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Pi, adventures with a 32 bit ARM development board
a techie view

by Ed Thelen

Also see Pi, my adventures with FreeBasic in a Windows PC
Van Snyder sent me this URL. He must be cruel, I don't even like numbers !!

Table of Contents:

Introduction
Long ago, about 1964, and far away, well - OK - Phoenix, Arizona
I was getting paid to play with computers :-)) how about that for fun ?? !!

So I got to enjoy computing PI

Recently there has been a flood of very low priced "evaluation" boards so that you can try out and become familiar with some vendor's computer chips. I finally decided to become modern and try one.

This Quick Start Guide with Lessons 0-10 on video seemed attractive. Lesson 0 tells how to obtain the free development software with control interface and simulator. The cheap USB powered $13 board with flashing lights caught my attention ;-)) Lesson 0 also tells how to order this board. (The control interface can also work with many other boards.)

Overview and results
So I ordered the board from Ti, and started watching and doing the lessons, using the simulator, until the hardware arrived.

When the hardware arrived, I had trouble getting the software to recognize the hardware. ( OK, I'm not real good at following directions.) Finally called up and was guided to some unobvious box to click. Magic happened - all became happy - the simulated code in the lessons worked just fine in the Ti hardware.

Lesson # 10 complete - time to compute Pi ;-))
But how to display/output the result???
The results of some experimentation yielded:

  • doing "printf("Hello World xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx\n");
    to the console output box in the development screen is slow, about 10 characters/second.
  • stdio.h seemed stripped of file I/O functions such as "fopen( );" and "fclose( );"
    so that path seemed out
  • I settled on displaying the decimal values as part of a HEX formatted display, example
    Each hex digit contains the decimal value of the decimal value in that position. That way the hex memory dump (available "instantly" is readable and comparable to decimal values for Pi in books and on the Internet :-)) - No need for a printer ;-))
    Unfortunately, the formatting is not ideal, nor is the formatting in the references ;-)

OK, we seem to be getting valid looking results compared to references :-))

Now, the usual thing is to
- optimize the code for faster results
- optimize memory for more possible digits in available memory

Tweaking things, and reusing one of the computing fields to display results, I was able to make this board with 32K bytes of SRAM output "decimal" 20,000 digits (see picture above for a little of the output) in under 3 minutes :-))
(The compiled software is executed out of e-prom, not part of the SRAM.)

And none of the chips on the board felt warm to the touch ;-))

Development environment
As mentioned above, I used Quick Start Guide. Which recommended a development system from IAR Systems. Use the long term but size limited option as mentioned in the Quick Start Guide.

The two main user displays are
edit

run

Compute the arctan (1/n)
We will use the 300 year old Machin formula for computing Pi.

which uses two arctangents.

A convenient formula for the arctan is the series

One of the arctangents is for the angle 1/5 radians, so the formula for that is
arctan(1/5) = 1/5 - 1/(3 x 5^3) + 1/(5 x 5^5) - ...
which can be rewritten
arctan(1/5) = 1/5 - 1/((2n+1) x 5^(2n+1) + ... - ...
   n=1 n=2 n=3
Lots of divisions, with alternating signs of successive terms.

This can conveniently computed by:

  1. zero arrays of unsigned integers called d1, d2, and acc
  2. set i to 1
  3. assume a decimal point to the right of the left (high order) integer in array d1
  4. place a 1 as the left integer in array d1
  5. unsigned divide array d1 by 5 giving array d1
  6. unsigned divide array d1 by i giving array d2
  7. add 2 to i
  8. add (alternately subtract) array d2 and array acc giving array acc
  9. unsigned divide array d1 by 5^2 or 25 giving array d1
  10. if array d1 not equal to zero, go to 6 above
The arctan of 1/5 (in this case) is in array acc

A slightly optimized version in "C" follows:


// --------------------------
//   int atan( divisor, initial_add_sub ), // does NOT clear accumulator !!
//                           division by 239^2  works just fine
void  atan( int  recip_radians, int sub1st_flag   );
void   atan( int recip_radians, int  sub1st_flag ){
     unsigned int loc_divisor = recip_radians*recip_radians ;
     int loc_flag = sub1st_flag;
     int all0s = 0;
     //  initialize atan_d1 field 
     zerof( atan_d1, f_len);  // zero most working fields
     zerof( atan_d2, f_len);
     atan_d1[0] = 1; // prepare for recip radians
     div( atan_d1, atan_d1, recip_radians, all0s, f_len ); // d1 correct
     if ( (loc_flag&1) == 0) 
           add( atan_d1, atan_acc, all0s, f_len ); // add
     else 
            sub( atan_d1, atan_acc, all0s, f_len ); // subtract
     loc_flag ^= 1;  // flip flag
     // fields correctly set for taylor series
     unsigned int divn = 3; // initial t2 divisor
     // divide d1 by divisor
     while (all0s != (f_len)){
      //void    div( unsigned short *fi, unsigned short *fo, 
            // unsigned int divisor, int start, int len );
       div( atan_d1, atan_d1, loc_divisor, all0s, f_len );  // divide d1 giving d1
       div( atan_d1, atan_d2, divn, all0s, f_len );  // divide d1 giving d2
       divn += 2;  // update, prep for next round
       if (divn > max_ndiv)  max_ndiv = divn;   // remember max size
       if ( (loc_flag&1) == 0) 
           add( atan_d2, atan_acc, all0s, f_len ); // add
       else 
            sub( atan_d2, atan_acc, all0s, f_len ); // subtract
       loc_flag ^= 1;  // flip flag
      // if ( (atan_d1[all0s] ^ (atan_d1[all0s+1] ) == 0) {
       if ( atan_d1[all0s] == 0) 
          ++all0s; //  start at next word next time
     }
}

Crippled Divide and work around
"In the goode olde dayze", when computers were simple, they had two basic computing registers, usually called "A" and "B" or "Q". They were often concatenated for shifting.

A multiply, which can yield about twice as many digits or bits as the inputs usually used the concatenated "A" and "B" registers for the results.

A divide usually used the concatenated "A" and "B" registers for the dividend, and left the quotient in the "A" register and the remainder in the "B" register.

Most Pi algorithms do lots of dividing, and to do extended precision work, you need the remainder.

Then two things happened, both inconvenient for Pi finders

  1. "Higher Level Languages" did not make the remainders available, or if they did, they caused another divide to get the remainder (again)
  2. Computers came with "register files", 6 or 14 or 29 or ... general purpose registers -
    Some of the computers placed the remainder in the next odd # register if the quotient was in an even numbered register
    However, modern RISC computers just trash the remainder, if you want it, compute it.
Oddly, since most people don't want remainders, and High Level Languages don't know what to do with 'em, the results are acceptable to most people.

Work Around

Assume your divisor is positive, and no larger than than 1/2 your register lengths
Example (32 bit registers with 16 bit divisors max, and an unsigned divide instruction)
then the following "C" code will work



// --------------------------
void div( unsigned short *fi, unsigned short *fo, unsigned int divisor, int start, int len );
   // "fi" is input array, "fo" is output array, may be same area, 
   //  "start" is where to start dividing, to save time because high order becomes all zeros
   //  "len" is total length of working array
void div( unsigned short *fi, unsigned short *fo, unsigned int divisor, int start, int len ){
    int index = start;  // get to register
    int endi = len; 
    unsigned int remainder = 0;  // storage for remainder 
    unsigned int t = 0;          // storage for quotient
    unsigned int dividend = 0;   // dividend 
                                 /// ( a 32 bit register, with register zeroing rules )
    while (index <= endi) {
        dividend = fi[index];    // low 16 bits of dividend, 
                                 //hi order zeroed because it is a register
        dividend |= remainder << (sizeof(unsigned short)*8); 
                                 // form hi order of dividend (register)
        t = dividend/divisor;    // divide to get quotient
        fo[index] = t;           // quotient, send to array of 16 bit unsigned ints
        remainder = dividend - (t*divisor); // form remainder
        ++index;                 // go hi order to low order
    }
 }

Notes

  1. unsigned int ;-)) The compiler seems to do a good job using unsigned multiplies and divides :-)) The high order bit is not treated as a special bit - so I can use all 16 bits using 239^2 = 57121 as a divisor :-))

  2. The available SRAM in the machine is 32K bytes, starting at 0x20000000 . (The program is stored and run from e-prom, in low memory.)
    - I wound up using 3 fields of 5006 16 bit words, reusing the zeroed d1 array as the display array.
    - This producing 20,000 decimal characters.
    - The last 8 characters matched values from the Internet which I accepted as valid.
    - The time with the big divisors was 178 seconds.

  3. Randy was wondering if he could multiply by the reciprocal of 5 instead of dividing by 5. The leading hex digits of 1/5 is 0.333333333333333333333333333333 ... in hex The leading hex digits of 1/25 is also a repeating fraction as are 1/239 and 1/(239^2) . I think a good consideration, but no cigar :-((

  4. > How many decimal digits is that array worth?

    I was figuring on outputting one decimal digit for every hex digit. Then I thought that wasteful - then thought some more and decided that I could really get 9 decimal digits (plus a little) for every 8 hex digits.

    Now - how did I figure that ?
         4 hex digits (unsigned) = 65,536 decimal which is 4 decimal digits plus a little over half a digit
         8 hex digits (unsigned) = 4,294,967,296 decimal which is 9 decimal digits plus a little under half a digit
         so, being truly lazy, I emit 4 decimal digits per 4 hex digits

    Experimenting with printing more digits until low order digits are incorrect indicate that it is "safe" to continue printing 14 % more digits, but 16 % more yields low order errors.

    Now - about getting those digits printed, and in my usual form
         nnnnn nnnnn nnnnn nnnnn nnnnn nnnnn nnnnn nnnnn nnnnn nnnnn * 10^-eeeeee0

  5. What about errors creeping in from the truncated least significant part?

    I think that the the division process sweeps digitization errors "out the low end", but the adding and subtracting into the accumulator causes error to accumulate. Using 5006 16 bit integers, there are about 16,000 iterations of dividing by 25 and adding/subtracting the division by (2n-1) and less than 10% of that in the process of dividing by 239^2. Assuming a maximum digitation error of 1 count per addition/subtraction would yield a maximum error of 1 16 bit int or less than 4 hex digits ( about 65,000 ).

    We are dropping off 1 decimal digit per 8 decimal digits "printed". As there are 5006/2 8 hex digit units, for our 20,000 decimal digit conversion, we are dropping of about 2,500 decimal digits - a much more than safe output truncation :-))

The total "C" code
/* Pi x  */ 
//  there is a limit of 32K bytes SRAM starting at 0x20000000
//     this version does over 20,000 decimal digits in 170 seconds
//                                                
// game plan 
//  Pi/4 = 4arctan(1/5)-arctan(1/239)
//  pi = 16arctan(1/5)-4arctan(1/239)
//     5^2 = 25    239^2 = 57121    2^16 = 65,536
//          the compiler generates unsigned divide and multiply which work well
// a) make 3 large areas, atan_d1[], atan_d2[], atan_acc[] 
//         area atan_d1[] is reused as an output field
// b) make subroutines div( *fi, *fo, divisor, start, len ),
//                 int    add( *fi, *fo,  start, len ),
//                         how test for overflow?
//                     sub( *fi, *fo, start, len  ),
//                         how test for underflow?
//                     atan( divisor, initial_add_sub ), // status is some error -
//                         how to handle division by 239^2  ??
//                           do /239 twice ??
//                     zerof( *fi, len ),

#define f_len 5006  // actual field length - 2, allow two guard ints
//   //   5006 = 4096 + 512 + 256 + 128 +16 - 2,  
//               time was 2 min +50 sec     max_divn = 33,499
//                  over 20,,000 decimal digits
   //   int is 16 bits -
static unsigned  int max_ndiv;
static int add_sub_error;
static unsigned short atan_d1[f_len+2], atan_d2[f_len+2], atan_acc[f_len+2]; //  guard at end
//                  note: atan_d1 is re-used as the decimal output field  
// --------------------------
void zerof( unsigned short *fi, int lim   );
void zerof( unsigned short *fi, int lim ){
  int counter = 0;
  while ( counter < lim) {
    fi[counter] = 0x0000;
    ++counter;
  }
}
// --------------------------
// add fi into fo
void   add( unsigned short *fi, unsigned short *fo, int zeros, int len );
void   add( unsigned short *fi, unsigned short *fo, int zeros, int len ){
  int carry = 0;     // yes, initialize carry
  int index = len;  // start at low end
  int t;
  while ( (index >= zeros) || (carry != 0) ) {
    t = carry + fi[index] + fo[index];
    fo[index] = t;
    carry = t >> (sizeof(unsigned short)*8);
    --index;
  }
  if ( index < -1 ) add_sub_error |= 1;  // bit 1 is add overflow
}

// --------------------------
// sub fi from fo  // error if return not 0
void    sub( unsigned short *fi, unsigned short *fo, int zeros, int len );
void    sub( unsigned short *fi, unsigned short *fo, int zeros, int len ){
  int borrow = 0;     // yes, initialize borrow
  int index = len;  // start at low end
  int t;
  while ( index >= zeros ) {
    t =  fo[index] - borrow - fi[index];
    fo[index] = t;
    borrow = t >> (sizeof(unsigned short)*8) & 1;
    --index;
  }
  if ( index < -1 ) add_sub_error |= 2;  // bit 2 is subtract underflow
}

// --------------------------
//    void mpy( *fio, start, multiplier  ), // multiply in place by multiplier
//            return is int part[0]
void    mpy( unsigned short *fio,  int start, int multiplier );
void    mpy( unsigned short *fio,   int start, int multiplier  ){
  unsigned int carry = 0;     // hi part of product
  int index = start;  // start at low end
  unsigned int t;
  while ( index >= 0 ) {
   // t = carry + fi[index] + fo[index];
    t = ( fio[index] * multiplier) + carry;
    fio[index] = t;
    carry = t >> (sizeof(unsigned short)*8);
    --index;
  }
}


// --------------------------
void div( unsigned short *fi, unsigned short *fo, unsigned int divisor, int start, int len );
   // "fi" is input array, "fo" is output array, may be same area, 
   //  "start" is where to start dividing, to save time because high order becomes all zeros
   //  "len" is total length of working array
void div( unsigned short *fi, unsigned short *fo, unsigned int divisor, int start, int len ){
    int index = start;  // get to register
    int endi = len; 
    unsigned int remainder = 0;  // storage for remainder 
    unsigned int t = 0;          // storage for quotient
    unsigned int dividend = 0;   // dividend 
                                 /// ( a 32 bit register, with register zeroing rules )
    while (index <= endi) {
        dividend = fi[index];    // low 16 bits of dividend, 
                                 //hi order zeroed because it is a register
        dividend |= remainder << (sizeof(unsigned short)*8); 
                                 // form hi order of dividend (register)
        t = dividend/divisor;    // divide to get quotient
        fo[index] = t;           // quotient, send to array of 16 bit unsigned ints
        remainder = dividend - (t*divisor); // form remainder
        ++index;                 // go hi order to low order
    }
 }


// --------------------------
// int atan( divisor, initial_add_sub ), // does NOT clear accumulator !!
//                           division by 239^2  works just fine
void  atan( int  recip_radians, int sub1st_flag   );
void   atan( int recip_radians, int  sub1st_flag ){
     unsigned int loc_divisor = recip_radians*recip_radians ;
     int loc_flag = sub1st_flag;
     int all0s = 0;
     //  initialize atan_d1 field 
     zerof( atan_d1, f_len);  // zero most working fields
     zerof( atan_d2, f_len);
     atan_d1[0] = 1; // prepare for recip radians
     div( atan_d1, atan_d1, recip_radians, all0s, f_len ); // d1 correct
     if ( (loc_flag&1) == 0) 
           add( atan_d1, atan_acc, all0s, f_len ); // add
     else 
            sub( atan_d1, atan_acc, all0s, f_len ); // subtract
     loc_flag ^= 1;  // flip flag
     // fields correctly set for taylor series
     unsigned int divn = 3; // initial t2 divisor
     // divide d1 by divisor
     while (all0s != (f_len)){
      //void    div( unsigned short *fi, unsigned short *fo,
           //  unsigned int divisor, int start, int len );
       div( atan_d1, atan_d1, loc_divisor, all0s, f_len );  
          // divide d1 giving d1
       div( atan_d1, atan_d2, divn, all0s, f_len );  // divide d1 giving d2
       divn += 2;  // update, prep for next round
       if (divn > max_ndiv)  max_ndiv = divn;   // remember max size
       if ( (loc_flag&1) == 0) 
           add( atan_d2, atan_acc, all0s, f_len ); // add
       else 
            sub( atan_d2, atan_acc, all0s, f_len ); // subtract
       loc_flag ^= 1;  // flip flag
      // if ( (atan_d1[all0s] ^ (atan_d1[all0s+1] ) == 0) {
       if ( atan_d1[all0s] == 0) 
          ++all0s; //  start at next word next time
     }
}


//   form ascii array from atan_acc, atan_acc is destroyed
//   conv_bin_char( int len);
void  conv_bin_char( int len);
void  conv_bin_char( int len){
      int t1,t2,t3,i,wdivisor;
      int index = 0;

      zerof( atan_d1, f_len);  // changed from ascii
      while ( index <= len) {
           t1 = 0;   // initialize display
           t2 = 0;  // initialize working
           wdivisor = 1000;
           i = 3;
           t3 = atan_acc[0];
           while (wdivisor > 0 ) {
              t2 = t3/wdivisor;     // form a "decimal" digit
              t1 |= t2 << (i*4);    // insert next "decimal" digit
              t3 = t3 - (t2*wdivisor); // remainder
              wdivisor /= 10;
              --i;
            } 
         atan_d1[index] = t1;  // insert "decimal" into output field
         atan_acc[0] = 0;      // clear hi order accum
         mpy( atan_acc,   f_len - (index/2), 10000 ); // multiply, shorting field
         ++index;  // go to next word
      }    
}
       


//---------------------------------------------------------------
int main() {
      max_ndiv = 0;      // clear warns and errors
      add_sub_error = 0;
      zerof( atan_acc, f_len);  // clear result
      atan( 5, 0  );  // form arctan (1/5)
       mpy(atan_acc, f_len, 4 );   // follow Machin
      atan( 239, 1  );  // form arctan (1/239)
       mpy(atan_acc, f_len, 4 );   // follow Machin
      conv_bin_char(   f_len);  // "print" it out 
   
    return 0;
}

Pi to 20,000 digits
Images of the "decimalized hex digits" in a memory dump of SRAM in the Ti board are available in 11 frames as follows:
1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th

I added a little formatting and printing code to the above "C" program which sent about 20,000 digits of Pi to the "Console" display of the user interface, at about 10 characters/second. And here they are :-))


                                                      00003 xE     0
14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 xE   -50
58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 xE  -100
82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 xE  -150
48111 74502 84102 70193 85211 05559 64462 29489 54930 38196 xE  -200
44288 10975 66593 34461 28475 64823 37867 83165 27120 19091 xE  -250
45648 56692 34603 48610 45432 66482 13393 60726 02491 41273 xE  -300
72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 xE  -350
78925 90360 01133 05305 48820 46652 13841 46951 94151 16094 xE  -400
33057 27036 57595 91953 09218 61173 81932 61179 31051 18548 xE  -450
07446 23799 62749 56735 18857 52724 89122 79381 83011 94912 xE  -500
98336 73362 44065 66430 86021 39494 63952 24737 19070 21798 xE  -550
60943 70277 05392 17176 29317 67523 84674 81846 76694 05132 xE  -600
00056 81271 45263 56082 77857 71342 75778 96091 73637 17872 xE  -650
14684 40901 22495 34301 46549 58537 10507 92279 68925 89235 xE  -700
42019 95611 21290 21960 86403 44181 59813 62977 47713 09960 xE  -750
51870 72113 49999 99837 29780 49951 05973 17328 16096 31859 xE  -800
50244 59455 34690 83026 42522 30825 33446 85035 26193 11881 xE  -850
71010 00313 78387 52886 58753 32083 81420 61717 76691 47303 xE  -900
59825 34904 28755 46873 11595 62863 88235 37875 93751 95778 xE  -950
18577 80532 17122 68066 13001 92787 66111 95909 21642 01989 xE -1000
38095 25720 10654 85863 27886 59361 53381 82796 82303 01952 xE -1050
03530 18529 68995 77362 25994 13891 24972 17752 83479 13151 xE -1100
55748 57242 45415 06959 50829 53311 68617 27855 88907 50983 xE -1150
81754 63746 49393 19255 06040 09277 01671 13900 98488 24012 xE -1200
85836 16035 63707 66010 47101 81942 95559 61989 46767 83744 xE -1250
94482 55379 77472 68471 04047 53464 62080 46684 25906 94912 xE -1300
93313 67702 89891 52104 75216 20569 66024 05803 81501 93511 xE -1350
25338 24300 35587 64024 74964 73263 91419 92726 04269 92279 xE -1400
67823 54781 63600 93417 21641 21992 45863 15030 28618 29745 xE -1450
55706 74983 85054 94588 58692 69956 90927 21079 75093 02955 xE -1500
32116 53449 87202 75596 02364 80665 49911 98818 34797 75356 xE -1550
63698 07426 54252 78625 51818 41757 46728 90977 77279 38000 xE -1600
81647 06001 61452 49192 17321 72147 72350 14144 19735 68548 xE -1650
16136 11573 52552 13347 57418 49468 43852 33239 07394 14333 xE -1700
45477 62416 86251 89835 69485 56209 92192 22184 27255 02542 xE -1750
56887 67179 04946 01653 46680 49886 27232 79178 60857 84383 xE -1800
82796 79766 81454 10095 38837 86360 95068 00642 25125 20511 xE -1850
73929 84896 08412 84886 26945 60424 19652 85022 21066 11863 xE -1900
06744 27862 20391 94945 04712 37137 86960 95636 43719 17287 xE -1950
46776 46575 73962 41389 08658 32645 99581 33904 78027 59009 xE -2000
94657 64078 95126 94683 98352 59570 98258 22620 52248 94077 xE -2050
26719 47826 84826 01476 99090 26401 36394 43745 53050 68203 xE -2100
49625 24517 49399 65143 14298 09190 65925 09372 21696 46151 xE -2150
57098 58387 41059 78859 59772 97549 89301 61753 92846 81382 xE -2200
68683 86894 27741 55991 85592 52459 53959 43104 99725 24680 xE -2250
84598 72736 44695 84865 38367 36222 62609 91246 08051 24388 xE -2300
43904 51244 13654 97627 80797 71569 14359 97700 12961 60894 xE -2350
41694 86855 58484 06353 42207 22258 28488 64815 84560 28506 xE -2400
01684 27394 52267 46767 88952 52138 52254 99546 66727 82398 xE -2450
64565 96116 35488 62305 77456 49803 55936 34568 17432 41125 xE -2500
15076 06947 94510 96596 09402 52288 79710 89314 56691 36867 xE -2550
22874 89405 60101 50330 86179 28680 92087 47609 17824 93858 xE -2600
90097 14909 67598 52613 65549 78189 31297 84821 68299 89487 xE -2650
22658 80485 75640 14270 47755 51323 79641 45152 37462 34364 xE -2700
54285 84447 95265 86782 10511 41354 73573 95231 13427 16610 xE -2750
21359 69536 23144 29524 84937 18711 01457 65403 59027 99344 xE -2800
03742 00731 05785 39062 19838 74478 08478 48968 33214 45713 xE -2850
86875 19435 06430 21845 31910 48481 00537 06146 80674 91927 xE -2900
81911 97939 95206 14196 63428 75444 06437 45123 71819 21799 xE -2950
98391 01591 95618 14675 14269 12397 48940 90718 64942 31961 xE -3000
56794 52080 95146 55022 52316 03881 93014 20937 62137 85595 xE -3050
66389 37787 08303 90697 92077 34672 21825 62599 66150 14215 xE -3100
03068 03844 77345 49202 60541 46659 25201 49744 28507 32518 xE -3150
66600 21324 34088 19071 04863 31734 64965 14539 05796 26856 xE -3200
10055 08106 65879 69981 63574 73638 40525 71459 10289 70641 xE -3250
40110 97120 62804 39039 75951 56771 57700 42033 78699 36007 xE -3300
23055 87631 76359 42187 31251 47120 53292 81918 26186 12586 xE -3350
73215 79198 41484 88291 64470 60957 52706 95722 09175 67116 xE -3400
72291 09816 90915 28017 35067 12748 58322 28718 35209 35396 xE -3450
57251 21083 57915 13698 82091 44421 00675 10334 67110 31412 xE -3500
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13384 13385 36842 11978 93890 01852 95691 96780 45544 82858 xE-17350
48370 11709 67212 53533 87586 21582 31013 31038 77668 27211 xE-17400
57269 49518 17958 97546 93992 64219 79155 23385 76623 16762 xE-17450
75475 70354 69941 48929 04130 18638 61194 39196 28388 70543 xE-17500
67774 32242 76809 13236 54494 85366 76800 00010 65262 48547 xE-17550
30558 61598 99914 01707 69838 54831 88750 14293 89089 95068 xE-17600
54530 76511 68033 37322 26517 56622 07526 95179 14422 52808 xE-17650
16517 16677 66727 93035 48515 42040 23817 46089 23283 91703 xE-17700
27542 57508 67655 11785 93950 02793 38959 20576 68278 96776 xE-17750
44531 84040 41855 40104 35134 83895 31201 32637 83692 83580 xE-17800
82719 37831 26549 61745 99705 67450 71833 20650 34556 64403 xE-17850
44904 53627 56001 12501 84335 60736 12227 65949 27839 37064 xE-17900
78426 45676 33881 88075 65612 16896 05041 61139 03906 39601 xE-17950
62022 15368 49410 92605 38768 87148 37989 55999 91120 99164 xE-18000
64644 11918 56827 70045 74243 43402 16722 76445 58933 01277 xE-18050
81586 86952 50694 99364 61017 56850 60167 14535 43158 14801 xE-18100
05458 86056 45501 33203 75864 54858 40324 02987 17093 48091 xE-18150
05562 11671 54684 84778 03944 75697 98042 63180 99175 64228 xE-18200
09873 99876 69732 37695 73701 58080 68229 04599 21236 61689 xE-18250
02596 27304 30679 31653 11494 01764 73769 38735 14093 36183 xE-18300
32161 42802 14976 33991 89835 48487 56252 98752 42387 30775 xE-18350
59555 95546 51963 94401 82184 09984 12489 82623 67377 14672 xE-18400
26061 63364 32964 06335 72810 70788 75816 40438 14850 18841 xE-18450
14318 85988 27694 49011 93212 96827 15888 41338 69434 68285 xE-18500
90066 64080 63140 77757 72570 56307 29400 49294 03024 20498 xE-18550
41656 54797 36705 48558 04458 65720 22763 78404 66823 37985 xE-18600
28271 05784 31975 35417 95011 34727 36257 74080 21347 68260 xE-18650
45022 85157 97957 97647 46702 28409 99561 60156 91089 03845 xE-18700
82450 26792 65942 05550 39587 92298 18526 48007 06837 65041 xE-18750
83656 20945 55434 61351 34152 57006 59748 81916 34135 95567 xE-18800
19649 65403 21872 71602 64859 30490 39787 48958 90661 27250 xE-18850
79482 82769 38953 52175 36218 50796 29778 51461 88432 71922 xE-18900
32238 10158 74445 05286 65238 02253 28438 91375 27384 58923 xE-18950
84422 53547 26530 98171 57844 78342 15822 32702 06902 87232 xE-19000
33005 38621 63479 88509 46954 72004 79523 11201 50432 93226 xE-19050
62827 27632 17790 88400 87861 48022 14753 76578 10581 97022 xE-19100
26309 71749 50721 27248 47947 81695 72961 42365 85957 82090 xE-19150
83073 32335 60348 46531 87302 93026 65964 50137 18375 42889 xE-19200
75579 71449 92465 40386 81799 21389 34692 44741 98509 73346 xE-19250
26793 32107 26868 70768 06263 99193 61965 04409 95421 67627 xE-19300
84091 46698 56925 71507 43157 40793 80532 39252 39477 55744 xE-19350
15918 45821 56251 81921 55233 70960 74833 29234 92103 45146 xE-19400
26437 44980 55961 03307 99414 53477 84574 69999 21285 99999 xE-19450
39961 22816 15219 31488 87693 88022 28108 30019 86016 54941 xE-19500
65426 16968 58678 83726 09587 74567 61825 07275 99295 08931 xE-19550
80521 87292 46108 67639 95891 61458 55058 39727 42098 09097 xE-19600
81729 32393 01067 66386 82404 01113 04024 70073 50857 82872 xE-19650
46271 34946 36853 18154 69690 46696 86939 25472 51941 39929 xE-19700
14652 42385 77625 50047 48529 54768 14795 46700 70503 47999 xE-19750
58886 76950 16124 97228 20403 03995 46327 88306 95976 24936 xE-19800
15101 02436 55535 22306 90612 94938 85990 15734 66102 37122 xE-19850
35478 91129 25476 96176 00504 79749 28060 72126 80392 26911 xE-19900
02777 22610 25441 49221 57650 45081 20677 17357 12027 18024 xE-19950
29681 06203 77657 88371 66909 10941 80744 87814 04907 55178 xE-20000
20385 65390 99104 77594 1413

Finding a little more memory for slightly larger arrays,
     and printing  9/8 x 4 decimal digits per 16 bit word in the array,

20385 65390 99104 77594 14132 15432 84406 25030 18027 57169 xE-20050
65082 09642 73484 14695 72639 78842 56008 45312 14065 93580 xE-20100
90412 71135 92004 19759 85136 25479 61606 32288 73618 13673 xE-20150
73244 50607 92441 17639 97597 46193 83584 57491 59880 97667 xE-20200
44709 30065 46342 42346 06342 37474 66608 04317 01260 05205 xE-20250
59284 93695 94143 40814 68529 81505 39471 78900 45183 57551 xE-20300
54125 22359 05906 87264 87863 57525 41911 28887 73717 66374 xE-20350
86027 66063 49603 53679 47026 92322 97186 83277 17393 23619 xE-20400
20077 74522 12624 75186 98334 95151 01986 42698 87847 17193 xE-20450
96649 76907 08252 17423 36566 27259 28440 62043 02141 13719 xE-20500
92278 52699 84698 84770 23238 23840 05565 55178 89087 66136 xE-20550
01304 77098 43861 16870 52310 55314 91625 17283 73272 86760 xE-20600
07248 17298 76375 69816 33541 50746 08838 66364 06934 70437 xE-20650
20668 86512 75688 26614 97307 88657 01568 50169 18647 48854 xE-20700
16791 54596 50723 42877 30699 85371 39043 00266 53078 39877 xE-20750
63850 32381 82155 35597 32353 06860 43010 67576 08389 08627 xE-20800
04984 18885 95138 09103 04235 95782 49514 39885 90113 18583 xE-20850
58406 67472 37029 71497 85084 14585 30857 81339 15627 07603 xE-20900
56390 76394 73114 55495 83226 69457 02494 13983 16343 32378 xE-20950
97595 56808 56836 29725 38679 13275 05554 25244 91943 58912 xE-21000
84050 45226 95381 21791 31914 51350 09938 46311 77401 79715 xE-21050
12283 78546 01160 35955 40286 44059 02496 46693 07077 69055 xE-21100
48102 88502 08085 80087 81157 73817 19174 17760 17330 73855 xE-21150
47580 06056 01433 77432 99012 72867 72530 43182 51975 79167 xE-21200
92969 96504 14607 06645 71258 88346 97979 64293 16229 65520 xE-21250
16879 73000 35646 30457 93088 40327 48077 18115 55330 90988 xE-21300
70255 05207 68046 30346 08658 16539 48769 51960 04408 48206 xE-21350
59673 79473 16808 64156 45650 53004 98816 16490 57883 11543 xE-21400
45485 05266 00698 23093 15777 65003 78070 46612 64706 02145 xE-21450
75057 93270 96204 78256 15247 14591 89652 23608 39664 56241 xE-21500
05195 51052 23572 39739 51288 18164 05978 59142 79148 16542 xE-21550
63289 20042 81609 13693 77737 22299 98332 70820 82969 95573 xE-21600
77273 75667 61552 71139 22588 05520 18988 76201 14168 00546 xE-21650
87365 58063 34716 03734 29170 39079 86396 52296 13128 01782 xE-21700
67971 72898 22936 07028 80690 87768 66059 32527 46378 40539 xE-21750
76918 48082 04102 19447 19713 86925 60841 62451 12398 06201 xE-21800
13184 54124 47820 50110 79876 07171 55683 15407 88654 39041 xE-21850
21087 30324 02010 68534 19472 30476 66672 17498 69868 54707 xE-21900
67812 05124 73679 24791 93150 85644 47753 79853 79973 22344 xE-21950
56122 78584 32968 46647 51333 65736 92387 20146 47236 79427 xE-22000
87004 25032 55589 92688 43495 92876 12400 75587 56946 41370 xE-22050
56251 40011 79713 31662 07153 71543 60068 76477 31867 55871 xE-22100
48783 98908 10742 95309 41060 59694 43158 47753 97009 43988 xE-22150
39491 44323 53668 53920 99468 79645 06653 39857 38887 86614 xE-22200
76294 43414 01049 88899 31600 51207 67810 35886 11660 20296 xE-22250
11936 39682 13496 07501 11649 83278 56353 16145 16845 76956 xE-22300
87109 00299 97698 41263 26650 23477 16728 65737 85790 85746 xE-22350
64607 72283 41540 31144 15294 18804 78254 38761 77079 04300 xE-22400
01566 98677 67957 60909 96693 60755 94965 15273 63498 11896 xE-22450
41304 33116 62774 71233 88174 06037 31743 97054 06703 10967 xE-22500
67657 48695 35878 96700 31925 86625 94105 10533 58438 46560 xE-22550
23391 79674 92678 44763 70847 49783 33655 57900 73841 91473 xE-22600
19886 27135 25954 62518 16043 42253 72996 28632 67496 82405 xE-22650
80602 96421 14638 64368 64224 72488 72834 34170 44157 34824 xE-22700
81833 30164 05669 59668 86676 95634 91416 32842 64149 74533 xE-22750
34999 94800 02669 98758 88159 35073 57815 19588 99005 39512 xE-22800

Work done in 2012 ??