Leading digits and quadmath

My previous post looked at a problem that requires repeatedly finding the first digit of kn where k is a single digit but n may be on the order of millions or billions.

The most direct approach would be to first compute kn as a very large integer, then find it’s first digit. That approach is slow, and gets slower as n increases. A faster way is to look at the fractional part of log kn = n log k and see which digit it corresponds to.

If n is not terribly big, this can be done in ordinary precision. But when n is large, multiplying log k by n and taking the fractional part brings less significant digits into significance. So for very large n, you need extra precision. I first did this in Python using SymPy, then switched to C++ for more speed. There I used the quadmath library for gcc. (If n is big enough, even quadruple precision isn’t enough. An advantage to SymPy over quadmath is that the former has arbitrary precision. You could, for example, set the precision to be 10 more than the number of decimal places in n in order to retain 10 significant figures in the fractional part of n log k.)

The quadmath.h header file needs to be wrapped in an extern C declaration. Otherwise gcc will give you misleading error messages.

The 128-bit floating point type __float128 has twice as many bits as a double. The quadmath functions have the same name as their standard math.h counterparts, but with a q added on the end, such as log10q and fmodq below.

Here’s code for computing the leading digit of kn that illustrates using quadmath.

#include <cmath >
extern "C" {
#include <quadmath.h>

__float128 logs[11];

for (int i = 2; i  <= 10; i++)
    logs[i] = log10q(i + 0.0);

int first_digit(int base, long long exponent)
    __float128 t = fmodq(exponent*logs[base], 1.0);
    for (int i = 2; i  <= 10; i++)
        if (t  < logs[i])
            return i-1;

The code always returns because t is less than 1.

Caching the values of log10q saves repeated calls to a relatively expensive function. So does using the search at the bottom rather than computing powq(10, t).

The linear search at the end is more efficient than it may seem. First, it’s only search a list of length 9. Second, because of Benford’s law, the leading digits are searched in order of decreasing frequency, i.e. most inputs will cause first_digit to return early in the search.

When you compile code using quadmath, be sure to add -lquadmath to the compile command.

Related posts

Porting Visual C++ code to Linux/gcc

Here are a few lessons learned from porting a numerical library recently from Windows/Visual C++ to Linux/gcc.

Some of our code only runs on Windows, and only needs to run on Windows. Our first thought was to put #ifdef WIN32 directives around that code. Clift Norris came up with the clever idea of using #ifndef EXCLUDE_WINDOWS_ONLY_CODE instead. That way we could do a preliminary test of the portable subset of the code while still working on Windows where we’re more comfortable. Also, by not referring specifically to 32-bit Windows, we’re OK moving the code to 64-bit Windows.

Visual C++ does not require source and header files to end with newline characters, but gcc does. We got hundreds of warnings of the form warning: no newline at end of file when we first attempted to compile our code on Linux. Apparently there’s no gcc switch to turn this off, and it may not be prudent to turn it off if you could. As I understand it, Visual Studio inserts a linebreak after including header files, but gcc may not and so gcc needs to issue a warning in this case while Visual Studio does not. We copy our source tree to a Linux box then run the following Python code on that box to insert the extra newline characters when needed.

import os 

# List of directories of files to add newlines to.
# Script must be in the same location as these directories.
directories = ["Banana", "Apple", "Peach"]

for dir in directories:
    for file in os.listdir(dir):
        if file.endswith(".h") or file.endswith(".cpp"):
            path = dir + "/" + file
            handle = open(path, "r")
            slurp = handle.read()

            if not slurp.endswith("n"):
                retcode = os.system("chmod +w " + path)
                if retcode != 0:
                    print "chmod returned " + retcode + " on " + path
                    handle = open(path, "a")

There were several places in our code where a variable was deliberately unused but retained in a function signature. Suppose a function has signature void foo(int a, int b) but b is unused. We had usually handled that by making b; the first line of the implementation. That would suppress unused variable warnings in Visual C++, but not in gcc. When we changed the function signature to void foo(int a, int /* b */), that made both compilers happy.

We started out using autoconf, but that was overkill for our project. Our build process became two orders of magnitude simpler when we switched over to a crude, old-fashioned make file. After porting the library to Linux, we built it on OS X without any issue.

This wasn’t an issue for us, but a potential problem when moving numerical code between Unix-like systems is that the function gamma computes different things on different systems. On Linux it computes the logarithm of the mathematical gamma function but on OS X it computes the gamma function itself. See the last two paragraphs of how to calculate binomial probabilities and Thomas Guest’s comment on that post for a full explanation.

This library had been ported to Linux years ago, but nobody used it on Linux and so development continued only on Windows. When we first ported the code, gcc and Visual C++ seemed to have incompatible requirements, especially with templates. The more recent port described here was much easier now that both compilers are more compliant with the C++ standard.

Regular expressions in C++ TR1

Regular expressions are not a part of the C++ Standard Library quite yet, but there is a document (Technical Report 1, or TR1) that includes among other things a specification for regular expression support that will probably be added to the C++ standard eventually.

The Boost library has supported TR1 for a while. Microsoft just released a feature pack for Visual Studio 2008 a month ago that includes support for most of TR1. (They’ve left out support for mathematical special functions.) And Dinkumware sells a complete TR1 implementation.

I’ve added some notes to my website for getting started with C++ TR1 regular expressions. I took my PowerShell regex notes as a starting point and implemented some of the same examples in C++. I changed the organization though, because the C++ implementation is fairly different from PowerShell.

Working with regular expressions is harder in C++ than in scripting languages such as Perl or Python, but not unnecessarily so. C++ is optimized for fine-grained control and efficiency rather than ease of use; that’s what C++ is for. The TR1 implementation is internally consistent and elegant in its own way.

It’s easy to find API-level documentation but harder to find examples for getting started. (I’ve heard good things about Pete Becker’s book The C++ Standard Library Extensions but I haven’t read it.) So I decided to keep some notes as I played with the Visual Studio implementation. I imagine most of the content applies to other implementations, but I’ve only tested the examples using Visual Studio.

Update: GCC just added support for C++ TR1 two days ago with their version 4.3 release.  However, it appears support for regular expressions is not included.

How to calculate binomial probabilities

Suppose you’re doing something that has probability of success p and probability of failure q = 1 − p. If you repeat what you’re doing m+n times, the probability of m successes and n failures is given by

\frac{(m+n)!}{m!\, n!} p^m q^n

Now suppose m and n are moderately large. The terms (m+n)! and m! n! will be huge, but the terms pm and qn will be tiny. The huge terms might overflow, and the tiny terms might underflow, even though the final result may be a moderate-sized number. The numbers m and n don’t have to be that large for this to happen since factorials grow very quickly. On a typical system, overflow happens if m+n > 170. How do you get to the end result without having to deal with impossibly large or small values along the way?

The trick is to work with logs. You want to first calculate

log( (m+n)! ) − log( m! ) − log( n! ) + m log( p ) + n log( q )

then exponentiate the result. This pattern comes up constantly in statistical computing.

Libraries don’t always include functions for evaluating factorial or log factorial. They’re more likely to include functions for Γ(x) and its log. For positive integers n, Γ(n+1) = n!. Now suppose you have a function lgamma that computes log Γ(x). You might write something like this.

    double probability(double p, double q, int m, int n)
        double temp = lgamma(m + n + 1.0);
        temp -=  lgamma(n + 1.0) + lgamma(m + 1.0);
        temp += m*log(p) + n*log(q);
        return exp(temp);

The function lgamma is not part of the ANSI standard library for C or C++, but it is part of the POSIX standard. On Unix-like systems, lgamma is included in the standard library. However, Microsoft does not include lgamma as part of the Visual C++ standard library. On Windows you have to either implement your own lgamma function or grab an implementation from somewhere like Boost.

Here’s something to watch out for with POSIX math libraries. I believe the POSIX standard does not require a function called gamma for evaluating the gamma function Γ(x). Instead, the standard requires functions lgamma for the log of the gamma function and tgamma for the gamma function itself. (The mnemonic is “t” for “true,” meaning that you really want the gamma function.) I wrote a little code that called gamma and tested it on OS X and Red Hat Linux this evening. In both cases gcc compiled the code without warning, even with the -Wall and -pedantic warning flags. But on the Mac, gamma was interpreted as tgamma and on the Linux box it was interpreted as lgamma. This means that gamma(10.0) would equal 362880 on the former and 12.8018 on the latter.

If you don’t have access to an implementation of log gamma, see How to compute log factorial.