the Floating-point error - c

Question

#include <stdio.h>
int main()
{
int n;
while ( scanf( "%d", &n ) != EOF ) {
double sum = 0,k;
if( n > 5000000 || n<=0 ) //the judgment of the arrange
break;
for ( int i = 1; i <= n; i++ ) {
k = (double) 1 / i;
sum += k;
}
/*
for ( int i = n; i > 0; i-- ) {
k = 1 / (double)i;
sum += k;
}
*/
printf("%.12lf\n", sum);
}
return 0;
}
Why in the different loop I get the different answer. Is there a float-error? When I input 5000000 the sum is 16.002164235299 but as I use the other loop of for (notation part) I get the sum 16.002164235300.

Answer 1

Because floating point math is not associative:
i.e. (a + b) + c is not necessarily equal to a + (b + c)

Answer 2

I also bumped into a + b + c issue. Totally agreed with ArjunShankar.
// Here A != B in general case
float A = ( (a + b) + c) );
float B = ( (a + c) + b) );
Most of floating point operations are performed with data loss in mantis, even when components are fit well in it (numbers like 0.5 or 0.25).
In fact I was quite happy to find out the cause of bug in my application. I have written short reminder article with detailed explanation:
http://stepan.dyatkovskiy.com/2018/04/machine-fp-partial-invariance-issue.html
Below is the C example. Good luck!
example.c
#include <stdio.h>
// Helpers declaration, for implementation scroll down
float getAllOnes(unsigned bits);
unsigned getMantissaBits();
int main() {
// Determine mantissa size in bits
unsigned mantissaBits = getMantissaBits();
// Considering mantissa has only 3 bits, we would then get:
// a = 0b10 m=1, e=1
// b = 0b110 m=11, e=1
// c = 0b1000 m=1, e=3
// a + b = 0b1000, m=100, e=1
// a + c = 0b1010, truncated to 0b1000, m=100, e=1
// a + b + c result: 0b1000 + 0b1000 = 0b10000, m=100, e=2
// a + c + b result: 0b1000 + 0b110 = 0b1110, m=111, e=1
float a = 2,
b = getAllOnes(mantissaBits) - 1,
c = b + 1;
float ab = a + b;
float ac = a + c;
float abc = a + b + c;
float acb = a + c + b;
printf("\n"
"FP partial invariance issue demo:\n"
"\n"
"Mantissa size = %i bits\n"
"\n"
"a = %.1f\n"
"b = %.1f\n"
"c = %.1f\n"
"(a+b) result: %.1f\n"
"(a+c) result: %.1f\n"
"(a + b + c) result: %.1f\n"
"(a + c + b) result: %.1f\n"
"---------------------------------\n"
"diff(a + b + c, a + c + b) = %.1f\n\n",
mantissaBits,
a, b, c,
ab, ac,
abc, acb,
abc - acb);
return 1;
}
// Helpers
float getAllOnes(unsigned bits) {
return (unsigned)((1 << bits) - 1);
}
unsigned getMantissaBits() {
unsigned sz = 1;
unsigned unbeleivableHugeSize = 1024;
float allOnes = 1;
for (;sz != unbeleivableHugeSize &&
allOnes + 1 != allOnes;
allOnes = getAllOnes(++sz)
) {}
return sz-1;
}