C++11 introduced very useful math functions in the standard like erf and erfc. There are mentions about "guaranteed underflow" for inputs greater or smaller than certain values, but I don't know enough about floating point representation to understand clearly what this means in terms of precision.
If this question makes sense; what precision (order of magnitude at least) can I expect from the approximation implemented by the standard library (if it is specified)?
This depends on the quality of the implementation, which is up to the vendor of the compiler (or runtime libraries, if acquired separately). In the best case, the precision will match the precision of the specific type you use (double, long double, and so on).
Notice that the precision of the returned value is not related to the guaranteed underflow. This is just an enforced postcondition that assures the return value is the special underflow FP value if the input is outside the expected domain.
Related
I noticed that the C++ standard library has separate functions for round and lround rather than just having you use long(round(x)) for the latter.
Looking into the implementation in glibc, I find that indeed, for platforms using IEEE754 floating point, the version that returns an integer will directly manipulate the bits from within the floating point representation, and not do the rounding using floating point operations (e.g. adding ±0.5).
What is the benefit of having a distinct implementation when you want the result as an integer type? Is this supposed to be faster, or more accurate? If it is better to use integer math on the underlying representation, why not just always do it that way even if returning the result as a double?
One reason is that adding .5 is insufficient. Let’s say you add .5 and then truncate to an integer. (How? Is there an instruction for that? Or are you doing more work?) If x is ½−2−54 (the greatest representable value less than ½), adding .5 yields 1, because the mathematical sum, 1−2−54, is exactly halfway between the nearest two representable values, 1−2−53 and 1, and the common default rounding mode, round-to-nearest-ties-to-even, rounds that to 1. But the correct result for lround(x) is 0.
And, of course, lround is specified to round ties away from zero, regardless of the current rounding mode. You could set the rounding mode, do some arithmetic, and restore the rounding mode, but there are problems with this.
One is that changing the rounding mode is a typically a time-consuming operation. The rounding mode is a global state that affects most floating-point instructions. So the processor has to ensure all pending instructions complete with the prior mode, change the global state, and ensure all later instructions start after that change.
If you are lucky, you might have a processor with per-instruction rounding modes or something similar, and then you can use any rounding mode you like without time penalty. Hewlett Packard has some processors like that. However, “round away from zero” is an uncommon mode. Most processors have round-to-nearest-ties-to-even, round toward zero, round down (toward −∞), and round up (toward +∞), and round-to-odd is becoming popular for its value in avoiding double-rounding errors. But round away from zero is rare.
Another reason is that doing floating-point instructions alters the floating-point status flags and may generate traps, but it is desired that library routines behave as single operations. For example, if we add .5 and rounding occurs, the inexact flag will be raised, since the floating-point addition with .5 produced a result different from the mathematical sum. But to the user of lround, no inexact condition ever occurs; lround is defined to return a value rounded to an integer, and it always does so—within the long range, it never returns a computed result different from its ideal mathematical definition. So if lround(x) raised the inexact flag, that would be incorrect behavior. To avoid it, an implementation that used floating-point instructions would have to save the current floating-point flags, do its work, and restore the flags before returning.
What happens if you divide by Zero on a Computer?
In any given programming languange (I worked with, at least) this raises an error.
But why? Is it built in the language, that this is prohibited? Or will it compile, and the hardware will figure out that an error must be returned?
I guess handling this by the language can only be done, if it is hard code, e.g. there is a line like double z = 5.0/0.0; If it is a function call, and the devisor is given from outside, the language could not even know that this is a division by zero (at least a compile time).
double divideByZero(double divisor){
return 5.0/divisor;
}
where divisor is called with 0.0.
Update:
According to the comments/answers it makes a difference whether you divide by int 0 or double 0.0.
I was not aware of that. This is interesting in itself and I'm interested in both cases.
Also one answer is, that the CPU throws an error. Now, how is this done? Also in software (doesn't make sense on a CPU), or are there some circuits which recognize this? I guess this happens on the Arithmetic Logic Unit (ALU).
When an integer is divided by 0 in the CPU, this causes an interrupt.¹ A programming language implementation can then handle that interrupt by throwing an exception or employing whichever other error-handling mechanisms the language has.
When a floating point number is divided by 0, the result is infinity, NaN or negative infinity (which are special floating point values). That's mandated by the IEEE floating point standard, which any modern CPU will adhere to. Programming languages generally do as well. If a programming language wanted to handle it as an error instead, it could just check for NaN or infinite results after every floating point operation and cause an error in that case. But, as I said, that's generally not done.
¹ On x86 at least. But I imagine it's the same on most other architectures as well.
I am developing a programming language, September, which uses a tagged variant type as its main value type. 3 bits are used for the type (integer, string, object, exception, etc.), and 61 bits are used for the actual value (the actual integer, pointer to the object, etc.).
Soon, it will be time to add a float type to the language. I almost have the space for a 64-bit double, so I wanted to make use of doubles for calculations internally. Since I'm actually 3 bits short for storage, I would have to round the doubles off after each calculation - essentially resulting in a 61-bit double with a mantissa or exponent shorter by 3 bits.
But! I know floating point is fraught with peril and doing things which sound sensible on paper can produce disastrous results with FP math, so I have an open-ended question to the experts out there:
Is this approach viable at all? Will I run into serious error-accumulation problems in long-running calculations by rounding at each step? Is there some specific way in which I could do the rounding in order to avoid that? Are there any special values that I won't be able to treat that way (subnormals come to mind)?
Ideally, I would like my floats to be as well-behaved as a native 61-bit double would be.
I would recommend borrowing bits from the exponent field of the double-precision format. This is the method described in this article (that you would modify to borrow 3 bits from the exponent instead of 1). With this approach, all computations that do not use very large or very small intermediate results behave exactly as the original double-precision computation would. Even computations that run into the subnormal region of the new format behave exactly as they would if a 1+8+52 61-bit format had been standardized by IEEE.
By contrast, naively borrowing any number of bits at all from the significand introduces many double-rounding problems, all the more frequent that you are rounding from a 52-bit significand to a significand with only a few bits removed. Borrowing one bit from the significand as you suggest in an edit to your question would be the worst, with half the operations statistically producing double-rounded results that are different from what the ideal “native 61-bit double” would have produced. This means that instead of being accurate to 0.5ULP, the basic operations would be accurate to 3/4ULP, a dramatic loss of accuracy that would derail many of the existing, finely-designed numerical algorithms that expect 0.5ULP.
Three is a significant number of bits to borrow from an exponent that only has 11, though, and you could also consider using the single-precision 32-bit format in your language (calling the single-precision operations from the host).
Lastly, I give visibility here to another solution found by Jakub: borrow the three bits from the significand, and simulate round-to-odd for the intermediate double-precision computation before converting to the nearest number in 49-explicit-significand-bit, 11-exponent-bit format. If this way is chosen, it may useful to remark that the rounding itself to 49 bits of significand can be achieved with the following operations:
if ((repr & 7) == 4)
repr += (repr & 8) >> 1); /* midpoint case */
else
repr += 4;
repr &= ~(uint64_t)7; /* round to the nearest */
Despite working on the integer having the same representation as the double being considered, the above snippet works even if the number goes from normal to subnormal, from subnormal to normal, or from normal to infinite. You will of course want to set a tag in the three bits that have been freed as above. To recover a standard double-precision number from its unboxed representation, simply clear the tag with repr &= ~(uint64_t)7;.
This is a summary of my own research and information from the excellent answer by #Pascal Cuoq.
There are two places where we can truncate the 3-bits we need: the exponent, and the mantissa (significand). Both approaches run into problems which have to be explicitly handled in order for the calculations to behave as if we used a hypothetical native 61-bit IEEE format.
Truncating the mantissa
We shorten the mantissa by 3 bits, resulting in a 1s+11e+49m format. When we do that, performing calculations in double-precision and then rounding after each computation exposes us to double rounding problems. Fortunately, double rounding can be avoided by using a special rounding mode (round-to-odd) for the intermediate computations. There is an academic paper describing the approach and proving its correctness for all doubles - as long as we truncate at least 2 bits.
Portable implementation in C99 is straightforward. Since round-to-odd is not one of the available rounding modes, we emulate it by using fesetround(FE_TOWARD_ZERO), and then setting the last bit if the FE_INEXACT exception occurs. After computing the final double this way, we simply round to nearest for storage.
The format of the resulting float loses about 1 significant (decimal) digit compared to a full 64-bit double (from 15-17 digits to 14-16).
Truncating the exponent
We take 3 bits from the exponent, resulting in a 1s+8e+52m format. This approach (applied to a hypothetical introduction of 63-bit floats in OCaml) is described in an article. Since we reduce the range, we have to handle out-of-range exponents on both the positive side (by simply 'rounding' them to infinity) and the negative side. Doing this correctly on the negative side requires biasing the inputs to any operation in order to ensure that we get subnormals in the 64-bit computation whenever the 61-bit result needs to be subnormal. This has to be done a bit differently for each operation, since what matters is not whether the operands are subnormal, but whether we expect the result to be (in 61-bit).
The resulting format has significantly reduced range since we borrow a whopping 3 out of 11 bits of the exponent. The range goes down from 10-308...10308 to about 10-38 to 1038. Seems OK for computation, but we still lose a lot.
Comparison
Both approaches yield a well-behaved 61-bit float. I'm personally leaning towards truncating the mantissa, for three reasons:
the "fix-up" operations for round-to-odd are simpler, do not differ from operation to operation, and can be done after the computation
there is a proof of mathematical correctness of this approach
giving up one significant digit seems less impactful than giving up a big chunk of the double's range
Still, for some uses, truncating the exponent might be more attractive (especially if we care more about precision than range).
If I run following line of code, I get DIVIDE BY ZERO error
1. System.out.println(5/0);
which is the expected behavior.
Now I run the below line of code
2. System.out.println(5/0F);
here there is no DIVIDE BY ZERO error, rather it shows INFINITY
In the first line I am dividing two integers and in the second two real numbers.
Why does dividing by zero for integers gives DIVIDE BY ZERO error while in the case of real numbers it gives INFINITY
I am sure it is not specific to any programming language.
(EDIT: The question has been changed a bit - it specifically referred to Java at one point.)
The integer types in Java don't have representations of infinity, "not a number" values etc - whereas IEEE-754 floating point types such as float and double do. It's as simple as that, really. It's not really a "real" vs "integer" difference - for example, BigDecimal represents real numbers too, but it doesn't have a representation of infinity either.
EDIT: Just to be clear, this is language/platform specific, in that you could create your own language/platform which worked differently. However, the underlying CPUs typically work the same way - so you'll find that many, many languages behave this way.
EDIT: In terms of motivation, bear in mind that for the infinity case in particular, there are ways of getting to infinity without dividing by zero - such as dividing by a very, very small floating point number. In the case of integers, there's obviously nothing between zero and one.
Also bear in mind that the cases in which integers (or decimal floating point types) are used typically don't need to concept of infinity, or "not a number" results - whereas in scientific applications (where float/double are more typically useful), "infinity" (or at least, "a number which is too large to sensibly represent") is still a potentially valid result.
This is specific to one programming language or a family of languages. Not all languages allow integers and floats to be used in the same expression. Not all languages have both types (for example, ECMAScript implementations like JavaScript have no notion of an integer type externally). Not all languages have syntax like this to convert values inline.
However, there is an intrinsic difference between integer arithmetic and floating-point arithmetic. In integer arithmetic, you must define that division by zero is an error, because there are no values to represent the result. In floating-point arithmetic, specifically that defined in IEEE-754, there are additional values (combinations of sign bit, exponent and mantissa) for the mathematical concept of infinity and meta-concepts like NaN (not a number).
So we can assume that the / operator in this programming language is generic, that it performs integer division if both operands are of the language's integer type; and that it performs floating-point division if at least one of the operands is of a float type of the language, whereas the other operands would be implicitly converted to that float type for the purpose of the operation.
In real-number math, division of a number by a number close to zero is equivalent to multiplying the first number by a number whose absolute is very large (x / (1 / y) = x * y). So it is reasonable that the result of dividing by zero should be (defined as) infinity as the precision of the floating-point value would be exceeded.
Implementation details were to be found in that programming language's specification.
I have a method that deals with some geographic coordinates in .NET, and I have a struct that stores a coordinate pair such that if 256 is passed in for one of the coordinates, it becomes 0. However, in one particular instance a value of approximately 255.99999998 is calculated, and thus stored in the struct. When it's printed in ToString(), it becomes 256, which should not happen - 256 should be 0. I wouldn't mind if it printed 255.9999998 but the fact that it prints 256 when the debugger shows 255.99999998 is a problem. Having it both store and display 0 would be even better.
Specifically there's an issue with comparison. 255.99999998 is sufficiently close to 256 such that it should equal it. What should I do when comparing doubles? use some sort of epsilon value?
EDIT: Specifically, my problem is that I take a value, perform some calculations, then perform the opposite calculations on that number, and I need to get back the original value exactly.
This sounds like a problem with how the number is printed, not how it is stored. A double has about 15 significant figures, so it can tell 255.99999998 from 256 with precision to spare.
You could use the epsilon approach, but the epsilon is typically a fudge to get around the fact that floating-point arithmetic is lossy.
You might consider avoiding binary floating-points altogether and use a nice Rational class.
The calculation above was probably destined to be 256 if you were doing lossless arithmetic as you would get with a Rational type.
Rational types can go by the name of Ratio or Fraction class, and are fairly simple to write
Here's one example.
Here's another
Edit....
To understand your problem consider that when the decimal value 0.01 is converted to a binary representation it cannot be stored exactly in finite memory. The Hexidecimal representation for this value is 0.028F5C28F5C where the "28F5C" repeats infinitely. So even before doing any calculations, you loose exactness just by storing 0.01 in binary format.
Rational and Decimal classes are used to overcome this problem, albeit with a performance cost. Rational types avoid this problem by storing a numerator and a denominator to represent your value. Decimal type use a binary encoded decimal format, which can be lossy in division, but can store common decimal values exactly.
For your purpose I still suggest a Rational type.
You can choose format strings which should let you display as much of the number as you like.
The usual way to compare doubles for equality is to subtract them and see if the absolute value is less than some predefined epsilon, maybe 0.000001.
You have to decide yourself on a threshold under which two values are equal. This amounts to using so-called fixed point numbers (as opposed to floating point). Then, you have to perform the round up manually.
I would go with some unsigned type with known size (eg. uint32 or uint64 if they're available, I don't know .NET) and treat it as a fixed point number type mod 256.
Eg.
typedef uint32 fixed;
inline fixed to_fixed(double d)
{
return (fixed)(fmod(d, 256.) * (double)(1 << 24))
}
inline double to_double(fixed f)
{
return (double)f / (double)(1 << 24);
}
or something more elaborated to suit a rounding convention (to nearest, to lower, to higher, to odd, to even). The highest 8 bits of fixed hold the integer part, the 24 lower bits hold the fractional part. Absolute precision is 2^{-24}.
Note that adding and substracting such numbers naturally wraps around at 256. For multiplication, you should beware.