Showing posts with label type casting. Show all posts
Showing posts with label type casting. Show all posts

Friday, November 28, 2014

Programming in C Chapter V - Typecasting

In its simplest sense Typecasting is altering a computer's interpretation of data by implicitly or explicitly changing its data type; for example, by changing an `int` to a `float` and vice verse.

To better understand typecasting, we must start with data types themselves. In programming languages like C, every variable has some kind of `type` that determines how the computer and the user interprets that variable. Each of these data types, for instance `int`, `long long`, `float` and `double` all have their own unique characteristics and are use to handle data types of various ranges and precision.

Typecasting allows us to take a floating point number, like 3.14, and specifying the number before the decimal - 3 - by parsing it to an `int`.

Let's us an example from the English language to better clarify what we mean.



Each carefully manipulated line in the example above forms a unique symbol. However, these symbols are immediately identifiable to those fluent in a Romance language as letters. We implicitly understand the data type `letter`.

Even more interesting, reviewing the string of `letter` data type symbols composing the example above, we can see that two very different, specific data types are formed. Each of the two words that are formed has a completely different meaning, connotation, pronunciation and history.

There is the noun wind, as in: "The wind blows outside". Yet there is also the verb wind, as in: "Wind up that spool".

This is a valuable analogy inasmuch as it leads us to understand that how we type the data determines how we use that data. The `noun` data type of WIND ought to be used in very different circumstances than the `verb` data type of WIND.

Setting aside more advanced topics such as Natural Language Processing for a moment, let's take for granted that computers do not care about English grammar. Computer programming languages, such as C, rely on the same idea - taking the same bit of data, and using it very differently based on how we cast the `type` of that data.

Here are most common data types of a 32 bit operating system:

1 byte  : char
4 bytes : int, float
8 bytes : long long, double

Each byte represents 8 bits of memory storage in a 32 bit OS. Thus, an variable of type `int` will use 32 bits of memory when it is stored. As long as that variable remains of type `int`, the processor will always be able to convert that variable back to its' relevant number. However, we could in theory cast those same 32 bits into a series of boolean operators. As a result, the computer would no longer see a number in that address space, but an altogether different sequence of binary characters. We could then try to read that data as a different numeric type, or even as a string of four characters.

When dealing with numbers and type casting, it is vital to understand how the *precision* of your value will be effected. Keep in mind that the precision can stay the same, or you can lose precision - as in our float 3.14 to int 3 example at the very beginning of our discussion. You cannot, however, gain precision. The data to do so simply does not exist in the addressed memory space you would be attempting to pull it from.

Let's review the 3 most common ways you can lose precision.

Casting a float to an int would cause truncation of everything after the decimal point, leaving us with a whole number.

To perform a float to int conversion, we can perform the following simple operation:


        float -> int
float x = 3.7;

In the scenario above, (int)x = 3, because we will have truncated all values after the decimal point.

We can also convert a long long to an int:

long long -> int

As before, this will lead to a loss of higher-order bits. a long long takes up 8 bytes or 64 bits in memory.

Similarly a double can be cast a a float:

double -> float

This will give you the closest possible float to the double without rounding. A double will allow you to store 53 bits or 16 significant digits, while a float has 24 significant bits.

Because floats can only store 24 significant bits, you can only store number up to the value of 2^24 AKA two to the power of twenty-four AKA 16777217.


Explicit casting is when we write the data type in parentheses before the variable name, like so:

(int)x -> explicit casting

Implicit casting is when the compiler automatically changes similar types to a "super-type", or performs some other form of casting without requiring any additional code from the user to perform the operation.

For example when we write the following:

5 + 1.1 -> implicit casting

The values already have types associated with them. 5 is an `int`, while 1.1 is a `float`. In order to add the two of them together, the computer implicitly casts the `int` 5 into a `float`:

(float)5.0 + (float)1.1 -> implicit casting

Implicit casting also allows us to assign variables of different types to each other. We can always assign a less precise type into a more precise one. For instance:


        double x;
int y;

We cannot take from this example that `x=y`, because a `double` has more precision than an `int`. On the other hand, it would also be problematic to say `y=x`, because `y` might have a larger value than `x`, and may not be able to hold all of the information stored in the double.

Type casting is also used in comparison operators such as:



        if (5.1 > 5)

The example above will be returned as one, because the compiler will implicitly cast 5 to a float in order to compare the two numbers. The same would be true of this example as well:

if(2.0 == 2)

Also, don't forget that `int's can be cast to `char's or ASCII values. `char's also need to be reduced to binary, which is why you can easily convert between char's and their respective ASCII values.

Wednesday, November 12, 2014

Programming in C - Chapter II - It Really IS Rocket Science

Problems arise with numerical expression in computing. In reality, there are an infinite number of real numbers. However there is clearly not an infinite amount of infinite memory even in the largest of super-computers, and memory that is addressable by an application is only a fraction of the total finite available memory. How to we deal with these obstacles? We will explain more in a moment.

First let's overview in more detail how the C compiler handles numeral types. Consider the application below:

#include <stdio.h>

int main (void)
    float f = 1 / 10;
    printf("%.2f\n", f);
    return 0;

Here we declare a float, 1/10 which should clearly resolve to 0.1 or 0.10 since I am declaring that printf provide a float with two digits after the decimal point. However, upon complation and excecution the program will stubbornly return a value of "0.00".


The issue is that I am declaring a float as an operation of two integers - 1 and 10. As a result, the compiler is performing an "implicit typeset" of the float "f" 0.10; as a result throwing out EVERYTHING after the decimal point (without so much as performing a round function). The truncation occurs before the storage in memory (note this is different from an implicit typeset); as a result the float f is able to store "0" as "0.00" without the compiler presenting an error about a symbol conflict or the wrong variable type. "0" isnt an int; it is a truncated float.

How do we resolve this? The easiest solution is to convert the "int" input into "float" input like so:

int main (void)
    float f = 1.0 / 10.0;
    printf("%.2f\n", f);
    return 0;

Which produces:

#: ./float0

That is not the only resolution, however. We can explicitly cast the input integer into floats like so, by providing the preferred variable type in parantheses () directly in front of the input to be "cast" into the correct type. This is called *explicit typecasting*:

#include <stdio.h>

int main (void)
    float f = (float) 1 / (float) 10;
    printf("%.2f\n", f);
    return 0;

Note that 1 and 10 could just as easily be actual variables in the above example.

Let's return to an earlier example, in which we tried to chan the int inputs 1 and 10 into float inputs 1.0 and 10.0. What happens when we look more closely at the output - for instance, by drawing out the decimal point to 20 places instead of 2, like so?

#include <stdio.h>

int main (void)
    float f = 1.0 / 10.0;
    printf("%.20f\n", f);
    return 0;

The output becomes imprecise:

#: ./float1

Why? Floats, like all of our other variables types, have a finite amount of memory with which they can be addressed and not an arbitrary amount. Thus there is a finite amount of real numbers that can be represented with each float, as we discussed above. The imprecision that we see here is a result of that limitation.

Put another way, we begin to see a core paradox in computing limitation:

There is an INFINITE amount of numbers that computers CANNOT represent with a FINITE amount of bits.

This limitation has real world consequences. The lesson proceeds with a video from the show "Modern Marvels: Engineering Disasters". The video described the June 4th, 1996 launch of an Un-manned Ariane 5 Rocket carrying satellites designed to determine precisely how the Earth's magnetic field interacts with Solar Winds. The rocket was built for the European Space Agency, and was launched from a facility in French Guyana.

37 seconds into the flight, engineers responsible for the launch first determined that something was wrong - the rocket's nozzles were swivelling in a way that they should not have been.

40 seconds into the flight it became clear that the vehicle was in trouble and might not survive the launch. Mission control at that point made a decision to destroy the rocket completely. The rocket, if it had been allowed to fail on its own, could have become a hazard to public safety as parts and components rained down unto homes and onlookers below.

This was the very first launch of the Ariane 5 class rocket, and the failure that occurred was the result of a software issue.

The Ariane 5's software contained a number that required 64 bits to express correctly. The developers intended to convert this into a 16 bit number. They assumed that the number would never be vey big; most of the digits in that 64 bit number would be zeroes. The assumptions about the size of the 64 bit number were wrong.

Most of the software from the Ariane 5 was originally designed for the Ariane 4; in which the software had been successful. The software was carried over to the new model as it had posed no problems in the old model. However, there was a key difference between the two rockets. The Ariane 5 accelerated much faster than the Ariane 4. The 64 bit number described above was a function of acceleration; numbers that remained "mostly zeroes" in the older model were no longer so in the new, faster model.

The Ariane 5 was not the first rocket in which data conversion (or "type casting") errors played a role in modern rocket technology. In 1991, with the start of the first Gulf War, the famous American PATRIOT Missile manufactured by Raytheon Corporation experienced a failure similar to that of the Ariane 5. As a result of the PATRIOT Missile failure, 28 American soldiers were killed and approximately 100 other wounded when a PATRIOT Missile designed to target and destroy incoming Iraqi SCUD missiles did not fire.

The PATRIOT Interceptor was deployed to protect Saudi Arabia and Israel from Iraqi SCUDs in 1991. The PATRIOT is a medium range Surface to Air rocket. The Orwellian acronym stands for Phased Array TRacking Intercept Of Target. The missile is often loaded into a carriage delivery system that is mounted onto the back of a truck. It is designed to be portable; the Patriot is a 20 feet long and weighs 2000 pounds - while the payload itself is a mere 150 pounds. That payload is a high explosive fragmentation device; the casing of the warhead is designed to act as buckshot. The missiles are packed in a container that holds four and loads onto the pack of a semi trailer.

PATRIOTs have a long history; 20 years before they obtained world-fame through the marketing efforts of George Bush Sr. Originally they were thought of as an air-defense battery, to shoot down planes. The anti-missle capability was a new-feature, one implemented to fulfill the military's unique mandates in the Gulf during Round 1 of the Hussein v Bush saga.

SCUDs fly much faster than the average plane: Mach 5. But reaching the right speed was only part of the problem of the sort of upgrade that Raytheon needed to pull off to keep their Pentagon paymasters happy. When the PATRIOT was rushed into service, Raytheon was unaware that the Iraqi military had modified their fleet of SCUDs; making interception much more difficult; close to impossible. Ironicallly enough, the Iraqi modifications were not intended to defeat interception, but to increase the range of the missle from their original 300km to 600km

The modified SCUDs would `wobble` as they flew inbound to their target, maintaining an unstable trajectory. To increase the range of the rockets, the Iraqis had taken weight out of the warhead, which was loaded onto the tip of the SCUD. This unstable trajectory meant that in the overwhelming majority of cases, PATRIOT missles would fly right past the SCUD, missing it entirely.

Once operators of the PATRIOT realized they had missed their mark (which was close to 100% of the time), they detonated the payload remotely as did the mission control for the Ariane 5 - except in this case the operators could not care less about the safety of those below the PATRIOT; they were concerned that allowing the PATRIOT to land would allow the Iraqis to salvage components. Some might view this concern as strange, given that the PATRIOT was at this point completely worthless. Perhaps the Pentagon feared that Iraqi military engineers could succeed where Raytheon corporation had failed? Not quite - the Pentagon filmed these remote detonations and provided them to the credulous Press, who released the footage of giant airborne fireballs while declaring breathlessly that the film represented yet another victorious interception by the PATRIOT missile of one of Saddam Hussein's diabolical SCUD warheads. These clips, filmed exclusively in night-vision, had a blinding glare during the PATRIOT remote detonation; it was during the moment, when the screen was all greenish-white, that the SCUD streaked past, completely undamaged. When the film began again, viewers saw the night sky, empty except for a shower of sparks and debris that they assumed was the mingled destruction of both missiles but contained only the destroyed PATRIOT.

What was in Iraq a failure rate of 100% became a success rate of 100% for the TV audience back in the United States. Support soared for the glorious campaign and its fearless leader, the command-in-chief.

But you can only fool all of the people some of the time. For Gulf War I, the fakery came to a startling and sudden end one night in the desert of Dhahran in Saudi Arabia. There, even the Pentagon spin doctors were not bold enough to replace the uniforms of the dead. There could be no mis-understanding; the PATRIOT missile was a complete and utter failure, a failure that left 28 young Americans dead.

In Dhahran a PATRIOT battery's radar system lost track of an incoming SCUD. The PATRIOT interceptor did worse than miss the incoming missle; it never launched at all. Because of a programming error.

It was clear to anyone paying attention and with the proper access that the Pentagon was completely un-bothered by the apparent madness of conducting a military campaign in which the primary armament was essentially a plastic pistol that when fired produced a flag printed with the cartoon word "BANG!". So it should be no surprise to us that it was not the Pentagon who pointed out and resolved the software glitch killing American soldiers, but the Israelis. The Israeli military first caught on that the longer a PATRIOT missile system remained on, the larger was the time discrepancies in the targeting systems became. The time discrepancies were the result of a clock application in the targeting computer.

At two weeks before the debacle in Dhahran, the Israelis reported to the Defense Department that the PATRIOT computers "lost" time. After about 8 hours of operation, the system became significantly less accurate. DoD, in their infinite wisdom, responded to the Israeli warning by telling all PATRIOT operators to regularly reboot their targeting computers. The DoD failed to specify how long the PATRIOT should remain online before a reboot; this crucial detail was left as an excerise for the reader. 8 hours? 10 hours? 1000 hours? The operators did not know what the Israelis knew - that there was a very specific window of operation after which the PATRIOT would fail to even appear functional.

The PATRIOT missle battery in Dhahran had been online continuously for 100 hours on the night of February 25th, 1991.

The targeting computer clock was designed to track time to an accuracy of 1/10th of a second. Unfortunately, fractions cannot directly and exactly be expressed in binary.

For example, lets consider the example of 1/3 (or one third). As a decimal number, we cannot exactly express 1/3 - it is a repeating decimal, 0.333333[...], where .3 repeats infinitely. Because computers have a finite amount of memory, the infinite decimal version of 1/3 cannot accurately be represented.

It is this issue that impacted the PATRIOT as it attempted to calculate values represented by fragments of a second. And due to the design of the PATRIOT's flawed application, the errors compounded over time.

After 100 hours of continuous operation, the time measurement flaw had compounded to a misrepresentation of time values by one third of a second. Not long under ordinary circumstances - but a lifetime for a SCUD missile travelling at Mach 5. At Mach 5 a targeting error of 1/3 of 1 second resulted in a trajectory miscalculation of 600 meters.

Immediately preceding the Dhahran catastrophe, a SCUD launch was detected by early warning satellites orbiting the Earth. The satellites were able to predict a general trajectory of Dhahran for the location of the SCUD's impact, but not exactly where it would hit. The PATRIOT's radar system was designed to calculate the missing part of the trajectory and fire an interception missile. The PATRIOT radar system would send a radar pulse, and based on that pulse was designed to format a prediction of the location of the SCUD at the time of the next radar pulse. Such a prediction simulation is referred to as a "range gate". The PATRIOT would then correct its next prediction based on the results, and do so several times; this prediction information would form the basis of the PATRIOT missile's own targeting solution. After all, the PATRIOT missile had to be pointed at where the SCUD *would* be after a given interval of time based on the speed of both the PATRIOT and SCUD; not where the SCUD *was* at the time of the launch of the PATRIOT missile.

It is not difficult to see how absolutely vital an accurate clock is to the proper function of such a system. Even worse, there was an additional "feature" of the PATRIOT radar system that ensured that the malfunctioning clock would result in disaster at Dhahar.

At any given time, there are a multitude of objects flying through the air. Birds. Clouds. In a war zone, there are friendly military aircraft. On top of this is the fact that radar is not, in itself, a flawless data input system. Radar systems are not unknown to produce blips that do not correspond to physical objects for all sorts of reasons.

As such, the PATRIOT system had to have a function for error detection - it just wouldn't do to launch a 2,000 pound missle at a flock of geese. To this end, the targeting computer had a window during which trajectory calculation would occur. If an object remained in this window, trajectory calculation would continue. If, however, an object did not appear within the window than the PATRIOT would dismiss further calculation and label prior input as a false positive. With such a function, again, correct time calculation is absolutely critical. At Dhahrar, the broken time calculator let the PATRIOT battery to look in the wrong areas to calculate its range gate. When the SCUD failed to appear within the range gate, the PATRIOT dismissed prior input as a false positive; when in fact the SCUD was still on its way, but one third of one second outside of where the PATRIOT was looking for it.

The incoming SCUD bypassed the PATRIOT defense battery, slamming directly into its intended target - a military barracks housing hundreds of American soldiers.

All because of incompetent type casting within the PATRIOT targeting application; and furthermore due to the absolutely unethical handling of those software issues by the Defense Department, who knew long in advance of the critical failures of the PATRIOT system but did not take proper action to inform the soldiers whose lives depended on the PATRIOT. Perhaps Pentagon officials feared that explicit orders to replace the software would cause a publicity disaster for those involved with the procurement of the PATRIOT system and their friends at Raytheon. There is no other explanation for the DoD's actions, besides perhaps acknowledging that the DoD cannot function in even the most basic and vital of bureaucratic tasks - that the Pentagon excels in buying war toys and marketing wars but is completely incompetent in the waging of war. Both options, as of the date of this writing, strike the author as equally feasible.

The SCUD missile that destroyed the Dhahran military barracks in Saudi Arabia was the last SCUD successfully fired by Saddam Hussein during the first Gulf War.

Perhaps I am being too harsh on the Pentagon and their pals at Raytheon. A software patch for the PATRIOT missile targeting system was in fact built and delivered to the front. It arrived in Dhahran on February 26th, 1991 - the day after the barracks were destroyed.

***NOTE: This is the Second Chapter of an ongoing series published here on Programming in C which is based upon the Harvard Class CS50. Content here from that class is published solely for educational purposes; I see no profit through the publication of this website including advertising. A good portion of today's post includes content from the show "Modern Marvels", parts of which are transcribed here - again only for the fair use of educational purpose. All barbs, jokes, political statements and assignations of responsibility for the Dharar disaster are entirely my own and not the opinions of Harvard or "Modern Marvels".

RAT Bastard

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