Floating Point Package, Part II

By Jörg Langowski, EMBL, c/o I.L.L., Grenoble, Cedex, France, Editorial Board

"Fast exp(x) and ln(x) in single precision"

We will continue with numerics this time, in order to give some examples how to put the 32 bit floating point package to practical use, and also because we got feedback that some more information about number crunching would be appreciated.

First, however, it is time for some apologies: the bugs have been creeping into the multiply routine, and when I noticed the last few traces they left, the article was already in press. The problem was that when the number on top of stack was zero, the routine would all of a sudden leave two numbers on the stack, one of which was garbage. This problem has been fixed in the revision, which is printed in Listing 1. I hope there will be no more errors, but please let me know if you find any. A reliable 32 bit package is so important for numerical applications on the Mac!

For many applications, the four basic operations +-*/ by themselves already help a lot in speeding up. However, alone they do not make a functional floating point package. For operations that are not used so frequently, like conversion between integer, single and extended or input/output on can still rely on the built-in SANE routines. But for the standard mathemetical functions you would want to have your own definitions that make full use of the speed of the 32 bit routines.

Developing a complete package of mathematical functions would be a project that is outside the scope of this column. I'll only give you two examples that serve to show that a very reasonable speed can be attained in Forth (here, Mach1) without making too much use of assembly language. The two examples, ln(x) and exp(x) are based on approximations taken from the Handbook of Mathematical Functions by M. Abramowitz and I.A. Stegun, Dover Publications, New York 1970. Furthermore, the routines given here profited a lot from ideas published in the April '86 issue of BYTE on number crunching.

First, we have to realize that a transcendental function like ln(x), using a finite number of calculation steps, can only be approximated over a certain range of input numbers to a certain maximum accuracy. It is intuitively clear that the wider the range of the argument x, the lengthier the calculation gets to achieve the desired accuracy. Therefore, approximation formulas for standard functions are usually given over a very restricted range of x. We have to see that we play some tricks on the input value x so that we can get a reliable approximation over the whole range of allowed floating point numbers, which is approximately 10-38 to 10+38 for the IEEE 32 bit format.

The handbook mentioned gives various approximations for ln(x) with different degrees of accuracy. The accuracy that we need for a 24 bit mantissa is 2-23 10-7, and a suitable approximation for this accuracy would be

ln(1+x)   a1x + a2x2 + a3x3 + a4x4 + a5x5+ a6x6 + a7x7 + a8x8  +  (error),
 [1]

where for 0 ¯ x ¯ 1 the error is less than 3.10-8. The coefficients a1 to a8 are:

a1 =  0.9999964239, a2 =  -0.4998741238, 
a3 =  0.3317990258, a4 =  -0.2407338084,
a5 =  0.1676540711, a6 =  -0.0953293897, 
a7 =  0.0360884937, a8 =  -0.0064535442 . 

To calculate eqn. [1] more rapidly, it is of course convenient to write it as

ln(1+x)   x.(a1 + x.(a2 + x.(a3 + x.(a4 + x.(a5 + x.(a6+ x.(a7 + x.a8))))))
 [2]

where by consecutive addition of coefficients and multiplication by the argument the polynomial may be evaluated with a minimum of operations. ln.base in Listing 2 calculates eqn. [2] and gives a good approximation for ln(x) in the range of x=1 2.

For numbers outside this range, we have to realize that

 ln(a.x) = ln(a) + ln(x),

and in the special case when a = 2n,

 ln(2n .x) = n.ln(2) + ln(x).

Now, all our floating point numbers are already split up in such a way; they contain a binary exponent n and a mantissa x such that x is between 1 and 2. So it remains to separate the exponent and mantissa, calculate eqn.[2] for the mantissa and add n times ln(2), which is a constant that we can calculate and store beforehand.

The separation of exponent and mantissa is done in get.exp, which will leave the biased exponent on top of stack, followed by the mantissa in the format of a 32-bit floating point number between 1 and 2. We now have to multiply the exponent by ln(2), an (integer) times (real) multiplication. Instead of writing another routine do do this, we use a faster method that, however, is a little memory consuming: we build a lookup table for all values of n.ln(2) with n between -127 and +128, the allowed range of exponents. Since the exponent is biased by +127, we can use it directly to index the table. The table consumes 1K of memory, so I wouldn't use it on a 48K CP/M system, but with 0.5 to 1 megabyte on a Mac, this can be justified. The lookup table is created using the SANE routines; this takes a couple of seconds, but it is done only for the initialization.

For faster indexing, I also defined the word 4* in assembly, which does not exist in Mach1 (it does, of course, in MacForth).

The final definition ln first separates exponent and mantissa and then computes ln(x) from those separate parts. Note that ln as well as ln.base are written completely in Forth. Fine-tuning of those routines, using assembler, should speed them up by another factor of 1.5 to 2 (wild guess). Still, you already gain a factor of 12 over the SANE routine (use speed.test to verify). The accuracy is reasonably good; the value calculated here differs from the 'exact' extended precision value by approximately 1 part in 107 to 108, just about the intrinsic precision of 32-bit floating point.

Let's now proceed to the inverse of the logarithm, the exponential. The handbook gives us the approximation

e-x    a1x + a2x2 + a3x3 + a4x4 + a5x5+ a6x6 + a7x7 + (error),

with the coefficients

a1 =  -0.9999999995, a2 =  0.4999999206, 
a3 =  -0.1666653019, a4 =  0.0416573745,
a5 =  -0.0083013598, a6 =  0.0013298820, 
a7 =  -0.0001413161 .

This approximation is valid to within 2.10-10 for x between 0 and ln(2) 0.6, and we use it for x = 0 1 for our purposes here, which still is sufficiently precise for a 24 bit mantissa.

Again, we have to scale down the input value of x in order to get it into the range of validity of the approximation. This time, we use the relationship

 e(N+f) = eN  . ef  ,

where N is the integer and f the fractional part of x. eN will be looked up in a table and ef calculated from the approximation. To get N, we need a real-to-integer conversion routine; this routine, together with its integer-to-real counterpart, is coded in assembler with some Forth code to get the signs correct (words s>i and i>s). The fractional part is calculated by subtracting the integer part from the input number; this is done in Forth without giving up too much in speed. exp puts it all together and calculates ex for the whole possible range of x values.

As before, the lookup table for the eN values is initialized separately, using the SANE routines.

The benchmark, speed.test, shows a 24 fold speed increase of this exponential function as compared to the 80-bit SANE version.

Other mathematical standard functions can be defined in a way very similar to the examples that I gave here. A good source of some approximations is the handbook mentioned above, also, many interesting ideas regarding numerical approximations can be found in BYTE 4/86.

Feedback dept.

Let's turn to some comments that I received through electronic mail on Bitnet and BIX.

Here comes a comment (through BIX) on the IC! bug in NEON, which leads to a very interesting observation regarding the 68000 instruction set:

Memo #82583

From: microprose

Date: Fri, 23 May 86 21:44:08 EDT

To: jlangowski

Cc: mactutor

Message-Id: <memo.82583>

Subject: "IC!" bug -- why it happens

Just got my April '86 MacTutor, and I thought I'd answer your question about the bug in the "IC!" word. Register A7 in the 68000 is always used as the stack pointer, and as such must always be kept word-aligned. As a special case, the pre-decrement and post-increment addressing modes, when used with a byte-sized operand, automatically push or pop an extra padding byte to keep the stack word-aligned. In the case of MOVE.B (A7)+,<dest>, this causes the most-significant byte of the word at the top of the stack to be transferred; then the stack pointer is adjusted by 2 (not 1). I would guess that a similar thing is happening with ADDQ #3,A7; since you mention nothing about a stack underflow, it seems that this instruction is adding 2 to A7, not 4 as I would have suspected. (Otherwise, in combination with the following instruction, an extra word is being removed from the stack.) Since the desired byte is at the bottom of the longword, your solution is the best one (assuming that D0 is a scratch register).

I should point out that this is based only on the material printed in your column, as I do not own Neon. I do, however, have Mach 1 (V1.2), and I am looking forward to more coverage of it in future issues of MacTutor.

Russell Finn

 NOP
 NOP
 MOVE.L A7,D0
>>>>> ADDQ.L #3,A7     <<<<<
 MOVE.L D0,A7
 etc. etc.

I didn't even get a chance to look at the registers! As soon as the program hits the ADDQ.L instruction, the screen goes dark, bing! reset! Also, running right through that piece of code (setting a breakpoint after the point where A7 was restored) resulted in the same crash. Therefore, this behavior should have nothing to do with A7 being used intermediately by Macsbug. I see two explanations: Either an interrupt occuring while A7 is set to a wrong value or a peculiarity of the 68000, which makes the machine go reset when this instruction is encountered (???). At any rate, the designers of NEON never seem to have tested their IC! definition, otherwise they would have noticed it]

A last comment: we have received a nicely laid out newsletter of the MacForth User's group, which can be contacted at

MFUG,

With the variety of threaded code systems for the Macintosh around and being actively used, I think it is a good idea to keep the topics dealt with in this column as general as possible; even though I am using Mach1 for my work at the moment, most of the things apply to other Forths as well.

What would help us a great deal, of course, is feedback from you readers 'out there'. If you have pieces of information, notes or even whole articles on Forth aspects that you think would be of interest to others (and if it interested you, it will interest others), please, send them in.

Listing 1: 32 bit FP multiply, first revision (and hopefully the last one)
CODE     S*     
         MOVE.L  (A6)+,D1
         BEQ     @zero
         MOVE.L  (A6)+,D0
         BEQ     @end
         MOVE.L  D0,D2
         MOVE.L  D1,D3
         SWAP.W  D2
         SWAP.W  D3
         CLR.W   D4
         CLR.W   D5
         MOVE.B  D2,D4
         MOVE.B  D3,D5
         BSET    #7,D4
         BSET    #7,D5
(        ANDI.W  #$FF80,D2 )
         DC.L    $0242FF80
(        ANDI.W  #$FF80,D3 )
         DC.L    $0243FF80
         ROL.W   #1,D2
         ROL.W   #1,D3
         SUBI.W  #$7F00,D2
         SUBI.W  #$7F00,D3
         ADD.W   D2,D3
         BVS     @ovflchk
         MOVE.W  D4,D2  
         MULU.W  D1,D2  
         MULU.W  D0,D1  
         MULU.W  D5,D0  
         MULU.W  D4,D5 
         ADD.L   D2,D0  
         MOVE.W  D5,D1 
         SWAP.W  D1
         ADD.L   D1,D0  
         BPL     @nohibit
     ADDI.W  #$100,D3
         BVC     @round
         BRA     @ovflchk
@nohibit ADD.L   D0,D0
@round   BTST    #7,D0
         BEQ     @blk.exp
         BTST    #6,D0
         BNE     @incr
         BTST    #8,D0
         BEQ     @blk.exp
@incr    ADDI.L  #$80,D0
         BCC     @blk.exp
         ADDI.W  #$100,D3
         BVC     @blk.exp
@ovflchk BPL     @makezero
         MOVE.L  #$7F800000,-(A6)  
         RTS
@makezero  CLR.L D0
         MOVE.L  D0,-(A6)
         RTS
@zero    CLR.L D0
         MOVE.L  D0,(A6)
         RTS
@blk.exp ADDI.W  #$7F00,D3
         BLE     @makezero
         ROR.W   #1,D3
(        ANDI.W  #$FF80,D3 )
         DC.L    $0243FF80
         LSR.L   #8,D0
         BCLR    #23,D0
         SWAP.W  D3
         CLR.W   D3
         OR.L    D3,D0
@end     MOVE.L  D0,-(A6)
         RTS     
END-CODE          
Listing 2: Example definitions for exponential and natural logarithm, Mach1 
only forth definitions also assembler also sane
include" add.sub"
include" mul.sp"
include" div.sp"
(  files  I keep my floating point routines )

CODE 4*
     MOVE.L (A6)+,D0
     ASL.L  #2,D0
     MOVE.L D0,-(A6)
     RTS
END-CODE MACH

( extract biased exponent & mantissa 
from 32 bit FP # )

CODE get.exp
     MOVE.L  (A6)+,D0
     MOVE.L  D0,D1
     SWAP.W  D0
     LSR.W   #7,D0
     ANDI.L  #$FF,D0
     MOVE.L  D0,-(A6)
     ANDI.L  #$7FFFFF,D1
     ORI.L   #$3F800000,D1
     MOVE.L  D1,-(A6)
     RTS
END-CODE
   
CODE stoi  
        MOVE.L  (A6)+,D0
        MOVE.L  D0,D1
        SWAP.W  D0
        LSR.W   #7,D0
        SUBI.B  #127,D0
        BMI     @zero
        BEQ     @one
        ANDI.L  #$7FFFFF,D1
        BSET    #23,D1
        CMP.B   #8,D0
        BCC     @long.shift
        LSL.L   D0,D1
        CLR.W   D1
        SWAP.W  D1
        LSR.L   #7,D1
        MOVE.L  D1,-(A6)
        RTS
@long.shift
        LSL.L   #7,D1
        SUBQ.B  #7,D0
        CLR.L   D2
@shifts LSL.L   #1,D1
        ROXL.L  #1,D2
        SUBQ.B  #1,D0
        BNE     @shifts
        CLR.W   D1
        SWAP.W  D1
        LSR.L   #7,D1
        LSL.L   #8,D2
        ADD.L   D2,D2
        OR.L    D2,D1
        MOVE.L  D1,-(A6)
        RTS
@zero   CLR.L   D0
        MOVE.L  D0,-(A6)
        RTS
@one    MOVEQ.L #1,D0
        MOVE.L  D0,-(A6)
        RTS
END-CODE

: s>i dup 0< if stoi negate else stoi then ;

CODE itos
        MOVE.L  (A6)+,D0
        BEQ     @zero
        CLR.L   D1
        MOVE.L  #$7F,D2
@shifts CMPI.L  #1,D0
        BEQ     @one
        LSR.L   #1,D0
        ROXR.L  #1,D1
        ADDQ.L  #1,D2
        BRA     @shifts
@one    LSR.L   #8,D1
        LSR.L   #1,D1
        SWAP.W  D2
        LSL.L   #7,D2
        BCLR    #31,D2
        OR.L    D2,D1
        MOVE.L  D1,-(A6)
        RTS
@zero   MOVE.L  D0,-(A6)
        RTS
END-CODE        
hex
: i>s dup 0< if negate itos 80000000 or
 else itos then ;
decimal
 
: s. s>f f. ;

vocabulary maths also maths definitions

decimal
fp 9 float

-inf f>s constant -infinity
 inf f>s constant  infinity

1.0  f>s constant one
10.  f>s constant ten
100. f>s constant hun
pi f>s constant pi.s
2.718281828  f>s constant eu

( exponential, natural log )

 .9999964239 f>s constant a1ln
-.4998741238 f>s constant a2ln
 .3317990258 f>s constant a3ln
-.2407338084 f>s constant a4ln
 .1676540711 f>s constant a5ln
-.0953293897 f>s constant a6ln
 .0360884937 f>s constant a7ln
-.0064535442 f>s constant a8ln

variable ln2table 1020 vallot
  2.0 fln    f>s constant ln2
: fill.ln2table
    256 0 do ln2 i 127 - i>s s*
             i 4* ln2table + !
          loop
;
: ln.base 
    one s- a8ln over s*
           a7ln s+ over s*
           a6ln s+ over s*
           a5ln s+ over s*
           a4ln s+ over s*
           a3ln s+ over s*
           a2ln s+ over s*
           a1ln s+ s*
;
: ln dup 0> if get.exp
               ln.base
               swap 4* ln2table + @
               s+
            else drop -infinity
            then
;
: lnacc
  1000 0 do 
    i . i i>s ln  dup s.
        i i>f fln fdup f.
          s>f f- f. cr
    loop
;
variable exptable 700 vallot
: fill.exptable
      176 0 do i 87 - i>f fe^x f>s
             i 4* exptable + !
          loop
;
  
-.9999999995 f>s constant a1exp
 .4999999206 f>s constant a2exp
-.1666653019 f>s constant a3exp
 .0416573745 f>s constant a4exp
-.0083013598 f>s constant a5exp
 .0013298820 f>s constant a6exp
-.0001413161 f>s constant a7exp

: exp.base a7exp over s*
           a6exp s+ over s*
           a5exp s+ over s*
           a4exp s+ over s*
           a3exp s+ over s*
           a2exp s+ over s*
           a1exp s+ s*
           one s+
           one swap s/
;
: exp dup s>i swap over i>s s- exp.base swap 
          dup -87 < if 2drop 0
     else dup  88 > if 2drop infinity
     else 87 + 4* exptable + @ 
           ( get exp of integer part ) s* then
     then
;
: expacc
  1000 0 do 
    i . i i>s hun  s/  exp  dup s.
        i i>f 100. f/ fe^x fdup f.
          s>f f- f. cr
    loop
;
:  emptyloop 0  1000 0 do  dup  drop loop  drop ;
: femptyloop 0. 1000 0 do fdup fdrop loop fdrop ;
: testexp  ten one s+ 1000 0 do  dup  exp  drop loop  drop ;
: testfexp        11. 1000 0 do fdup fe^x fdrop loop fdrop ;
: testln  ten one s+ 1000 0 do  dup  ln  drop loop  drop ;
: testfln        11. 1000 0 do fdup fln fdrop loop fdrop ;
: speed.test cr
  ." Testing 32 bit routines..." cr
 ."    empty..." counter emptyloop timer cr
."      exp..." counter testexp timer cr
 ."       ln..." counter testln timer cr cr
    ." Testing SANE routines..." cr
    ."    empty..." counter femptyloop timer cr
    ."      exp..." counter testfexp timer cr
    ."       ln..." counter testfln timer cr
;