is the floating-point value whose hyperbolic tangent is r1

fceil   ( r1 -- r2 )       return the smallest integral value that is not less than r1
ffloor  ( r1 -- r2 )       Round r1 to an integral value using the "round toward negative infinity" rule, giving r2

fexp    ( r1 -- r2 )       raise e to the power r1, giving r2.
f**     ( r1 r2 -- r3 )    raise r1 to the power r2, giving the product r3

fln     ( r1 -- r2 )       r2 is the natural logarithm of r1
flog    ( r1 -- r2 )       r2 is the base-ten logarithm of r1

Fixed-Point Words

Fixed-point numbers (s31.32) are stored ( n-comma n-whole ) and can be handled like signed double numbers. Because of the name conflict with the floating-point words I changed the names of the fixed-point word and use for fixed-point words x instead of f.

All angles are in degrees.

d+      ( r1 r2 -- r3 )     add r1 to r2 giving the sum r3
d-      ( r1 r2 -- r3 )     subtract r2 from r1, giving r3
x*      ( r1 r2 -- r3 )     multiply r1 by r2 giving r3
x/      ( r1 r2 -- r3 )     divide r1 by r2, giving the quotient r3
x.      ( r --  )           display, with a trailing space, the fixed-point number r
x.n 	( r n -- ) 	    print a fixed-point number r with n fractional digits (truncated)
x#S 	( n1 -- n2 ) 	    Adds 32 comma-digits to number output
x# 	( n1 -- n2 ) 	    Adds one comma-digit to number output

sqrt    ( r1 -- r2 )        r2 is the square root of r1
sin
cos
tan
asin
acos
atan
log2
log10
ln
pow2
pow10
exp

floor
deg2rad ( deg -- rad )
rad2deg ( rad -- deg )

pi
pi/2
pi/4
+inf
-inf

*) fixpt-mat-lib.fs

How to Use

: f|| ( r1 r2 -- r3) 
  2dup f* -rot f+ f/ 
;

27k 100k f|| fm.  21.3k  ok.

: f. ( r -- )  \ display, with a trailing space, the floating-point number r in fixed-point notation
  $3F000000 \ .5
  precision 0 do
    $41200000 f/ \ 10.0 / 
  loop
  f+            \ round
  f>x
  <# 
     0 #s 2drop    \ integer part
     46 hold<       \ decimal point
     precision 0 do
       x#             \ fract digit
     loop
     dup
  #> 
  type space
;