I'm using GCC to compile a program which adds floats, longs, ints and chars. When it runs, the result is bad. The following program unexpectedly prints the value of 34032.101562.
Recompiling with a Microsoft compiler gives the right result.
#include <stdio.h>
int main (void) {
const char val_c = 10;
const int val_i = 20;
const long val_l = 34000;
const float val_f = 2.1;
float result;
result = val_c + val_i + val_l + val_f;
printf("%f\n", result);
return 0;
}
What do you think the "right result" is? I'm guessing that you believe it is 34032.1. It isn't.
2.1 is not representable as a float, so val_f instead is initialized with the closest representable float value. In binary, 2.1 is:
10.000110011001100110011001100110011001100110011001...
a float has 24 binary digits, so the value of val_f in binary is:
10.0001100110011001100110
The expression resultat = val_c + val_i + val_l + val_f computes 34030 + val_f, which is evaluated in single-precision and causes another rounding to occur.
1000010011101110.0
+ 10.0001100110011001100110
-----------------------------------------
1000010011110000.0001100110011001100110
rounds to 24 digits:
-----------------------------------------
1000010011110000.00011010
In decimal, this result is exactly 34032.1015625. Because the %f format prints 6 digits after the decimal point (unless specified otherwise), this is rounded again, and printf prints 34032.101562.
Now, why do you not get this result when you compile with MSVC? The C and C++ standard allow floating-point calculations to be carried out in a wider type if the compiler chooses to do so. MSVC does this with your calculation, which means that the result of 34030 + val_f is not rounded before being passed to printf. In that case, the exact floating-point value being printed is 34032.099999999991268850862979888916015625, which is rounded to 34032.1 by printf.
Why don't all compilers do what MSVC does? A few reasons. First, it's slower on some processors. Second, and more importantly, although it can give more accurate answers, the programmer cannot depend on that -- seemingly unrelated code changes can cause the answer to change in the presence of this behavior. Because of this, carrying extra precision often causes more problems than it solves.
Google David Goldberg's paper "What Every Computer Scientist Should Know About
Floating-Point Arithmetic".
The float format has only about 6-7 digits of precision. Use %7.1f or some other reasonable format and you will like your results better.
I don't see any problem here. 2.1 has no exact representation in IEEE floating-point format, and as such, it is converting the entire answer to a floating-point number with around 6-7 (correct) sig-figs. If you need more precision, use a double.
Related
Could someone give me an explanation why I get two different
numbers, resp. 14 and 15, as an output from the following code?
#include <stdio.h>
int main()
{
double Vmax = 2.9;
double Vmin = 1.4;
double step = 0.1;
double a =(Vmax-Vmin)/step;
int b = (Vmax-Vmin)/step;
int c = a;
printf("%d %d",b,c); // 14 15, why?
return 0;
}
I expect to get 15 in both cases but it seems I'm missing some fundamentals of the language.
I am not sure if it's relevant but I was doing the test in CodeBlocks. However, if I type the same lines of code in some on-line compiler ( this one for example) I get an answer of 15 for the two printed variables.
... why I get two different numbers ...
Aside from the usual float-point issues, the computation paths to b and c are arrived in different ways. c is calculated by first saving the value as double a.
double a =(Vmax-Vmin)/step;
int b = (Vmax-Vmin)/step;
int c = a;
C allows intermediate floating-point math to be computed using wider types. Check the value of FLT_EVAL_METHOD from <float.h>.
Except for assignment and cast (which remove all extra range and precision), ...
-1 indeterminable;
0 evaluate all operations and constants just to the range and precision of the
type;
1 evaluate operations and constants of type float and double to the
range and precision of the double type, evaluate long double
operations and constants to the range and precision of the long double
type;
2 evaluate all operations and constants to the range and precision of the
long double type.
C11dr ยง5.2.4.2.2 9
OP reported 2
By saving the quotient in double a = (Vmax-Vmin)/step;, precision is forced to double whereas int b = (Vmax-Vmin)/step; could compute as long double.
This subtle difference results from (Vmax-Vmin)/step (computed perhaps as long double) being saved as a double versus remaining a long double. One as 15 (or just above), and the other just under 15. int truncation amplifies this difference to 15 and 14.
On another compiler, the results may both have been the same due to FLT_EVAL_METHOD < 2 or other floating-point characteristics.
Conversion to int from a floating-point number is severe with numbers near a whole number. Often better to round() or lround(). The best solution is situation dependent.
This is indeed an interesting question, here is what happens precisely in your hardware. This answer gives the exact calculations with the precision of IEEE double precision floats, i.e. 52 bits mantissa plus one implicit bit. For details on the representation, see the wikipedia article.
Ok, so you first define some variables:
double Vmax = 2.9;
double Vmin = 1.4;
double step = 0.1;
The respective values in binary will be
Vmax = 10.111001100110011001100110011001100110011001100110011
Vmin = 1.0110011001100110011001100110011001100110011001100110
step = .00011001100110011001100110011001100110011001100110011010
If you count the bits, you will see that I have given the first bit that is set plus 52 bits to the right. This is exactly the precision at which your computer stores a double. Note that the value of step has been rounded up.
Now you do some math on these numbers. The first operation, the subtraction, results in the precise result:
10.111001100110011001100110011001100110011001100110011
- 1.0110011001100110011001100110011001100110011001100110
--------------------------------------------------------
1.1000000000000000000000000000000000000000000000000000
Then you divide by step, which has been rounded up by your compiler:
1.1000000000000000000000000000000000000000000000000000
/ .00011001100110011001100110011001100110011001100110011010
--------------------------------------------------------
1110.1111111111111111111111111111111111111111111111111100001111111111111
Due to the rounding of step, the result is a tad below 15. Unlike before, I have not rounded immediately, because that is precisely where the interesting stuff happens: Your CPU can indeed store floating point numbers of greater precision than a double, so rounding does not take place immediately.
So, when you convert the result of (Vmax-Vmin)/step directly to an int, your CPU simply cuts off the bits after the fractional point (this is how the implicit double -> int conversion is defined by the language standards):
1110.1111111111111111111111111111111111111111111111111100001111111111111
cutoff to int: 1110
However, if you first store the result in a variable of type double, rounding takes place:
1110.1111111111111111111111111111111111111111111111111100001111111111111
rounded: 1111.0000000000000000000000000000000000000000000000000
cutoff to int: 1111
And this is precisely the result you got.
The "simple" answer is that those seemingly-simple numbers 2.9, 1.4, and 0.1 are all represented internally as binary floating point, and in binary, the number 1/10 is represented as the infinitely-repeating binary fraction 0.00011001100110011...[2] . (This is analogous to the way 1/3 in decimal ends up being 0.333333333... .) Converted back to decimal, those original numbers end up being things like 2.8999999999, 1.3999999999, and 0.0999999999. And when you do additional math on them, those .0999999999's tend to proliferate.
And then the additional problem is that the path by which you compute something -- whether you store it in intermediate variables of a particular type, or compute it "all at once", meaning that the processor might use internal registers with greater precision than type double -- can end up making a significant difference.
The bottom line is that when you convert a double back to an int, you almost always want to round, not truncate. What happened here was that (in effect) one computation path gave you 15.0000000001 which truncated down to 15, while the other gave you 14.999999999 which truncated all the way down to 14.
See also question 14.4a in the C FAQ list.
An equivalent problem is analyzed in analysis of C programs for FLT_EVAL_METHOD==2.
If FLT_EVAL_METHOD==2:
double a =(Vmax-Vmin)/step;
int b = (Vmax-Vmin)/step;
int c = a;
computes b by evaluating a long double expression then truncating it to a int, whereas for c it's evaluating from long double, truncating it to double and then to int.
So both values are not obtained with the same process, and this may lead to different results because floating types does not provides usual exact arithmetic.
The following code:
float numberForFactorial;
float floatingPart = 1.400000;
int integralPart = 1;
numberForFactorial = ((floatingPart) - (float)integralPart) * 10;
printf("%d", (int)numberForFactorial);
Returns 3 instead of 4. Can you explain me why?
The float closest to 1.400000 is slightly less than 1.4. You can verify that by doing
printf("%0.8hf\n", floatingPart);
The result from ideone is 1.39999998. This means that 10 times the first digit after the decimal point is 3.
To avoid this issue, use rounding instead of truncation. One easy way to round is by adding half before truncation:
printf("%d", (int)(numberForFactorial + 0.5f));
will print 4 as you were expecting. You can also use round, rint, lround, or modf to get the same result: https://www.gnu.org/software/libc/manual/html_node/Rounding-Functions.html. Rounding is a complex topic, so choose the method whose constraints match your situation best.
This is due to binary representation of floating-point values. More specifically, the 0.4 or 2/5 cannot be expressed with mantissa as sum of any combination like 1/2 + 1/4 + 1/8 + ...
The literal 1.400000 is stored as something closer to 1.399999976158142 in its binary representation. The cast to int truncates non-integer part, giving three as the final result.
To be pedantic, the C standard does not require binary-based representation of floating-point data type, however IEEE 754 is de facto the standad one in today's computing.
Is there a reason why converting from a double to an int performs as expected in this case:
double value = 45.33;
double multResult = (double) value*100.0; // assign to double
int convert = multResult; // assign to int
printf("convert = %d\n", convert); // prints 4533 as expected
But not in this case:
double value = 45.33;
int multResultInt = (double) value*100.0; // assign directly to int
printf("multResultInt = %d\n", multResultInt); // prints 4532??
It seems to me there should be no difference. In the second case the result is still first stored as a double before being converted to an int unless I am not understanding some difference between casts and hard assignments.
There is indeed no difference between the two, but compilers are used to take some freedom when it comes down to floating point computations. For example compilers are free to use higher precision for intermediate results of computations but higher still means different so the results may vary.
Some compilers provide switches to always drop extra precision and convert all intermediate results to the prescribed floating point numbers (say 64bit double-precision numbers). This will make the code slower, however.
In the specific the number 45.33 cannot be represented exactly with a floating point value (it's a periodic number when expressed in binary and it would require an infinite number of bits). When multiplying by 100 this value may be you don't get an integer, but something very close (just below or just above).
int conversion or cast is performed using truncation and something very close to 4533 but below will become 4532, when above will become 4533; even if the difference is incredibly tiny, say 1E-300.
To avoid having problems be sure to account for numeric accuracy problems. If you are doing a computation that depends on exact values of floating point numbers then you're using the wrong tool.
#6502 has given you the theory, here's how to look at things experimentally
double v = 45.33;
int x = v * 100.0;
printf("x=%d v=%.20lf v100=%.20lf\n", x, v, v * 100.0 );
On my machine, this prints
x=4533 v=45.32999999999999829470 v100=4533.00000000000000000000
The value 45.33 does not have an exact representation when encoded as a 64-bit IEEE-754 floating point number. The actual value of v is slightly lower than the intended value due to the limited precision of the encoding.
So why does multiplying by 100.0 fix the problem on some machines? One possibility is that the multiplication is done with 80-bits of precision and then rounded to fit into a 64-bit result. The 80-bit number 4532.999... will round to 4533 when converted to 64-bits.
On your machine, the multiplication is evidently done with 64-bits of precision, and I would expect that v100 will print as 4532.999....
My textbook - C in a Nutshell, ISBN 978-0596006976
The part of casting, the code in an example showing C rounding error:
Code:
#include <stdio.h>
int
main()
{
long l_var = 123456789L;
float f_var = l_var;
printf("The rounding error (f_var - l_var) is %f\n", f_var - l_var);
return 0;
}
then the value it output with nothing but 0.000000
seems it made no precision problem while casting those literal
with gcc(v4.4.7) command
gcc -Wall file.c -o exec
did GNU make a better way to get around the problem which mentioned in that chapter
or just some setting not strictly close to the issue of rounding error?
I don't know what this chapter is telling you, but:
float f_var = l_var;
We can tell that f_var is (float)l_var. Now the expression:
f_var - l_var
As this operates on a long and a float, the long will be converted into a float. So the compiler will do:
f_var - (float)l_var
Which is the same as:
(float)l_var - (float)l_var
Which is zero, regardless of any rounding of the conversion.
I don't have access to this book.
My guess is that the example is trying to tell you that if you assign a 32 bit integer to a 32 bit float, you may lose bits due to truncation (rounding errors): A 32 bit float has only 23 bit significand and some bits may be lost during the assignment accordingly.
Apparently, the example code is bogus in the book though. Here is the code to demonstrate the truncation error:
#include <stdint.h>
#include <stdio.h>
int main() {
int32_t l_var = 123456789L;
/* 32 bit variable, 23 bit significand, approx. 7 decimals */
float f_var = l_var;
double err = (double) f_var - (double) l_var;
printf("The rounding error (f_var - l_var) is %f\n", err);
return 0;
}
This prints
The rounding error (f_var - l_var) is 3.000000
on my machine.
0 is the value you get if both values are converted to float, you'll get something else if they are converted to something else. And there is an allowance in the standard to use wider floating point representation that required by the type for computation (*). Using it here is especially tempting here as the result has to be converted to a double for passing to printf.
My version of gcc is not using that allowance when compiling for x86_64 (-m64 argument for gcc) and it is using it when compiling for x86 (-m32 argument). That make sense when you know that for 64 bits, it is using sse instructions which can easily do the computation in float, while when compiling for 32 bits it is using the older "8087" stack model which can't do that easily.
(*) Last paragraph of 6.2.1.5 in C90, 6.3.1.8/2 in C99, 6.3.1.8/2 in C11. I give the text of the latest (as in n1539)
The values of floating operands and of the results of floating expressions may be
represented in greater precision and range than that required by the type; the types are not changed thereby.
As pointed by Pascal Cuoq, starting from C99, you can test with FLT_EVAL_METHOD.
In C89, floor() returns a double. Is the following guaranteed to work?
double d = floor(3.0 + 0.5);
int x = (int) d;
assert(x == 3);
My concern is that the result of floor might not be exactly representable in IEEE 754. So d gets something like 2.99999, and x ends up being 2.
For the answer to this question to be yes, all integers within the range of an int have to be exactly representable as doubles, and floor must always return that exactly represented value.
All integers can have exact floating point representation if your floating point type supports the required mantissa bits. Since double uses 53 bits for mantissa, it can store all 32-bit ints exactly. After all, you could just set the value as mantissa with zero exponent.
If the result of floor() isn't exactly representable, what do you expect the value of d to be? Surely if you've got the representation of a floating point number in a variable, then by definition it's exactly representable isn't it? You've got the representation in d...
(In addition, Mehrdad's answer is correct for 32 bit ints. In a compiler with a 64 bit double and a 64 bit int, you've got more problems of course...)
EDIT: Perhaps you meant "the theoretical result of floor(), i.e. the largest integer value less than or equal to the argument, may not be representable as an int". That's certainly true. Simple way of showing this for a system where int is 32 bits:
int max = 0x7fffffff;
double number = max;
number += 10.0;
double f = floor(number);
int oops = (int) f;
I can't remember offhand what C does when conversions from floating point to integer overflow... but it's going to happen here.
EDIT: There are other interesting situations to consider too. Here's some C# code and results - I'd imagine at least similar things would happen in C. In C#, double is defined to be 64 bits and so is long.
using System;
class Test
{
static void Main()
{
FloorSameInteger(long.MaxValue/2);
FloorSameInteger(long.MaxValue-2);
}
static void FloorSameInteger(long original)
{
double convertedToDouble = original;
double flooredToDouble = Math.Floor(convertedToDouble);
long flooredToLong = (long) flooredToDouble;
Console.WriteLine("Original value: {0}", original);
Console.WriteLine("Converted to double: {0}", convertedToDouble);
Console.WriteLine("Floored (as double): {0}", flooredToDouble);
Console.WriteLine("Converted back to long: {0}", flooredToLong);
Console.WriteLine();
}
}
Results:
Original value: 4611686018427387903
Converted to double:
4.61168601842739E+18Floored (as double): 4.61168601842739E+18
Converted back to long:
4611686018427387904
Original value: 9223372036854775805
Converted to double:
9.22337203685478E+18Floored (as double): 9.22337203685478E+18
Converted back to long:
-9223372036854775808
In other words:
(long) floor((double) original)
isn't always the same as original. This shouldn't come as any surprise - there are more long values than doubles (given the NaN values) and plenty of doubles aren't integers, so we can't expect every long to be exactly representable. However, all 32 bit integers are representable as doubles.
I think you're a bit confused about what you want to ask. floor(3 + 0.5) is not a very good example, because 3, 0.5, and their sum are all exactly representable in any real-world floating point format. floor(0.1 + 0.9) would be a better example, and the real question here is not whether the result of floor is exactly representable, but whether inexactness of the numbers prior to calling floor will result in a return value different from what you would expect, had all numbers been exact. In this case, I believe the answer is yes, but it depends a lot on your particular numbers.
I invite others to criticize this approach if it's bad, but one possible workaround might be to multiply your number by (1.0+0x1p-52) or something similar prior to calling floor (perhaps using nextafter would be better). This could compensate for cases where an error in the last binary place of the number causes it to fall just below rather than exactly on an integer value, but it will not account for errors which have accumulated over a number of operations. If you need that level of numeric stability/exactness, you need to either do some deep analysis or use an arbitrary-precision or exact-math library which can handle your numbers correctly.