mjhjmtu

I am mildly curious that though the 6502 provides BCD arithmetic which would be useful for implementing decimal floating point, Commodore BASIC uses, like all (?) Micro-Soft BASIC, binary floating point instead.

Are there any easy example that show precision errors in Commodore BASIC, that would not be present if it would be based on decimal FP?

A classic test of the difference is 0.1 + 0.2 = 0.3; this evaluates to false in pretty much every modern language (since almost all of them use the IEEE floating point that is built into modern hardware). There is even a website devoted to this oddity: 0.30000000000000004.com

I tried this on a C64 emulator (which uses the same BASIC as the PET) and to my astonishment, it correctly evaluated to true. So did some other obvious tests like 0.1 * 10 = 1 and 0.1 + 0.9 = 1 but they worked as well.

What test would give a wrong answer on Commodore BASIC? That is, I'm not asking for a way to get it to demonstrate rounding errors per se; that much is trivial. I'm asking for a way to get it to give a wrong answer, not because it lacks infinite precision, but specifically for(simple) cases where decimal arithmetic would give the right answer. Some Commodore BASIC (MS-BASIC) equivalent to the 0.1 + 0.2 = 0.3 test on IEEE 754.

This example reveals a rounding error under Commodore BASIC V2.0:

 A=0.3:B=0.6:IF A+B<>0.9 THEN PRINT A+B-0.9

Running this on a C64 yields a difference of 2.32830644e-10. Other pairs that fail are 0.4+0.5, 0.6+0.1 and 0.8+0.1. Please note that also the order in which the numbers are summed up affects the result. 0.6+0.1-0.7 yields a difference, while 0.1+0.6-0.7 results to 0.

Here is my favourite example for this problem. I often use it to show Excel's mathematical shortcomings, but not surprisingly it works the same in the C64:

10 A = 0.1
20 B = 0.1
30 FOR I = 1 TO 10
40 D = B
50 B = 20 * A - 19 * B
60 PRINT B
70 A = D
80 NEXT I

In every iteration, the algorithm should be doing 20 * 0.1 - 19 * 0.1 = 0.1, but the output on this simulator is

 .0999999999
 .100000002
 .0999999578
 .100000845
 .0999831052
 .100337895
 .0932421037
 .235157925
-2.60315849
 54.1631699

Might I suggest you try 0.11+0.12?

I believe IEEE754 will in fact give the right answer on 0.1+0.2=0.3, using standard single precision. It is, however, not difficult to provoke IEEE754 failures, for instance on 0.11+0.12. The C program below show the raw bin32 representations of the relevant IEEE754 numbers, the program output is:

a :3dcccccd
b :3e4ccccd
a+b:3e99999a
c :3e99999a
IEEE754 copes
a :3de147ae
b :3df5c28f
a+b:3e6b851e
c :3e6b851f
IEEE754 fails

Program:

#include <stdio.h>
#include <stdint.h>
int main( void ) 
 float a = 0.1;
 float b = 0.2;
 float c = 0.3;
 float apb = a+b;
 printf( "a :%xn", *(uint32_t *)&a);
 printf( "b :%xn", *(uint32_t *)&b);
 printf( "a+b:%xn", *(uint32_t *)&apb);
 printf( "c :%xn", *(uint32_t *)&c);
 if ( a + b == c ) 
 printf( "IEEE754 copesn" );
 else 
 printf( "IEEE754 failsn" );
 

 a = 0.11;
 b = 0.12;
 c = 0.23;
 apb = a+b;
 printf( "a :%xn", *(uint32_t *)&a);
 printf( "b :%xn", *(uint32_t *)&b);
 printf( "a+b:%xn", *(uint32_t *)&apb);
 printf( "c :%xn", *(uint32_t *)&c);
 if ( a + b == c ) 
 printf( "IEEE754 copesn" );
 else 
 printf( "IEEE754 failsn" );
 

 return 0;