# 13.10 <cmath>

The <cmath> header declares a number of mathematical functions (from the C standard <math.h>). In addition to the standard C function, most functions have overloaded versions for different parameter types; each function's syntax shows all the overloaded versions.

If an argument is out of range, a domain error occurs. The function sets errno to EDOM and returns an error value. The value is defined by the implementation, so the only portable way to test for a domain error is to check errno. If the function's result is an overflow, a range error occurs. The function returns HUGE_VAL and sets errno to ERANGE. If underflow occurs, the function returns 0 and may or may not set errno to ERANGE. (See <cerrno> for more information about errno.)

 HUGE_VAL is defined to be a double, and the C++ standard does not define a suitable value for the float and long double versions of the math functions. If you are using a system that has infinity as an explicit floating-point value (such as IEC 60559/IEEE 754, which is found on PCs, Macintoshes, and modern workstations), the overloaded versions of a function probably return infinity for overflow, so there is no problem with the float and long double versions of the functions. For maximum portability, however, use only the double versions of the math functions.

All the trigonometric functions use radians. The descriptions of these functions use the common mathematical notation for ranges of values. [x, y) represents all values z such that x z < ythat is, the square bracket denotes an inclusive endpoint of a range, and the parenthesis denotes an exclusive endpoint of a range.

<cfloat>

Declares macros for the limits of floating-point types

<climits>

Declares macros for the limits of integer types

<complex>

Declares types and functions for working with complex numbers

<cstdlib>

Declares integer absolute value functions and functions that compute a quotient and remainder in a single operation

<limits>

Declares the numeric_limits class template for the limits of the numerical typese.g., the largest float, the precision of double, and so on

<numeric>

Declares generic numerical algorithms

<valarray>

Declares types and functions for computation with arrays of numbers

 abs function Computes absolute value

 ```float abs(float x) double abs(double x) long double abs(long double x)```

The abs function returns the absolute value of its argument: if x < 0, it returns -x; otherwise, it returns x.

The abs function in <cmath> is the same as fabs. The <cstdlib> header declares integer versions of the abs function.

fabs function, abs function in <cstdlib>

 acos function Computes inverse cosine

 ```float acos(float x) double acos(double x) long double acos(long double x)```

The acos function returns the inverse cosine of its argument. The parameter x must be in the range [-1, 1], or a domain error occurs. The return value is in the range [0, ].

 asin function Computes inverse sine

 ```float asin(float x) double asin(double x) long double asin(long double x)```

The asin function returns the inverse sine of its argument. The parameter x must be in the range [-1, 1], or a domain error occurs. The return value is in the range [-/2, /2].

 atan function Computes inverse tangent

 ```float atan(float x) double atan(double x) long double atan(long double x)```

The atan function returns the inverse tangent of its argument. The return value is in the range [-/2, /2].

 atan2 function Computes inverse tangent

 ```float atan2(float y, float x) double atan2(double y, double x) long double atan2(long double y, long double x)```

The atan2 function returns the inverse tangent of y/x using the sign of both numbers to determine the quadrant for the return value. It correctly handles the case in which x is 0. (That is, it returns /2 times the sign of y for nonzero y; if y is 0, the result is implementation-defined and might be a range error). The return value is in the range [-, ].

 ceil function Computes ceiling

 ```float ceil(float x) double ceil(double x) long double ceil(long double x)```

The ceil function returns the smallest integer that is greater than or equal to x.

floor function

 cos function Computes cosine

 ```float cos(float x) double cos(double x) long double cos(long double x)```

The cos function returns the cosine of its argument, in radians. The return value is in the range [-1, 1].

 cosh function Computes hyperbolic cosine

 ```float cosh(float x) double cosh(double x) long double cosh(long double x)```

The cosh function returns the hyperbolic cosine of its argument. Note that <cmath> has no inverse hyperbolic trigonometric functions; the Boost project fills that gap. See Appendix B for information about Boost.

 exp function Computes exponential

 ```float exp(float x) double exp(double x) long double exp(long double x)```

The exp function returns ex. If x is too large, a range error occurs.

log function, pow function

 fabs function Computes absolute value

 ```float fabs(float x) double fabs(double x) long double fabs(long double x)```

The fabs function returns the absolute value of its argument: if x < 0, it returns -x; otherwise, it returns x.

The fabs function is the same as abs for floating-point numbers. It exists only for compatibility with C.

abs function, abs function in <cstdlib>

 floor function Computes floor

 ```float floor(float x) double floor(double x) long double floor(long double x)```

The floor function returns the largest integer that is less than or equal to x.

ceil function

 fmod function Computes modulus

 ```float fmod(float x, float y) double fmod(double x, double y) long double fmod(long double x, long double y)```

The fmod function returns the floating-point remainder of dividing x by y. If y is 0, the behavior is implementation-defined: the return value might be 0, or a domain error can occur. If y is nonzero, the return value is x - k x y for some integer k, such that the result has the same sign as x and an absolute value less than the absolute value of y.

 frexp function Computes binary fraction and exponent

 ```float frexp(float x, int* exp) double frexp(double x, int* exp) long double frexp(long double x, int* exp)```

The frexp function separates a floating-point number into a fraction and an exponent (with a base of 2) such that x = frac x 2e, in which frac is in the range [1/2, 1) or is 0 if x is 0. The exponent, e, is stored in *exp. The return value is frac. If x is 0, the return value and *exp are 0.

ldexp function, modf function

 HUGE_VAL macro Range error value

 `double HUGE_VAL`

When an overflow occurs, most functions set errno to ERANGE and return HUGE_VAL with the correct sign of the result. The exact value of HUGE_VAL is implementation-defined and is not necessarily a compile-time constant. It might even be a value that can be returned as a valid result from the function. In that case, the only way to discover whether an overflow occurred is to test errno, as shown in Example 13-5.

#### Example

##### Example 13-5. Computing a logarithm to any base
```// Return the logarithm of x to the base n.

template<typename T>

T logn(T x, T n)

{

errno = 0;

T logx = log(x);

if (errno == ERANGE)

return logx;    // Should be HUGE_VAL

else if (errno != 0)

return logx;    // Implementation defined

T logn = log(n);

if (errno == ERANGE)

return logn;    // Should be HUGE_VAL

else if (errno != 0)

return logn;    // Implementation defined

if (logn == 0) {

errno = EDOM;

return 0;

}

return logx / logn;

}```

<cerrno>

 ldexp function Makes floating point from binary fraction and exponent

 ```float ldexp(float frac, int exp) double ldexp(double frac, int exp) long double ldexp(long double frac, int exp)```

The ldexp function returns a floating-point number that it constructs from a fractional part and an exponent (base 2). The return value is frac x 2exp.

frexp function, modf function

 log function Computes natural logarithm

 ```float log(float x) double log(double x) long double log(long double x)```

The log function returns the natural (base e) logarithm of its argument. A domain error occurs if x is negative. A range error might occur if x is 0.

 log10 function Computes common logarithm

 ```float log10(float x) double log10(double x) long double log10(long double x)```

The log10 function returns the common (base 10) logarithm of its argument. A domain error occurs if x is negative. A range error might occur if x is 0.

 modf function Separates integer and fraction parts

 ```float modf(float x, float* iptr) double modf(double x, double* iptr) long double modf(long double x, long double* iptr)```

The modf function splits a floating-point number into integral and fractional parts. Both parts have the same sign as x. The integral part is stored in *iptr; the return value is the fractional part.

frexp function, ldexp function

 pow function Computes power

 ```float pow(float x, float y) float pow(float x, int y) double pow(double x, double y) double pow(double x, int y) long double pow(long double x, long double y) long double pow(long double x, int y)```

The pow function raises x to the y power. If x is negative, and y is not an integral value, a domain error occurs. If x is 0, and y is less than or equal to 0, and the result cannot be represented as a real number, a domain error occurs. A range error can occur if the result is out of range.

exp function

 sin function Computes sine

 ```float sin(float x) double sin(double x) long double sin(long double x)```

The sin function returns the sine of its argument, in radians. The return value is in the range [-1, 1].

 sinh function Computes hyperbolic sine

 ```float sinh(float x) double sinh(double x) long double sinh(long double x)```

The sinh function returns the hyperbolic sine of its argument. Note that <cmath> has no inverse hyperbolic trigonometric functions; the Boost project fills that gap. See Appendix B for information about Boost.

 sqrt function Computes square root

 ```float sqrt(float x) double sqrt(double x) long double sqrt(long double x)```

The sqrt function returns the square root or its argument. If x is negative, a domain error occurs. The return value is always positive or 0.

 tan function Computes tangent

 ```float tan(float x) double tan(double x) long double tan(long double x)```

The tan function returns the tangent of its argument. The standard does not specify the result when the tangent is undefined (that is, when x is k +/2 for any integer k), but a reasonable result is a range error. Due to the nature of the tangent function, the sign of the return value (HUGE_VAL) can be positive or negative.

 tanh function Computes hyperbolic tangent

 ```float tanh(float x) double tanh(double x) long double tanh(long double x)```

The tanh function returns the hyperbolic tangent of its argument. Note that <cmath> has no inverse hyperbolic trigonometric functions; the Boost project fills that gap. See Appendix B for information about Boost.

 Chapter 1. Language Basics
 Chapter 2. Declarations
 Chapter 3. Expressions
 Chapter 4. Statements
 Chapter 5. Functions
 Chapter 6. Classes
 Chapter 7. Templates
 Chapter 8. Standard Library
 Chapter 9. Input and Output
 Chapter 10. Containers, Iterators, and Algorithms
 Chapter 11. Preprocessor Reference
 Chapter 12. Language Reference
 Appendix A. Compiler Extensions
 Appendix B. Projects
 Glossary