Mathematical Functions¶
This chapter describes basic mathematical functions. Some of these functions are present in system libraries, but the alternative versions given here can be used as a substitute when the system functions are not available.
The functions and macros described in this chapter are defined in the
header file cml/math.h
.
Mathematical Constants¶
The library ensures that the standard BSD mathematical constants are defined. For reference, here is a list of the constants:
M_E 
The base of exponentials, 
M_LOG2E 
The base2 logarithm of , 
M_LOG10E 
The base10 logarithm of , 
M_SQRT2 
The square root of two, 
M_SQRT1_2 
The square root of onehalf, 
M_SQRT3 
The square root of three, 
M_PI 
The constant pi, 
M_PI_2 
Pi divided by two, 
M_PI_4 
Pi divided by four, 
M_SQRTPI 
The square root of pi, 
M_2_SQRTPI 
Two divided by the square root of pi, 
M_1_PI 
The reciprocal of pi, 
M_2_PI 
Twice the reciprocal of pi, 
M_LN10 
The natural logarithm of ten, 
M_LN2 
The natural logarithm of two, 
M_LNPI 
The natural logarithm of pi, 
M_EULER 
Euler’s constant, 
Infinities and Notanumber¶

POSINF
¶ This macro contains the IEEE representation of positive infinity, . It is computed from the expression
+1.0/0.0
.

NEGINF
¶ This macro contains the IEEE representation of negative infinity, . It is computed from the expression
1.0/0.0
.

NAN
¶ This macro contains the IEEE representation of the NotaNumber symbol,
NaN
. It is computed from the ratio0.0/0.0
.

bool
cml_isnan
(double x)¶ This function returns 1 if
x
is notanumber.

bool
cml_isinf
(double x)¶ This function returns if
x
is positive infinity, ifx
is negative infinity and 0 otherwise. [1]

bool
cml_isfinite
(double x)¶ This function returns 1 if
x
is a real number, and 0 if it is infinite or notanumber.
Elementary Functions¶
The following routines provide portable implementations of functions
found in the BSD math library. When native versions are not available
the functions described here can be used instead. The substitution can
be made automatically if you use autoconf
to compile your
application (see portabilityfunctions).

double
cml_log1p
(double x)¶ This function computes the value of in a way that is accurate for small
x
. It provides an alternative to the BSD math functionlog1p(x)
.

double
cml_expm1
(double x)¶ This function computes the value of in a way that is accurate for small
x
. It provides an alternative to the BSD math functionexpm1(x)
.

double
cml_hypot
(double x, double y)¶ This function computes the value of in a way that avoids overflow. It provides an alternative to the BSD math function
hypot(x,y)
.

double
cml_hypot3
(double x, double y, double cml_z)¶ This function computes the value of in a way that avoids overflow.

double
cml_acosh
(double x)¶ This function computes the value of . It provides an alternative to the standard math function
acosh(x)
.

double
cml_asinh
(double x)¶ This function computes the value of . It provides an alternative to the standard math function
asinh(x)
.

double
cml_atanh
(double x)¶ This function computes the value of . It provides an alternative to the standard math function
atanh(x)
.

double
cml_ldexp
(double x, int e)¶ This function computes the value of . It provides an alternative to the standard math function
ldexp(x,e)
.

double
cml_frexp
(double x, int *e)¶ This function splits the number
x
into its normalized fraction and exponent , such that and . The function returns and stores the exponent in . If is zero, both and are set to zero. This function provides an alternative to the standard math functionfrexp(x, e)
.
Small integer powers¶
A common complaint about the standard C library is its lack of a function for calculating (small) integer powers. CML provides some simple functions to fill this gap. For reasons of efficiency, these functions do not check for overflow or underflow conditions.

double
cml_pow_int
(double x, int n)¶ 
double
cml_pow_uint
(double x, unsigned int n)¶ These routines computes the power for integer
n
. The power is computed efficiently—for example, is computed as , requiring only 3 multiplications.

double
cml_pow_2
(double x)¶ 
double
cml_pow_3
(double x)¶ 
double
cml_pow_4
(double x)¶ 
double
cml_pow_5
(double x)¶ 
double
cml_pow_6
(double x)¶ 
double
cml_pow_7
(double x)¶ 
double
cml_pow_8
(double x)¶ 
double
cml_pow_9
(double x)¶ These functions can be used to compute small integer powers , , etc. efficiently. The functions will be inlined when
HAVE_INLINE
is defined, so that use of these functions should be as efficient as explicitly writing the corresponding product expression:#include <cml.h> [...] double y = pow_4(3.141); /* compute 3.141**4 */
Testing the Sign of Numbers¶

double
cml_sgn
(double x)¶ This macro returns the sign of
x
. It is defined as((x) >= 0 ? 1 : 1)
. Note that with this definition the sign of zero is positive (regardless of its IEEE sign bit).
Maximum and Minimum functions¶
Note that the following macros perform multiple evaluations of their arguments, so they should not be used with arguments that have side effects (such as a call to a random number generator).

CML_MAX
(a, b)¶ This macro returns the maximum of
a
andb
. It is defined as((a) > (b) ? (a):(b))
.

CML_MIN
(a, b)¶ This macro returns the minimum of
a
andb
. It is defined as((a) < (b) ? (a):(b))
.
Approximate Comparison of Floating Point Numbers¶
It is sometimes useful to be able to compare two floating point numbers approximately, to allow for rounding and truncation errors. The following function implements the approximate floatingpoint comparison algorithm proposed by D.E. Knuth in Section 4.2.2 of “Seminumerical Algorithms” (3rd edition).

bool
cml_cmp
(double x, double y, double epsilon)¶ This function determines whether
x
andy
are approximately equal to a relative accuracyepsilon
.The relative accuracy is measured using an interval of size , where and is the maximum base2 exponent of and as computed by the function
frexp()
.If and lie within this interval, they are considered approximately equal and the function returns 0. Otherwise if , the function returns , or if , the function returns .
Note that and are compared to relative accuracy, so this function is not suitable for testing whether a value is approximately zero.
The implementation is based on the package
fcmp
by T.C. Belding.
Footnotes
[1]  Note that the C99 standard only requires the
system isinf() function to return a nonzero value, without the
sign of the infinity. The implementation in some earlier versions of
CML used the system isinf() function and may have this behavior
on some platforms. Therefore, it is advisable to test the sign of
x separately, if needed, rather than relying the sign of the
return value from isinf() . 