An Abbreviated Table of binary64 Values

In IEEE Floats and Python, I chose to use a tiny "binary6" type because it's easy to show a table of values that helps clarify things.

Someone suggested that you could just as easily write a table of binary64 values, ellipsizing large chunks of it, and it would also be useful for clarifying things. And I think he's right.

The Table

    sign  exponent significand    binary scientific              decimal
       0 000...000 0000...0000    0.0...0000e-1022 = 0           0.0
       0 000...000 0000...0001    0.0...0001e-1022 = 1.0e-1074   5e-324
       0 000...000 0000...0010    0.0...0010e-1022 = 1.0e-1073   1e-323
       0 000...000 0000...0011    0.0...0011e-1022 = 1.1e-1073   1.5e-323
       0 000...000 0000...0100    0.0...0100e-1022 = 1.0e-1072   2e-323
       0 000...000 0000...0101    0.0...0101e-1022 = 1.01e-1072  2.5e-323
       0 000...000 0000...0110    0.0...0110e-1022 = 1.10e-1072  3e-323
       0 000...000 0000...0111    0.0...0111e-1022 = 1.11e-1072  3.5e-323
       0 000...000 0000...1100    0.0...1100e-1022 = 1.0e-1071   4e-323
       0 000...000 0000...1101    0.0...1101e-1022 = 1.01e-1071  4.4e-323
       0 000...000 0000...1110    0.0...1110e-1022 = 1.10e-1072  5e-323
       0 000...000 0000...1111    0.0...1111e-1022 = 1.11e-1072  5.4e-323
       ...
       0 000...000 1111...1110    0.1...1110e-1022 = 1.11e-1023  2.2250738585072004e-308
       0 000...000 1111...1111    0.1...1111e-1022 = 1.11e-1023  2.225073858507201e-308
       0 000...001 0000...0000    1.0...0000e-1022 = 1.0e-1022   2.2250738585072014e-308
       0 000...001 0000...0001    1.0...0001e-1022               2.225073858507202e-308
       ...
       0 000...001 1111...1111    1.1...1111e-1022               4.4501477170144023e-308
       0 000...010 0000...0000    1.0...0000e-1021               4.450147717014403e-308
       ...
       0 011...110 1111...1111    1.1...111e-1                   0.9999999999999999
       0 011...111 0000...0000    1.0...000e0                    1.0
       0 011...111 0000...0001    1.0...001e0                    1.0000000000000002
       ...
       0 111...101 1111...1111    1.1...111e+1022                8.988465674311579e+307
       0 111...110 0000...0000    1.0...000e+1023                8.98846567431158e+307
       0 111...110 0000...0001    1.0...001e+1023                8.988465674311582e+307
       ...
       0 111...110 1111...1110    1.1...110e+1023                1.7976931348623155e+308
       0 111...110 1111...1111    1.1...111e+1023                1.7976931348623157e+308
       0 111...111 0000...0000    inf                            inf
       0 111...111 0000...0001    snan1                          snan1
       0 111...111 0000...0010    snan10                         snan2
       ...
       0 111...111 0111...1111    snan1...1                      snan2251799813685247
       0 111...111 1000...0000    qnan0                          qnan0
       0 111...111 1000...0001    qnan1                          qnan1
       0 111...111 1000...0010    qnan10                         qnan2
       0 111...111 1111...1111    qnan1...1                      qnan2251799813685247
       1 000...000 0000...0000    -0.0...0000e-1022 = -0         -0.0
       0 000...000 0000...0001    -0.0...0001e-1022 = -1.0e-1074 -5e-324
       ...
       1 111...111 0000...0000    -inf                           -inf
       1 111...111 0000...0001    -snan1                         -snan1
       ...
       1 111...111 1000...0000    -qnan0                         -qnan0
       1 111...111 1111...1111    -qnan1...1                     -qnan2251799813685247

Notes

For finite numbers, I've displayed the decimal representation the same way Python 3.5 does (the shorted decimal fraction that rounds to the same binary fraction). For NaN values, Python displays them all as just "nan". Some C libraries will display something like "QNaN(1)" or "qnan1" or the like, so I've chosen to do that for clarity.

As mentioned in the previous post, this is for IEEE 754-2008 binary64, even though your platform (assuming you're not reading this in some far-future era where C11 and Python 3.5 are distant memories but blogspot posts are still relevant) probably uses the older IEEE 754-1985 double type, which is a slightly looser version of the same thing. So if you're on some older platform, a few things may be different (e.g., Irix on MIPS has snan and qnan reversed).

Generating the table yourself

Using the floatextras module, you can generate the values yourself. If it's not quite as easy as maybe it could be, suggestions are welcome.

One trick that helps a bit is that the make_tuple function can take any sequence of digits, or an int, which will be interpreted as an unsigned 52-bit integer:

    >>> from_tuple((0, -1, 0))
    1.9999999999999998

Unfortunately, you still have to turn that into bits to print the significand column of the table. I guess you could recover them from the float with as_tuple, but that seems silly. (I used the bitstring library again.) Maybe a function for generating table rows given either a float or a tuple and a +/- range would be useful?

It's been more than a decade since Typical Programmer Greg Jorgensen taught the word about Abject-Oriented Programming.

Much of what he said still applies, but other things have changed. Languages in the Abject-Oriented space have been borrowing ideas from another paradigm entirely—and then everyone realized that languages like Python, Ruby, and JavaScript had been doing it for years and just hadn't noticed (because these languages do not require you to declare what you're doing, or even to know what you're doing). Meanwhile, new hybrid languages borrow freely from both paradigms.

This other paradigm—which is actually older, but was largely constrained to university basements until recent years—is called Functional Addiction.

A Functional Addict is someone who regularly gets higher-order—sometimes they may even exhibit dependent types—but still manages to retain a job.

Retaining a job is of course the goal of all programming. This is why some of these new hybrid languages, like Rust, check all borrowing, from both paradigms, so extensively that you can make regular progress for months without ever successfully compiling your code, and your managers will appreciate that progress. After all, once it does compile, it will definitely work.

Closures

It's long been known that Closures are dual to Encapsulation.

As Abject-Oriented Programming explained, Encapsulation involves making all of your variables public, and ideally global, to let the rest of the code decide what should and shouldn't be private.

Closures, by contrast, are a way of referring to variables from outer scopes. And there is no scope more outer than global.

Immutability

One of the reasons Functional Addiction has become popular in recent years is that to truly take advantage of multi-core systems, you need immutable data, sometimes also called persistent data.

Instead of mutating a function to fix a bug, you should always make a new copy of that function. For example:

function getCustName(custID)
{
    custRec = readFromDB("customer", custID);
    fullname = custRec[1] + ' ' + custRec[2];
    return fullname;
}

When you discover that you actually wanted fields 2 and 3 rather than 1 and 2, it might be tempting to mutate the state of this function. But doing so is dangerous. The right answer is to make a copy, and then try to remember to use the copy instead of the original:

function getCustName(custID)
{
    custRec = readFromDB("customer", custID);
    fullname = custRec[1] + ' ' + custRec[2];
    return fullname;
}

function getCustName2(custID)
{
    custRec = readFromDB("customer", custID);
    fullname = custRec[2] + ' ' + custRec[3];
    return fullname;
}

This means anyone still using the original function can continue to reference the old code, but as soon as it's no longer needed, it will be automatically garbage collected. (Automatic garbage collection isn't free, but it can be outsourced cheaply.)

Higher-Order Functions

In traditional Abject-Oriented Programming, you are required to give each function a name. But over time, the name of the function may drift away from what it actually does, making it as misleading as comments. Experience has shown that people will only keep once copy of their information up to date, and the CHANGES.TXT file is the right place for that.

Higher-Order Functions can solve this problem:

function []Functions = [
    lambda(custID) {
        custRec = readFromDB("customer", custID);
        fullname = custRec[1] + ' ' + custRec[2];
        return fullname;
    },
    lambda(custID) {
        custRec = readFromDB("customer", custID);
        fullname = custRec[2] + ' ' + custRec[3];
        return fullname;
    },
]

Now you can refer to this functions by order, so there's no need for names.

Parametric Polymorphism

Traditional languages offer Abject-Oriented Polymorphism and Ad-Hoc Polymorphism (also known as Overloading), but better languages also offer Parametric Polymorphism.

The key to Parametric Polymorphism is that the type of the output can be determined from the type of the inputs via Algebra. For example:

function getCustData(custId, x)
{
    if (x == int(x)) {
        custRec = readFromDB("customer", custId);
        fullname = custRec[1] + ' ' + custRec[2];
        return int(fullname);
    } else if (x.real == 0) {
        custRec = readFromDB("customer", custId);
        fullname = custRec[1] + ' ' + custRec[2];
        return double(fullname);
    } else {
        custRec = readFromDB("customer", custId);
        fullname = custRec[1] + ' ' + custRec[2];
        return complex(fullname);
    }
}

Notice that we've called the variable x. This is how you know you're using Algebraic Data Types. The names y, z, and sometimes w are also Algebraic.

Type Inference

Languages that enable Functional Addiction often feature Type Inference. This means that the compiler can infer your typing without you having to be explicit:

function getCustName(custID)
{
    // WARNING: Make sure the DB is locked here or
    custRec = readFromDB("customer", custID);
    fullname = custRec[1] + ' ' + custRec[2];
    return fullname;
}

We didn't specify what will happen if the DB is not locked. And that's fine, because the compiler will figure it out and insert code that corrupts the data, without us needing to tell it to!

By contrast, most Abject-Oriented languages are either nominally typed—meaning that you give names to all of your types instead of meanings—or dynamically typed—meaning that your variables are all unique individuals that can accomplish anything if they try.

Memoization

Memoization means caching the results of a function call:

function getCustName(custID)
{
    if (custID == 3) { return "John Smith"; }
    custRec = readFromDB("customer", custID);
    fullname = custRec[1] + ' ' + custRec[2];
    return fullname;
}

Non-Strictness

Non-Strictness is often confused with Laziness, but in fact Laziness is just one kind of Non-Strictness. Here's an example that compares two different forms of Non-Strictness:

/****************************************
*
* TO DO:
*
* get tax rate for the customer state
* eventually from some table
*
****************************************/
// function lazyTaxRate(custId) {}

function callByNameTextRate(custId)
{
    /****************************************
    *
    * TO DO:
    *
    * get tax rate for the customer state
    * eventually from some table
    *
    ****************************************/
}

Both are Non-Strict, but the second one forces the compiler to actually compile the function just so we can Call it By Name. This causes code bloat. The Lazy version will be smaller and faster. Plus, Lazy programming allows us to create infinite recursion without making the program hang:

/****************************************
*
* TO DO:
*
* get tax rate for the customer state
* eventually from some table
*
****************************************/
// function lazyTaxRateRecursive(custId) { lazyTaxRateRecursive(custId); }

Laziness is often combined with Memoization:

function getCustName(custID)
{
    // if (custID == 3) { return "John Smith"; }
    custRec = readFromDB("customer", custID);
    fullname = custRec[1] + ' ' + custRec[2];
    return fullname;
}

Outside the world of Functional Addicts, this same technique is often called Test-Driven Development. If enough tests can be embedded in the code to achieve 100% coverage, or at least a decent amount, your code is guaranteed to be safe. But because the tests are not compiled and executed in the normal run, or indeed ever, they don't affect performance or correctness.

Conclusion

Many people claim that the days of Abject-Oriented Programming are over. But this is pure hype. Functional Addiction and Abject Orientation are not actually at odds with each other, but instead complement each other.