The following C code
int main(){
int n=10;
int t1=pow(10,2);
int t2=pow(n,2);
int t3=2*pow(n,2);
printf("%d\n",t1);
printf("%d\n",t2);
printf("%d\n",t3);
return (0);
}
gives the following output
100
99
199
I am using a devcpp compiler.
It does not make any sense, right?
Any ideas?
(That pow(10,2) is maybe something
like 99.9999 does not explain the first
output. Moreover, I got the same
output even if I include math.h)
You are using a poor-quality math library. A good math library returns exact results for values that are exactly representable.
Generally, math library routines must be approximations both because floating-point formats cannot exactly represent the exact mathematical results and because computing the various functions is difficult. However, for pow, there are a limited number of results that are exactly representable, such as 102. A good math library will ensure that these results are returned correctly. The library you are using fails to do that.
Store the result computations as doubles. Print as double, using %f instead of %d. You will see that the 99 is really more like 99.999997, and this should make more sense.
In general, when working with any floating point math, you should assume results will be approximate; that is, a little off in either direction. So when you want exact results - like you did here - you're going to have trouble.
You should always understand the return type of functions before you use them. See, e.g. cplusplus.com:
double pow (double base, double exponent); /* C90 */
From other answers I understand there are situations when you can expect pow or other floating-point math to be precise. Once you understand the necessary imprecision that plagues floating point math, please consult these.
Your variables t1, t2 and t3 must be of type double because pow() returns double.
But if you do want them to be of type int, use round() function.
int t1 = pow(10,2);
int t2 = round(pow(n,2));
int t3 = 2 * round(pow(n,2));
It rounds the returned values 99.9... and 199.9... to 100.0 and 200.0. And then t2 == 100 because it is of type int and so does t3.
The output will be:
100
100
200
Because the round function returns the integer value nearest to x rounding half-way cases away from zero, regardless of the current rounding direction.
UPDATE: Here is comment from math.h:
/* Excess precision when using a 64-bit mantissa for FPU math ops can
cause unexpected results with some of the MSVCRT math functions. For
example, unless the function return value is stored (truncating to
53-bit mantissa), calls to pow with both x and y as integral values
sometimes produce a non-integral result. ... */
Related
I'm wondering if there are any circumstances where code like this will be incorrect due to floating point inaccuracies:
#include <math.h>
// other code ...
float f = /* random but not NAN or INF */;
int i = (int)floorf(f);
// OR
int i = (int)ceilf(f);
Are there any guarantees about these values? If I have a well-formed f (not NAN or INF) will i always be the integer that it rounds to, whichever way that is.
I can image a situation where (with a bad spec/implementation) the value you get is the value just below the true value rather than just above/equal but is actually closer. Then when you truncate it actually rounds down to the next lower value.
It doesn't seem possible to me given that integers can be exact values in ieee754 floating point but I don't know if float is guaranteed to be that standard
The C standard is sloppy in specifying floating-point behavior, so it is technically not completely specified that floorf(f) produces the correct floor of f or that ceilf(f) produces the correct ceiling of f.
Nonetheless, no C implementations I am aware of get this wrong.
If, instead of floorf(some variable), you have floorf(some expression), there are C implementations that may evaluate the expression in diverse ways that will not get the same result as if IEEE-754 arithmetic were used throughout.
If the C implementation defines __STDC_IEC_559__, it should evaluate the expressions using IEEE-754 arithmetic.
Nonetheless, int i = (int)floorf(f); is of course not guaranteed to set i to the floor of f if the floor of f is out of range of int.
While running the following lines of code:
int i,a;
for(i=0;i<=4;i++)
{
a=pow(10,i);
printf("%d\t",a);
}
I was surprised to see the output, it comes out to be 1 10 99 1000 9999 instead of 1 10 100 1000 10000.
What could be the possible reason?
Note
If you think it's a floating point inaccuracy that in the above for loop when i = 2, the values stored in variable a is 99.
But if you write instead
a=pow(10,2);
now the value of a comes out to be 100. How is that possible?
You have set a to be an int. pow() generates a floating point number, that in SOME cases may be just a hair less than 100 or 10000 (as we see here.)
Then you stuff that into the integer, which TRUNCATES to an integer. So you lose that fractional part. Oops. If you really needed an integer result, round may be a better way to do that operation.
Be careful even there, as for large enough powers, the error may actually be large enough to still cause a failure, giving you something you don't expect. Remember that floating point numbers only carry so much precision.
The function pow() returns a double. You're assigning it to variable a, of type int. Doing that doesn't "round off" the floating point value, it truncates it. So pow() is returning something like 99.99999... for 10^2, and then you're just throwing away the .9999... part. Better to say a = round(pow(10, i)).
This is to do with floating point inaccuracy. Although you are passing in ints they are being implicitly converted to a floating point type since the pow function is only defined for floating point parameters.
Mathematically, the integer power of an integer is an integer.
In a good quality pow() routine this specific calculation should NOT produce any round-off errors. I ran your code on Eclipse/Microsoft C and got the following output:
1 10 100 1000 10000
This test does NOT indicate if Microsoft is using floats and rounding or if they are detecting the type of your numbers and choosing the appropriate method.
So, I ran the following code:
#include <stdio.h>
#include <math.h>
main ()
{
double i,a;
for(i=0.0; i <= 4.0 ;i++)
{
a=pow(10,i);
printf("%lf\t",a);
}
}
And got the following output:
1.000000 10.000000 100.000000 1000.000000 10000.000000
No one spelt out how to actually do it correctly - instead of pow function, just have a variable that tracks the current power:
int i, a, power;
for (i = 0, a = 1; i <= 4; i++, a *= 10) {
printf("%d\t",a);
}
This continuing multiplication by ten is guaranteed to give you the correct answer, and quite OK (and much better than pow, even if it were giving the correct results) for tasks like converting decimal strings into integers.
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....
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
int main()
{
int n,i,ele;
n=5;
ele=pow(n,2);
printf("%d",ele);
return 0;
}
The output is 24.
I'm using GNU/GCC in Code::Blocks.
What is happening?
I know the pow function returns a double , but 25 fits an int type so why does this code print a 24 instead of a 25? If n=4; n=6; n=3; n=2; the code works, but with the five it doesn't.
Here is what may be happening here. You should be able to confirm this by looking at your compiler's implementation of the pow function:
Assuming you have the correct #include's, (all the previous answers and comments about this are correct -- don't take the #include files for granted), the prototype for the standard pow function is this:
double pow(double, double);
and you're calling pow like this:
pow(5,2);
The pow function goes through an algorithm (probably using logarithms), thus uses floating point functions and values to compute the power value.
The pow function does not go through a naive "multiply the value of x a total of n times", since it has to also compute pow using fractional exponents, and you can't compute fractional powers that way.
So more than likely, the computation of pow using the parameters 5 and 2 resulted in a slight rounding error. When you assigned to an int, you truncated the fractional value, thus yielding 24.
If you are using integers, you might as well write your own "intpow" or similar function that simply multiplies the value the requisite number of times. The benefits of this are:
You won't get into the situation where you may get subtle rounding errors using pow.
Your intpow function will more than likely run faster than an equivalent call to pow.
You want int result from a function meant for doubles.
You should perhaps use
ele=(int)(0.5 + pow(n,2));
/* ^ ^ */
/* casting and rounding */
Floating-point arithmetic is not exact.
Although small values can be added and subtracted exactly, the pow() function normally works by multiplying logarithms, so even if the inputs are both exact, the result is not. Assigning to int always truncates, so if the inexactness is negative, you'll get 24 rather than 25.
The moral of this story is to use integer operations on integers, and be suspicious of <math.h> functions when the actual arguments are to be promoted or truncated. It's unfortunate that GCC doesn't warn unless you add -Wfloat-conversion (it's not in -Wall -Wextra, probably because there are many cases where such conversion is anticipated and wanted).
For integer powers, it's always safer and faster to use multiplication (division if negative) rather than pow() - reserve the latter for where it's needed! Do be aware of the risk of overflow, though.
When you use pow with variables, its result is double. Assigning to an int truncates it.
So you can avoid this error by assigning result of pow to double or float variable.
So basically
It translates to exp(log(x) * y) which will produce a result that isn't precisely the same as x^y - just a near approximation as a floating point value,. So for example 5^2 will become 24.9999996 or 25.00002
Suppose I have an irrational number like \sqrt{3}. As it is irrational, it has no decimal representation. So when you try to express it with a IEEE 754 double, you will introduce an error.
A decimal representation with a lot of digits is:
1.7320508075688772935274463415058723669428052538103806280558069794519330169088
00037081146186757248575675...
Now, when I calculate \sqrt{3}, I get 1.732051:
#include <stdio.h> // printf
#include <math.h> // needed for sqrt
int main() {
double myVar = sqrt (3);
printf("as double:\t%f\n", myVar);
}
According to Wolfram|Alpha, I have an error of 1.11100... × 10^-7.
Is there any way I can calculate the error myself?
(I don't mind switching to C++, Python or Java. I could probably also use Mathematica, if there is no simple alternative)
Just to clarify: I don't want a solution that works only for sqrt{3}. I would like to get a function that gives me the error for any number. If that is not possible, I would at least like to know how Wolfram|Alpha gets more values.
My try
While writing this question, I found this:
#include <stdio.h> // printf
#include <math.h> // needed for sqrt
#include <float.h> // needed for higher precision
int main() {
long double r = sqrtl(3.0L);
printf("Precision: %d digits; %.*Lg\n",LDBL_DIG,LDBL_DIG,r);
}
With this one, I can get the error down to 2.0 * 10^-18 according to Wolfram|Alpha. So I thought this might be close enough to get a good estimation of the error. I wrote this:
#include <stdio.h> // printf
#include <math.h> // needed for sqrt
#include <float.h>
int main() {
double myVar = sqrt (3);
long double r = sqrtl(3.0L);
long double error = abs(r-myVar) / r;
printf("Double:\t\t%f\n", myVar);
printf("Precision:\t%d digits; %.*Lg\n",LDBL_DIG,LDBL_DIG,r);
printf("Error:\t\t%.*Lg\n", LDBL_DIG, error);
}
But it outputs:
Double: 1.732051
Precision: 18 digits; 1.73205080756887729
Error: 0
How can I fix that to get the error?
What every Programmer should know about Floating Point Arithmetic by Goldberg is the definite guide you are looking for.
https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/02Numerics/Double/paper.pdf
printf rounds doubles to 6 places when you use %f without a precision.
e.g.
double x = 1.3;
long double y = 1.3L;
long double err = y - (double) x;
printf("Error %.20Lf\n", err);
My output: -0.00000000000000004445
If the result is 0, your long double and double are the same.
One way to obtain an interval that is guaranteed to contain the real value of the computation is to use interval arithmetic. Then, comparing the double result to the interval tells you how far the double computation is, at worst, from the real computation.
Frama-C's value analysis can do this for you with option -all-rounding-modes.
double Frama_C_sqrt(double x);
double sqrt(double x)
{
return Frama_C_sqrt(x);
}
double y;
int main(){
y = sqrt(3.0);
}
Analyzing the program with:
frama-c -val t.c -float-normal -all-rounding-modes
[value] Values at end of function main:
y ∈ [1.7320508075688772 .. 1.7320508075688774]
This means that the real value of sqrt(3), and thus the value that would be in variable y if the program computed with real numbers, is within the double bounds [1.7320508075688772 .. 1.7320508075688774].
Frama-C's value analysis does not support the long double type, but if I understand correctly, you were only using long double as reference to estimate the error made with double. The drawback of that method is that long double is itself imprecise. With interval arithmetic as implemented in Frama-C's value analysis, the real value of the computation is guaranteed to be within the displayed bounds.
You have a mistake in printing Double: 1.732051 here printf("Double:\t\t%f\n", myVar);
The actual value of double myVar is
1.732050807568877281 //18 digits
so 1.732050807568877281-1.732050807568877281 is zero
According to the C standard printf("%f", d) will default to 6 digits after the decimal point. This is not the full precision of your double.
It might be that double and long double happen to be the same on your architecture. I have different sizes for them on my architecture and get a non-zero error in your example code.
You want fabsl instead of abs when calculating the error, at least when using C. (In C, abs is integer.) With this substitution, I get:
Double: 1.732051
Precision: 18 digits; 1.73205080756887729
Error: 5.79643049346087304e-17
(Calculated on Mac OS X 10.8.3 with Apple clang 4.0.)
Using long double to estimate the errors in double is a reasonable approach for a few simple calculations, except:
If you are calculating the more accurate long double results, why bother with double?
Error behavior in sequences of calculations is hard to describe and can grow to the point where long double is not providing an accurate estimate of the exact result.
There exist perverse situations where long double gets less accurate results than double. (Mostly encountered when somebody constructs an example to teach students a lesson, but they exist nonetheless.)
In general, there is no simple and efficient way to calculate the error in a floating-point result in a sequence of calculations. If there were, it would be effectively a means of calculating a more accurate result, and we would use that instead of the floating-point calculations alone.
In special cases, such as when developing math library routines, the errors resulting from a particular sequence of code are studied carefully (and the code is redesigned as necessary to have acceptable error behavior). More often, error is estimated either by performing various “experiments” to see how much results fluctuate with varying inputs or by studying general mathematical behavior of systems.
You also asked “I would like to get a function that gives me the error for any number.” Well, that is easy, given any number x and the calculated result x', the error is exactly x' – x. The actual problem is you probably do not have a description of x that can be used to evaluate that expression easily. In your example, x is sqrt(3). Obviously, then, the error is sqrt(3) – x, and x is exactly 1.732050807568877193176604123436845839023590087890625. Now all you need to do is evaluate sqrt(3). In other words, numerically evaluating the error is about as hard as numerically evaluating the original number.
Is there some class of numbers you want to perform this analysis for?
Also, do you actually want to calculate the error or just a good bound on the error? The latter is somewhat easier, although it remains hard for sequences of calculations. For all elementary operations, IEEE 754 requires the produced result to be the result that is nearest the mathematically exact result (in the appropriate direction for the rounding mode being used). In round-to-nearest mode, this implies that each result is at most 1/2 ULP (unit of least precision) away from the exact result. For operations such as those found in the standard math library (sine, logarithm, et cetera), most libraries will produce results within a few ULP of the exact result.