math.h

The math.h header file provides a number of mathematical functions.

Many math functions return values of type double, but offer both float and long double versions, for example the pow()` function has both powf() and ``powl() versions.

double pow(double x, double y); 
float powf(float x, float y);
long double powl(long double x, long double y);

For brevity, the f suffixed (float type) and l suffixed (long double) versions of the functions are omitted below.

Types and macros

Two new type aliases are defined in math.h.

float_t: the type that (on the current system) most efficiently performs float operations, and is at least as wide as float.
double_t: the type that (on the current system) most efficiently performs double operations, and is at least as wide as double.

Their specific types can be found with the macro FLT_EVAL_METHOD.

The value of FLT_EVAL_METHOD	The type corresponding to float_t	The type corresponding to double_t
0	float	double
1	double	double
2	long double	long double
other	determined by implementation	determined by implementation

math.h also defines a number of macros.

INFINITY: means positive infinity, returns a float type value.
NAN: indicates a non-number (Not-A-Number) and returns a float type value.

Error types

The following types of errors are reported for mathematical functions.

Range errors: The result of an operation cannot be represented by the function return type.
Domain errors: the function argument does not apply to the current function.
Pole errors: The argument causes the limit of the function to become infinite.
Overflow errors: The result of an operation is too large, resulting in an overflow.
Underflow errors: The result of an operation is too small, resulting in an overflow.

The variable math_errhandling indicates how the current system handles mathematical errors.

value of math_errhandling	description
MATH_ERRNO	The system uses errno to indicate a mathematical error.
MATH_ERREXCEPT	The system uses exceptions to indicate math errors
MATH_ERREXCEPT	The system uses both to indicate a maths error

Numeric Types

The arguments to mathematical functions can be divided into the following categories: normal, infinite, finite and non-numeric.

The following function is used to determine the type of a value.

fpclassify(): returns the classification of the given floating point number.
isfinite(): true if the argument is not infinite or NaN.
isinf(): true if the argument is infinite.
isnan(): true if the argument is not a number.
isnormal(): true if the argument is a normal number.

Here is an example.

isfinite(1.23) // 1
isinf(1/tan(0)) // 1
isnan(sqrt(-1)) // 1
isnormal(1e-310)) // 0

signbit()

signbit() determines if the argument is signed. Returns 1 if the argument is negative, 0 otherwise.

signbit(3490.0) // 0
signbit(-37.0) // 1

Trigonometric functions

The following are trigonometric functions, with arguments in radians.

acos()：反余弦。
asin()：反正弦。
atan()：反正切
atan2()：反正切。
cos()：余弦。
sin()：正弦。
tan()：正切。

不要忘了，上面所有函数都有 float 版本（函数名加上 f 后缀）和 long double 版本（函数名加上 l 后缀）。

下面是一个例子。

cos(PI/4) // 0.707107

双曲函数

以下是双曲函数，参数都为浮点数。

acosh()：反双曲余弦。
asinh()：反双曲正弦。
atanh()：反双曲正切。
cosh()：双曲余弦。
tanh()：双曲正切。
sinh()：双曲正弦。

指数函数和对数函数

以下是指数函数和对数函数，它们的返回值都是 double 类型。

exp()：计算欧拉数 e 的乘方，即 e^x。
exp2()：计算 2 的乘方，即 2^x。
expm1()：计算 e^x - 1。
log()：计算自然对数，exp()的逆运算。
log2()：计算以2为底的对数。
log10()：计算以10为底的对数。
logp1()：计算一个数加 1 的自然对数，即ln(x + 1)。
logb()：计算以宏FLT_RADIX（一般为2）为底的对数，但只返回整数部分。

下面是一些例子。

exp(3.0) // 20.085500
log(20.0855) // 3.000000
log10(10000) // 3.000000

如果结果值超出了 C 语言可以表示的最大值，函数将返回HUGE_VAL，它是一个在math.h中定义的 double 类型的值。

如果结果值太小，无法用 double 值表示，函数将返回0。以上这两种情况都属于出错。

frexp()

frexp()将参数分解成浮点数和指数部分（2为底数），比如 1234.56 可以写成 0.6028125 * 2¹¹，这个函数就能分解出 0.6028125 和 11。

double frexp(double value, int* exp);

它接受两个参数，第一个参数是用来分解的浮点数，第二个参数是一个整数变量指针。

它返回小数部分，并将指数部分放入变量exp。如果参数为0，则返回的小数部分和指数部分都为0。

下面是一个例子。

double frac;
int expt;

// expt 的值是 11
frac = frexp(1234.56, &expt);

// 输出 1234.56 = 0.6028125 x 2^11
printf("1234.56 = %.7f x 2^%d\n", frac, expt);

ilogb()

ilogb()返回一个浮点数的指数部分，指数的基数是宏FLT_RADIX（一般是2）。

int ilogb(double x);

它的参数为x，返回值是 log_r|x|，其中r为宏FLT_RADIX。

下面是用法示例。

ilogb(257) // 8
ilogb(256) // 8
ilogb(255) // 7

ldexp()

ldexp()将一个数乘以2的乘方。它可以看成是frexp()的逆运算，将小数部分和指数部分合成一个f * 2^n形式的浮点数。

double ldexp(double x, int exp);

It takes two arguments, the first being the multiplier x and the second being the exponential part of 2, exp, and returns "x * 2^exp".

ldexp(1, 10) // 1024.000000
ldexp(3, 2) // 12.000000
ldexp(0.75, 4) // 12.000000
ldexp(0.5, -1) // 0.250000

modf()

modf()函数提取一个数的整数部分和小数部分。

 double modf(double value, double* iptr);

它接受两个参数，第一个参数value表示待分解的数值，第二个参数是浮点数变量iptr。返回值是value的小数部分，整数部分放入变量double。

下面是一个例子。

// int_part 的值是 3.0
modf(3.14159, &int_part); // 返回 0.14159

scalbn()

scalbn()用来计算“x * rⁿ”，其中r是宏FLT_RADIX。

double scalbn(double x, int n);

它接受两个参数，第一个参数x是乘数部分，第二个参数n是指数部分，返回值是“x * rⁿ”。

下面是一些例子。

scalbn(2, 8) // 512.000000

这个函数有多个版本。

scalbn()：指数 n 是 int 类型。
scalbnf()：float 版本的 scalbn()。
scalbnl()：long double 版本的 scalbn()。
scalbln()：指数 n 是 long int 类型。
scalblnf()：float 版本的 scalbln()。
scalblnl(): the long double version of scalbln().

round()

The round()` function rounds in the traditional way, e.g. 1.5` rounds to 2 and -1.5 rounds to -2.

``c double round(double x);

It returns a floating point number.

Here are some examples.

```c
round(3.14) // 3.000000
round(3.5) // 4.000000
round(-1.5) // -2.000000
round(-1.14) // -1.000000

It also has some other versions.

lround(): the return value is of type long int.
llround(): the return value is of type long long int.

trunc()

`trunc() is used to truncate the fractional part of a floating point number and return the remaining integer part as a floating point number.

double trunc(double x);

Here are some examples.

trunc(3.14) // 3.000000
trunc(3.8) // 3.000000
trunc(-1.5) // -1.000000
trunc(-1.14) // -1.000000

ceil()

`ceil() returns the smallest integer (of type double) that is not less than its argument, which is "rounded up".

double ceil(double x);

Here are some examples.

ceil(7.1) // 8.0
ceil(7.9) // 8.0
ceil(-7.1) // -7.0
ceil(-7.9) // -7.0

floor()

`floor() returns the largest integer that is not greater than its argument, which is "rounded down".

double floor(double x);

Here are some examples.

floor(7.1) // 7.0
floor(7.9) // 7.0
floor(-7.1) // -8.0
floor(-7.9) // -8.0

The following function does the "rounding".

double round_nearest(double x) {
  return x < 0.0 ? ceil(x - 0.5) : floor(x + 0.5);
}

fmod()

fmod()` returns the remainder of the first argument divided by the second argument, which is the floating point version of the remainder operator %, since ``% can only be used for integer operations.

double fmod(double x, double y);

The calculation it performs behind the scenes is x - trunc(x / y) * y, and the return value has the same sign as x.

fmod(5.5, 2.2) // 1.100000
fmod(-9.2, 5.1) // -4.100000
fmod(9.2, 5.1) // 4.100000

Floating point comparison functions

The following functions are used to compare two floating point numbers, with the return value being of type integer.

isgreater(): returns the result of x > y.
isgreaterequal(): returns x >= y.
isless(): returns x < y.
islessequal(): returns x <= y.
islessgreater(): returns the result of (x < y) || (x > y).

Here are some examples.

isgreater(10.0, 3.0) // 1
isgreaterequal(10.0, 10.0) // 1
isless(10.0, 3.0) // 0
islessequal(10.0, 3.0) // 0
islessgreater(10.0, 3.0) // 1
islessgreater(10.0, 30.0) // 1
islessgreater(10.0, 10.0) // 0

isunordered()

`isunordered() returns whether NAN is present in one of the two arguments.

int isunordered(any_floating_type x, any_floating_type y);

Here are some examples.

isunordered(1.0, 2.0) // 0
isunordered(1.0, sqrt(-1)) // 1
isunordered(NAN, 30.0) // 1
isunordered(NAN, NAN) // 1

Other functions

The following are the other functions included in math.h.

pow(): computes the yth power of the argument x.
sqrt(): calculates the square root of a number.
cbrt(): calculates the cube root.
fabs(): calculates the absolute value.
hypot(): calculates the hypotenuse, based on the two right-angled sides of a right triangle.
fmax(): returns the maximum of the two arguments.
fmin(): returns the minimum value of the two arguments.
remainder(): returns the remainder of the IEC 60559 standard, similar to fmod(), but the remainder ranges from -y/2 to y/2 rather than from 0 to y.
remquo(): returns both the remainder and the quotient, the remainder is calculated in the same way as remainder().
copysign(): returns a value whose size is equal to the first argument and whose sign is equal to the second argument.
nan(): returns NAN.
nextafter(): gets the next (or previous, depending on the direction of the second argument, y) floating point value that the current system can represent.
nextoward(): same as nextafter(), except that the second argument is of type long double.
fdim(): returns the difference if the first argument minus the second argument is greater than 0, otherwise it returns 0.
fma(): returns the result of x * y + z in a quick calculation.
nearbyint(): rounds to the nearest integer in the current rounding direction. The current rounding direction can be set using the fesetround() function.
rint(): rounds to the nearest integer in the current rounding direction, same as nearbyint(). The difference is that it throws an INEXACT exception for floating point numbers.
lrint(): rounds to the nearest integer in the current rounding direction, same as rint(). The difference is that the return value is an integer, not a floating point number.
erf(): error function to calculate a value.
erfc(): computes a complementary error function for a value.
tgamma(): calculates the Gamma function.
lgamma(): calculates the natural logarithm of the absolute value of the Gamma function.

Here are some examples.

pow(3, 4) // 81.000000
sqrt(3.0) // 1.73205
cbrt(1729.03) // 12.002384
fabs(-3490.0) // 3490.000000
hypot(3, 4) // 5.000000
fmax(3.0, 10.0) // 10.000000
fmin(10.0, 3.0) // 3.000000
```''

math.h

Types and macros​

Error types​

Numeric Types​

signbit()​

Trigonometric functions​

双曲函数​

指数函数和对数函数​

frexp()​

ilogb()​

ldexp()​

modf()​

scalbn()​

round()​

trunc()​

ceil()​

floor()​

fmod()​

Floating point comparison functions​

isunordered()​

Other functions​

Types and macros

Error types

Numeric Types

signbit()

Trigonometric functions

双曲函数

指数函数和对数函数

frexp()

ilogb()

ldexp()

modf()

scalbn()

round()

trunc()

ceil()

floor()

fmod()

Floating point comparison functions

isunordered()

Other functions