3.3. Math Functions

In mathematics, you have probably seen functions like sin and log, and you have learned to evaluate expressions like \(\sin(\pi/2)\) and \(\log(1/x)\). First, you evaluate the expression in parentheses, which is called the argument of the function. For example, \(\pi/2\) is approximately 1.571, and \(1/x\) is 0.1 (if \(x\) happens to be 10).

Then you can evaluate the function itself, either by looking it up in a table or by performing various computations. The \(\sin\) of 1.571 is 1, and the \(\log\) of 0.1 is -1 (assuming that \(\log\) indicates the logarithm base 10).

This process can be applied repeatedly to evaluate more complicated expressions like \(\log(1/\sin(\pi/2))\). First we evaluate the argument of the innermost function, then evaluate the function, and so on.

C++ provides a set of built-in functions that includes most of the mathematical operations you can think of. The math functions are invoked using a syntax that is similar to mathematical notation:

This program performs calculations using some of the built-in functions from the cmath library.

The first example sets log to the logarithm of 17, base \(e\). There is also a function called log10 that takes logarithms base 10.

The second example finds the sine of the value of the variable radians. C++ assumes that the values you use with sin and the other trigonometric functions (cos, tan) are in radians.

Note

To convert from degrees to radians, you can multiply by \(\pi/180\).

If you don’t happen to know \(\pi\) to 15 digits, you can calculate it using the atan function. The arctangent (or inverse tangent) of 1 is \(\pi/4\), and multiplying by 4 evaluates to \(\pi\).

This program also uses built-in functions from the cmath library, specifically the functions that deal with angles. As you can see, we have a line of code that converts the default radians value to degrees.

The value pi can’t be declared constexpr here, because the functions in the math library are not constexpr. We can only use constant expressions to initialize something as constexpr. This is a part of the language that is actively changing! Starting in C++17, many more parts of the standard library have started to include constexpr versions of types and functions.

Note that some compilers do return constexpr values for the math library functions, but this is generally considered a bug and should not be counted on.

Note

Constants introduced in C++20.

C++20 defines several numeric constants useful in engineering and math. The full list of constants is available on cppreference.com and are located in the header numbers. For example:

#include <numbers>

int main() {
   double radians = 90 * std::numbers::pi / 180.0;
}

Prior to the official inclusion of these constants, programmers needed to either define their own, or use a third party library such as https::boost.org/. Some compilers define a macro M_PI, but it is not universal and should not be relied on.

Before you can use any of the math functions, you have to include the math header file. Header files contain information the compiler needs about functions that are defined in other source files. For example, in the “Hello, world!” program we included a header file named iostream using an include statement:

#include <iostream>
using namespace std;

We have been using header files since chapter 1. iostream contains information about input and output (I/O) streams, including the object named cout. C++ has a powerful feature called namespaces, that are used to avoid naming conflicts between function or types that might otherwise have the same name, like cout. Technically cout is declared in the namespace std, making its fully qualified name std::cout. We can pull all of the names from the standard name space into the global namespace with the statement:

using namespace std;

All of the facilities defined in the C++ Standard Library are defined in the namespace std. More on namespaces in a few sections.

Similarly, the math header file contains information about the math functions. You can include it at the beginning of your program along with iostream:

#include <cmath>

Such header files have an initial ‘c’ to signify that these header files have been derived from the C language. The original C version of this header is math.h

Use C++ headers in C++

Use the C++ versions of C header files when writing C++. Although the language allows you to use either, it is considered a best practice to use the C++ headers when writing C++ code.

There are some subtle differences in the guarantees that the different headers support.

For compatibility reasons, the functions in cmath are also in the std namespace, however, they are also in the global namespace. For this reason, you often see math functions used without the std:: namespace prefix.

Note

Fun fact!

All of the original C headers, such as math.h have been deprecated since the first ISO standard, C++98. However, they are now slated to be ‘undeprecated’ as of C++23.

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