These two programs — compiled with Microsoft Visual C++ and run on a 32-bit Intel Core Duo processor — demonstrate an anomaly that occurs when using single-precision floating point variables:

Program 1

#include "stdio.h"
int main (void)
{
 float f1 = 0.1f, f2 = 3.0f, f3;

 f3 = f1 * f2;
 if (f3 != f1 * f2)
   printf("Not equal\n");
}

Prints “Not equal”.

Program 2

#include "stdio.h"
int main (void)
{
 float f1 = 0.7f, f2 = 10.0f, f3;
 int i1, i2;

 f3 = f1 * f2;
 i1 = (int)f3;
 i2 = (int)(f1 * f2);
 if (i1 != i2)
   printf("Not equal\n");
}

Prints “Not equal”.

In each case, f3 and f1 * f2 differ. But why? I’ll explain what’s going on.

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In my article “Patterns in the Last Digits of the Positive Powers of Five” I showed that the cycles of ending digits of the positive powers of five could be represented with a binary tree:

Binary Tree Showing Nested Ending Digit Patterns of the Positive Powers of Five

The tree layout shows that certain pairs of ending digits are related, and that these pairs differ by five in their starting digits. I will show why this is true.

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In my article “Counting Binary and Hexadecimal Palindromes” I derived formulas for counting binary palindromes and hexadecimal palindromes. For each type of palindrome, I derived two pairs of formulas: one pair to count n-digit palindromes, and one pair to count palindromes of n digits or less.

In this article, I will derive similar formulas to count binary/hexadecimal palindromes — multi-base palindromes I’ve shown to have an algorithmically defined structure.

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To perform arbitrary-precision arithmetic in C and C++ programs on Windows, I use GMP. In particular, I use MPIR, a Windows port of GMP. MPIR is a simple alternative to using GMP under Cygwin or MinGW.

I will show you how to install MPIR in Microsoft Visual C++ as a static, 32-bit library. I will also show you how to install the optional C++ interface — also as a static, 32-bit library. I will provide two example C programs that call the GMP integer and floating-point functions, and two equivalent C++ programs — programs that use the same GMP functions, only indirectly through the C++ interface.

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