The Cygnus C Math Library

                                                                    libm 1.4

December 1995

Steve Chamberlain Roland Pesch Cygnus Support

Cygnus Support sac@cygnus.com pesch@cygnus.com

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1992, 1993 Cygnus Support ‘libm’ includes software developed at SunPro, a Sun Microsystems, Inc. business. Permission to use, copy, modify, and distribute this software is freely granted, provided that this notice is preserved. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, subject to the terms of the GNU General Public License, which includes the provision that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions.

1 Mathematical Functions (‘math.h’)

This chapter groups a wide variety of mathematical functions. The corresponding definitions and declarations are in ‘math.h’. Two definitions from ‘math.h’ are of particular interest.

1. The representation of infinity as a double is defined as HUGE_VAL; this number is returned on overflow by many functions.

2. The structure exception is used when you write customized error handlers for the mathematical functions. You can customize error handling for most of these functions by defining your own version of matherr; see the section on matherr for details.

Since the error handling code calls fputs, the mathematical subroutines require stubs or minimal implementations for the same list of OS subroutines as fputs: close, fstat, isatty, lseek, read, sbrk, write. See section “System Calls” in The Cygnus C Support Library, for a discussion and for sample minimal implementations of these support subroutines. Alternative declarations of the mathematical functions, which exploit specific machine capabilities to operate faster—but generally have less error checking and may reflect additional limitations on some machines—are available when you include ‘fastmath.h’ instead of

‘math.h’.

1.1 Version of library

There are four different versions of the math library routines: IEEE, POSIX, X/Open, or SVID. The version may be selected at runtime by setting the global variable _LIB_VERSION, defined in ‘math.h’. It may be set to one of the following constants defined in ‘math.h’:

_IEEE_, _POSIX_, _XOPEN_, or _SVID_. The _LIB_VERSION variable is not specific to any

thread, and changing it will affect all threads. The versions of the library differ only in how errors are handled. In IEEE mode, the matherr function is never called, no warning messages are printed, and errno is never set. In POSIX mode, errno is set correctly, but the matherr function is never called and no warning messages are printed. In X/Open mode, errno is set correctly, and matherr is called, but warning message are not printed. In SVID mode, functions which overflow return 3.40282346638528860e+38, the maximum single precision floating point value, rather than infinity. Also, errno is set correctly, matherr is called, and, if matherr returns 0, warning messages are printed for some errors. For example, by default ‘log(-1.0)’ writes this message on standard error output:

     log: DOMAIN error

The library is set to X/Open mode by default.

1.2 acos, acosf—arc cosine

Synopsis

     #include <math.h>
     double acos(double x);
     float acosf(float x);

Description

acos computes the inverse cosine (arc cosine) of the input value. Arguments to acos must be in the range −1 to 1. acosf is identical to acos, except that it performs its calculations on floats.

Returns acos and acosf return values in radians, in the range of 0 to π. If x is not between −1 and 1, the returned value is NaN (not a number) the global variable errno is set to EDOM, and a DOMAIN error message is sent as standard error output. You can modify error handling for these functions using matherr.

1.3 acosh, acoshf—inverse hyperbolic cosine

Synopsis

     #include <math.h>
     double acosh(double x);
     float acoshf(float x);

Description acosh calculates the inverse hyperbolic cosine of x. acosh is defined as

ln(x + \sqrt{x^{2} − 1})

x must be a number greater than or equal to 1. acoshf is identical, other than taking and returning floats.

Returns acosh and acoshf return the calculated value. If x less than 1, the return value is NaN and errno is set to EDOM. You can change the error-handling behavior with the non-ANSI matherr function.

Portability Neither acosh nor acoshf are ANSI C. They are not recommended for portable programs.

1.4 asin, asinf—arc sine

Synopsis

     #include <math.h>
     double asin(double x);
     float asinf(float x);

Description

asin computes the inverse sine (arc sine) of the argument x. Arguments to asin must be in the range −1 to 1. asinf is identical to asin, other than taking and returning floats. You can modify error handling for these routines using matherr.

Returns asin returns values in radians, in the range of −π/2 to π/2. If x is not in the range −1 to 1, asin and asinf return NaN (not a number), set the global variable errno to EDOM, and issue a DOMAIN error message. You can change this error treatment using matherr.

1.5 asinh, asinhf—inverse hyperbolic sine

Synopsis

     #include <math.h>
     double asinh(double x);
     float asinhf(float x);

Description asinh calculates the inverse hyperbolic sine of x. asinh is defined as

sign(x) × ln(|x| + \sqrt{1 + x^{2}})

asinhf is identical, other than taking and returning floats.

Returns asinh and asinhf return the calculated value.

Portability Neither asinh nor asinhf are ANSI C.

1.6 atan, atanf—arc tangent

Synopsis

     #include <math.h>
     double atan(double x);
     float atanf(float x);

Description

atan computes the inverse tangent (arc tangent) of the input value. atanf is identical to atan, save that it operates on floats.

Returns atan returns a value in radians, in the range of −π/2 to π/2.

Portability atan is ANSI C. atanf is an extension.

1.7 atan2, atan2f—arc tangent of y/x

Synopsis

     #include <math.h>
     double atan2(double y,double x);
     float atan2f(float y,float x);

Description

atan2 computes the inverse tangent (arc tangent) of y/x. atan2 produces the correct result even for angles near π/2 or −π/2 (that is, when x is near 0). atan2f is identical to atan2, save that it takes and returns float.

Returns atan2 and atan2f return a value in radians, in the range of −π to π. If both x and y are 0.0, atan2 causes a DOMAIN error. You can modify error handling for these functions using matherr.

Portability atan2 is ANSI C. atan2f is an extension.

1.8 atanh, atanhf—inverse hyperbolic tangent

Synopsis

     #include <math.h>
     double atanh(double x);
     float atanhf(float x);

Description atanh calculates the inverse hyperbolic tangent of x. atanhf is identical, other than taking and returning float values.

Returns atanh and atanhf return the calculated value. If |x| is greater than 1, the global errno is set to EDOM and the result is a NaN. A DOMAIN error is reported. If |x| is 1, the global errno is set to EDOM; and the result is infinity with the same sign as x. A SING error is reported. You can modify the error handling for these routines using matherr.

Portability Neither atanh nor atanhf are ANSI C.

1.9 jN,jNf,yN,yNf—Bessel functions

Synopsis

     #include <math.h>
     double j0(double x);
     float j0f(float x);
     double j1(double x);
     float j1f(float x);
     double jn(int n, double x);
     float jnf(int n, float x);
     double y0(double x);
     float y0f(float x);
     double y1(double x);
     float y1f(float x);
     double yn(int n, double x);
     float ynf(int n, float x);

Description The Bessel functions are a family of functions that solve the differential equation

x^{2} \frac{d^{2}y}{dx^{2}} + x\frac{dy}{dx} + (x^{2} − p^{2})y = 0

These functions have many applications in engineering and physics. jn calculates the Bessel function of the first kind of order n. j0 and j1 are special cases for order 0 and order 1 respectively. Similarly, yn calculates the Bessel function of the second kind of order n, and y0 and y1 are special cases for order 0 and 1. jnf, j0f, j1f, ynf, y0f, and y1f perform the same calculations, but on float rather than double values.

Returns The value of each Bessel function at x is returned.

Portability None of the Bessel functions are in ANSI C.

1.10 cosh, coshf—hyperbolic cosine

Synopsis

     #include <math.h>
     double cosh(double x);
     float coshf(float x)

Description

cosh computes the hyperbolic cosine of the argument x. cosh(x) is defined as

\frac{(e^{x} + e^{−x})}{2}

Angles are specified in radians. coshf is identical, save that it takes and returns float.

Returns The computed value is returned. When the correct value would create an overflow, cosh returns the value HUGE_VAL with the appropriate sign, and the global value errno is set to

ERANGE.

You can modify error handling for these functions using the function matherr.

Portability cosh is ANSI. coshf is an extension.

1.11 erf, erff, erfc, erfcf—error function

Synopsis

     #include <math.h>
     double erf(double x);
     float erff(float x);
     double erfc(double x);
     float erfcf(float x);

Description erf calculates an approximation to the “error function”, which estimates the probability that an observation will fall within x standard deviations of the mean (assuming a normal distribution). The error function is defined as

\frac{2}{\sqrt{π}} × ∫_{0}^{x} e^{−t^{2}}dt

erfc calculates the complementary probability; that is, erfc(x) is 1 - erf(x). erfc is computed directly, so that you can use it to avoid the loss of precision that would result from subtracting large probabilities (on large x) from 1. erff and erfcf differ from erf and erfc only in the argument and result types.

Returns For positive arguments, erf and all its variants return a probability—a number between 0 and 1.

Portability None of the variants of erf are ANSI C.

1.12 exp, expf—exponential

Synopsis

     #include <math.h>
     double exp(double x);
     float expf(float x);

Description exp and expf calculate the exponential of x, that is, e^{x} (where e is the base of the natural system of logarithms, approximately 2.71828). You can use the (non-ANSI) function matherr to specify error handling for these functions.

Returns On success, exp and expf return the calculated value. If the result underflows, the returned value is 0. If the result overflows, the returned value is HUGE_VAL. In either case, errno is

set to ERANGE.

Portability exp is ANSI C. expf is an extension.

1.13 fabs, fabsf—absolute value (magnitude)

Synopsis

     #include <math.h>
     double fabs(double x);
     float fabsf(float x);

Description fabs and fabsf calculate |x|, the absolute value (magnitude) of the argument x, by direct manipulation of the bit representation of x.

Returns The calculated value is returned. No errors are detected.

Portability fabs is ANSI. fabsf is an extension.

1.14 floor, floorf, ceil, ceilf—floor and ceiling

Synopsis

     #include <math.h>
     double floor(double x);
     float floorf(float x);
     double ceil(double x);
     float ceilf(float x);

Description floor and floorf find ⌊x⌋, the nearest integer less than or equal to x. ceil and ceilf find ⌈x⌉, the nearest integer greater than or equal to x.

Returns floor and ceil return the integer result as a double. floorf and ceilf return the integer result as a float.

Portability floor and ceil are ANSI. floorf and ceilf are extensions.

1.15 fmod, fmodf—floating-point remainder (modulo)

Synopsis

     #include <math.h>
     double fmod(double x, double y)
     float fmodf(float x, float y)

Description The fmod and fmodf functions compute the floating-point remainder of x/y (x modulo y).

Returns The fmod function returns the value x − i × y, for the largest integer i such that, if y is nonzero, the result has the same sign as x and magnitude less than the magnitude of y. fmod(x,0) returns NaN, and sets errno to EDOM. You can modify error treatment for these functions using matherr.

Portability fmod is ANSI C. fmodf is an extension.

1.16 frexp, frexpf—split floating-point number

Synopsis

     #include <math.h>
     double frexp(double val, int *exp);
     float frexpf(float val, int *exp);

Description All non zero, normal numbers can be described as m * 2**p. frexp represents the double val as a mantissa m and a power of two p. The resulting mantissa will always be greater than or equal to 0.5, and less than 1.0 (as long as val is nonzero). The power of two will be stored in *exp. m and p are calculated so that val = m × 2^{p}. frexpf is identical, other than taking and returning floats rather than doubles.

Returns frexp returns the mantissa m. If val is 0, infinity, or Nan, frexp will set *exp to 0 and return val.

Portability frexp is ANSI. frexpf is an extension.

1.17 gamma, gammaf, lgamma, lgammaf, gamma_r,

Synopsis

     #include <math.h>
     double gamma(double x);
     float gammaf(float x);
     double lgamma(double x);
     float lgammaf(float x);
     double gamma_r(double x, int *signgamp);
     float gammaf_r(float x, int *signgamp);
     double lgamma_r(double x, int *signgamp);
     float lgammaf_r(float x, int *signgamp);

Description gamma calculates ln(Γ (x)), the natural logarithm of the gamma function of x. The gamma function (exp(gamma(x))) is a generalization of factorial, and retains the property that Γ (N) ≡ N × Γ (N − 1). Accordingly, the results of the gamma function itself grow very quickly. gamma is defined as ln(Γ (x)) rather than simply Γ (x) to extend the useful range of results representable. The sign of the result is returned in the global variable signgam, which is declared in math.h. gammaf performs the same calculation as gamma, but uses and returns float values. lgamma and lgammaf are alternate names for gamma and gammaf. The use of lgamma instead of gamma is a reminder that these functions compute the log of the gamma function, rather than the gamma function itself.

The functions gamma_r, gammaf_r, lgamma_r, and lgammaf_r are just like gamma, gammaf,

lgamma, and lgammaf, respectively, but take an additional argument. This additional argument is a pointer to an integer. This additional argument is used to return the sign of the result, and the global variable signgam is not used. These functions may be used for reentrant calls (but they will still set the global variable errno if an error occurs).

Returns Normally, the computed result is returned. When x is a nonpositive integer, gamma returns HUGE_VAL and errno is set to EDOM. If the result overflows, gamma returns HUGE_VAL and errno is set to ERANGE. You can modify this error treatment using matherr.

Portability Neither gamma nor gammaf is ANSI C.

1.18 hypot, hypotf—distance from origin

Synopsis

     #include <math.h>
     double hypot(double x, double y);
     float hypotf(float x, float y);

Description hypot calculates the Euclidean distance \sqrt{x^{2} + y^{2}} between the origin (0,0) and a point represented by the Cartesian coordinates (x,y). hypotf differs only in the type of its arguments and result.

Returns Normally, the distance value is returned. On overflow, hypot returns HUGE_VAL and sets

errno to ERANGE.

You can change the error treatment with matherr.

Portability hypot and hypotf are not ANSI C.

1.19 isnan,isnanf,isinf,isinff,finite,finitef—test for

exceptional numbers

Synopsis

     #include <ieeefp.h>
     int isnan(double arg);
     int isinf(double arg);
     int finite(double arg);
     int isnanf(float arg);
     int isinff(float arg);
     int finitef(float arg);

Description These functions provide information on the floating point argument supplied. There are five major number formats -

zero a number which contains all zero bits.

subnormal

Is used to represent number with a zero exponent, but a non zero fraction.

normal A number with an exponent, and a fraction

infinity A number with an all 1’s exponent and a zero fraction.

NAN A number with an all 1’s exponent and a non zero fraction.

isnan returns 1 if the argument is a nan. isinf returns 1 if the argument is infinity. finite returns 1 if the argument is zero, subnormal or normal. The isnanf, isinff and finitef perform the same operations as their isnan, isinf and finite counterparts, but on single precision floating point numbers.

1.20 ldexp, ldexpf—load exponent

Synopsis

     #include <math.h>
     double ldexp(double val, int exp);
     float ldexpf(float val, int exp);

Description ldexp calculates the value val × 2^{exp}. ldexpf is identical, save that it takes and returns float rather than double values.

Returns ldexp returns the calculated value. Underflow and overflow both set errno to ERANGE. On underflow, ldexp and ldexpf return 0.0. On overflow, ldexp returns plus or minus HUGE_VAL.

Portability ldexp is ANSI, ldexpf is an extension.

1.21 log, logf—natural logarithms

Synopsis

     #include <math.h>
     double log(double x);
     float logf(float x);

Description Return the natural logarithm of x, that is, its logarithm base e (where e is the base of the natural system of logarithms, 2.71828…). log and logf are identical save for the return and argument types. You can use the (non-ANSI) function matherr to specify error handling for these functions.

Returns Normally, returns the calculated value. When x is zero, the returned value is -HUGE_VAL and errno is set to ERANGE. When x is negative, the returned value is -HUGE_VAL and errno is set to EDOM. You can control the error behavior via matherr.

Portability log is ANSI, logf is an extension.

1.22 log10, log10f—base 10 logarithms

Synopsis

     #include <math.h>
     double log10(double x);
     float log10f(float x);

Description log10 returns the base 10 logarithm of x. It is implemented as log(x) / log(10). log10f is identical, save that it takes and returns float values.

Returns log10 and log10f return the calculated value. See the description of log for information on errors.

Portability log10 is ANSI C. log10f is an extension.

1.23 pow, powf—x to the power y

Synopsis

     #include <math.h>
     double pow(double x, double y);
     float pow(float x, float y);

Description pow and powf calculate x raised to the exp1.0nt y. (That is, x^{y}.)

Returns On success, pow and powf return the value calculated. When the argument values would produce overflow, pow returns HUGE_VAL and set errno to ERANGE. If the argument x passed to pow or powf is a negative noninteger, and y is also not an integer, then errno is set to EDOM. If x and y are both 0, then pow and powf return

1.

You can modify error handling for these functions using matherr.

Portability pow is ANSI C. powf is an extension.

1.24 remainder, remainderf—round and remainder

Synopsis

     #include <math.h>
     double remainder(double x, double y);
     float remainderf(float x, float y);

Description remainder and remainderf find the remainder of x/y; this value is in the range -y/2 .. +y/2.

Returns remainder returns the integer result as a double.

Portability remainder is a System V release 4. remainderf is an extension.

1.25 sqrt, sqrtf—positive square root

Synopsis

     #include <math.h>
     double sqrt(double x);
     float  sqrtf(float x);

Description sqrt computes the positive square root of the argument. You can modify error handling for this function with matherr.

Returns On success, the square root is returned. If x is real and positive, then the result is positive. If x is real and negative, the global value errno is set to EDOM (domain error).

Portability sqrt is ANSI C. sqrtf is an extension.

1.26 sin, sinf, cos, cosf—sine or cosine

Synopsis

     #include <math.h>
     double sin(double x);
     float  sinf(float x);
     double cos(double x);
     float cosf(float x);

Description sin and cos compute (respectively) the sine and cosine of the argument x. Angles are specified in radians. sinf and cosf are identical, save that they take and return float values.

Returns The sine or cosine of x is returned.

Portability sin and cos are ANSI C. sinf and cosf are extensions.

1.27 sinh, sinhf—hyperbolic sine

Synopsis

     #include <math.h>
     double sinh(double x);
     float  sinhf(float x);

Description sinh computes the hyperbolic sine of the argument x. Angles are specified in radians. sinh(x) is defined as

\frac{e^{x} − e^{−x}}{2}

sinhf is identical, save that it takes and returns float values.

Returns The hyperbolic sine of x is returned. When the correct result is too large to be representable (an overflow), sinh returns HUGE_ VAL with the appropriate sign, and sets the global value errno to ERANGE. You can modify error handling for these functions with matherr.

Portability sinh is ANSI C. sinhf is an extension.

1.28 tan, tanf—tangent

Synopsis

     #include <math.h>
     double tan(double x);
     float tanf(float x);

Description tan computes the tangent of the argument x. Angles are specified in radians. tanf is identical, save that it takes and returns float values.

Returns The tangent of x is returned.

Portability tan is ANSI. tanf is an extension.

1.29 tanh, tanhf—hyperbolic tangent

Synopsis

     #include <math.h>
     double tanh(double x);
     float tanhf(float x);

Description

tanh computes the hyperbolic tangent of the argument x. Angles are specified in radians. tanh(x) is defined as

      sinh(x)/cosh(x)

tanhf is identical, save that it takes and returns float values.

Returns The hyperbolic tangent of x is returned.

Portability tanh is ANSI C. tanhf is an extension.

1.30 cbrt, cbrtf—cube root

Synopsis

     #include <math.h>
     double cbrt(double x);
     float  cbrtf(float x);

Description cbrt computes the cube root of the argument.

Returns The cube root is returned.

Portability cbrt is in System V release 4. cbrtf is an extension.

1.31 copysign, copysignf—sign of y, magnitude of x

Synopsis

     #include <math.h>
     double copysign (double x, double y);
     float copysignf (float x, float y);

Description copysign constructs a number with the magnitude (absolute value) of its first argument, x, and the sign of its second argument, y. copysignf does the same thing; the two functions differ only in the type of their arguments and result.

Returns copysign returns a double with the magnitude of x and the sign of y. copysignf returns a float with the magnitude of x and the sign of y.

Portability copysign is not required by either ANSI C or the System V Interface Definition (Issue 2).

1.32 expm1, expm1f—exponential minus 1

Synopsis

     #include <math.h>
     double expm1(double x);
     float expm1f(float x);

Description expm1 and expm1f calculate the exponential of x and subtract 1, that is, e^{x} − 1 (where e is the base of the natural system of logarithms, approximately 2.71828). The result is accurate even for small values of x, where using exp(x)-1 would lose many significant digits.

Returns e raised to the power x, minus 1.

Portability Neither expm1 nor expm1f is required by ANSI C or by the System V Interface Definition (Issue 2).

1.33 ilogb, ilogbf—get exponent of floating point number

Synopsis

     #include <math.h>
     int ilogb(double val);
     int ilogbf(float val);

Description

All non zero, normal numbers can be described as m * 2**p. ilogb and ilogbf examine the argument val, and return p. The functions frexp and frexpf are similar to ilogb and ilogbf, but also return m.

Returns

ilogb and ilogbf return the power of two used to form the floating point argument. If val is 0, they return - INT_MAX (INT_MAX is defined in limits.h). If val is infinite, or NaN, they

return INT_MAX.

Portability Neither ilogb nor ilogbf is required by ANSI C or by the System V Interface Definition (Issue 2).

1.34 infinity, infinityf—representation of infinity

Synopsis

     #include <math.h>
     double infinity(void);
     float infinityf(void);

Description infinity and infinityf return the special number IEEE infinity in double and single precision arithmetic respectivly.

1.35 log1p, log1pf—log of 1 + x

Synopsis

     #include <math.h>
     double log1p(double x);
     float log1pf(float x);

Description log1p calculates ln(1 + x), the natural logarithm of 1+x. You can use log1p rather than ‘log(1+x)’ for greater precision when x is very small. log1pf calculates the same thing, but accepts and returns float values rather than double.

Returns log1p returns a double, the natural log of 1+x. log1pf returns a float, the natural log of

1+x.

Portability Neither log1p nor log1pf is required by ANSI C or by the System V Interface Definition (Issue 2).

1.36 matherr—modifiable math error handler

Synopsis

     #include <math.h>
     int matherr(struct exception *e);

Description matherr is called whenever a math library function generates an error. You can replace matherr by your own subroutine to customize error treatment. The customized matherr must return 0 if it fails to resolve the error, and non-zero if the error is resolved. When matherr returns a nonzero value, no error message is printed and the value of errno is not modified. You can accomplish either or both of these things in your own matherr using the information passed in the structure *e. This is the exception structure (defined in ‘math.h’):

     struct exception {
             int type;

             char *name;
             double arg1, arg2, retval;
     int err;
     };

The members of the exception structure have the following meanings:

type The type of mathematical error that occured; macros encoding error types are also defined in ‘math.h’.

name a pointer to a null-terminated string holding the name of the math library function where the error occurred.

arg1, arg2

The arguments which caused the error.

retval The error return value (what the calling function will return).

err If set to be non-zero, this is the new value assigned to errno.

The error types defined in ‘math.h’ represent possible mathematical errors as follows:

DOMAIN An argument was not in the domain of the function; e.g. log(-1.0).

SING The requested calculation would result in a singularity; e.g. pow(0.0,-2.0)

OVERFLOW A calculation would produce a result too large to represent; e.g. exp(1000.0).

UNDERFLOW

A calculation would produce a result too small to represent; e.g. exp(-1000.0).

TLOSS Total loss of precision. The result would have no significant digits; e.g.

          sin(10e70).

PLOSS Partial loss of precision.

Returns The library definition for matherr returns 0 in all cases.

You can change the calling function’s result from a customized matherr by modifying e- >retval, which propagates backs to the caller.

If matherr returns 0 (indicating that it was not able to resolve the error) the caller sets errno to an appropriate value, and prints an error message.

Portability matherr is not ANSI C.

1.37 modf, modff—split fractional and integer parts

Synopsis

     #include <math.h>
     double modf(double val, double *ipart);
     float modff(float val, float *ipart);

Description modf splits the double val apart into an integer part and a fractional part, returning the fractional part and storing the integer part in *ipart. No rounding whatsoever is done; the sum of the integer and fractional parts is guaranteed to be exactly equal to val. That is, if . realpart = modf(val, &intpart); then ‘realpart+intpart’ is the same as val. modff is identical, save that it takes and returns float rather than double values.

Returns The fractional part is returned. Each result has the same sign as the supplied argument val.

Portability modf is ANSI C. modff is an extension.

1.38 nan, nanf—representation of infinity

Synopsis

     #include <math.h>
     double nan(void);
     float nanf(void);

Description nan and nanf return an IEEE NaN (Not a Number) in double and single precision arithmetic respectivly.

1.39 nextafter, nextafterf—get next number

Synopsis

     #include <math.h>
     double nextafter(double val, double dir);
     float nextafterf(float val, float dir);

Description nextafter returns the double) precision floating point number closest to val in the direction toward dir. nextafterf performs the same operation in single precision. For example, nextafter(0.0,1.0) returns the smallest positive number which is representable in double precision.

Returns Returns the next closest number to val in the direction toward dir.

Portability Neither nextafter nor nextafterf is required by ANSI C or by the System V Interface Definition (Issue 2).

1.40 scalbn, scalbnf—scale by integer

Synopsis

     #include <math.h>
     double scalbn(double x, int y);
     float scalbnf(float x, int y);

Description scalbn and scalbnf scale x by n, returning x times 2 to the power n. The result is computed by manipulating the exponent, rather than by actually performing an exponentiation or multiplication.

Returns x times 2 to the power n.

Portability Neither scalbn nor scalbnf is required by ANSI C or by the System V Interface Definition (Issue 2).

2 Reentrancy Properties of libm

When a libm function detects an exceptional case, errno may be set, the matherr function may be called, and a error message may be written to the standard error stream. This behavior may not be reentrant. With reentrant C libraries like the Cygnus C library, errno is a macro which expands to the per-thread error value. This makes it thread safe. When the user provides his own matherr function it must be reentrant for the math library as a whole to be reentrant. In normal debugged programs, there are usually no math subroutine errors—and therefore no assignments to errno and no matherr calls; in that situation, the math functions behave reentrantly.

38 Cygnus C Math Library Index 39

Index

A

acos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 acosf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 acosh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 acoshf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 asin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 asinf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 asinh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 asinhf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 atan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 atan2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 atan2f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 atanf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 atanh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 atanhf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

C

cbrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 cbrtf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ceil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 ceilf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 copysign. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 copysignf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 cos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 cosf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

E

erf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 erfc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 erfcf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 erff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 expf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 expm1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 expm1f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

F

fabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 fabsf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 finite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 finitef. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 floorf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 fmod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 fmodf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 frexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 frexpf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

G

gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gamma_r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gammaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 gammaf_r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

H

hypot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 hypotf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

I

ilogb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ilogbf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 infinityf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 isinf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 isinff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 isnan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 isnanf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

J

j0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 j0f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 j1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 j1f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 jn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 jnf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

L

ldexp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 ldexpf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 lgamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 lgamma_r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 lgammaf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 lgammaf_r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 log10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 log10f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 log1p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 log1pf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

40 Cygnus C Math Library

logf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

M

matherr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 matherr and reentrancy . . . . . . . . . . . . . . . . . . . . . . 37 modf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 modff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

N

nan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 nanf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 nextafter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 nextafterf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

O

OS stubs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

P

pow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 powf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

R

reentrancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 remainderf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

S

scalbn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 scalbnf. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 sin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 sinf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 sinh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 sinhf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 sqrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 sqrtf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 stubs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 support subroutines . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 system calls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

T

tan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 tanf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 tanh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 tanhf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Y

y0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 y0f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 y1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 y1f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 yn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 ynf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

The body of this manual is set in

cmr10 at 10.95pt, with headings in cmb10 at 10.95pt

and examples in cmtt10 at 10.95pt.

cmti10 at 10.95pt and

cmsl10 at 10.95pt are used for emphasis.

42 Cygnus C Math Library

Table of Contents

1 Mathematical Functions (‘math.h’) . . . . . . . . . . 1

1.1 Version of library. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 acos, acosf—arc cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 acosh, acoshf—inverse hyperbolic cosine . . . . . . . . . . . . . . . . . 4 1.4 asin, asinf—arc sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 asinh, asinhf—inverse hyperbolic sine . . . . . . . . . . . . . . . . . . . 6 1.6 atan, atanf—arc tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.7 atan2, atan2f—arc tangent of y/x . . . . . . . . . . . . . . . . . . . . . . . 8 1.8 atanh, atanhf—inverse hyperbolic tangent . . . . . . . . . . . . . . . . 9 1.9 jN,jNf,yN,yNf—Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.10 cosh, coshf—hyperbolic cosine . . . . . . . . . . . . . . . . . . . . . . . . 11 1.11 erf, erff, erfc, erfcf—error function . . . . . . . . . . . . . . . . . 12 1.12 exp, expf—exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.13 fabs, fabsf—absolute value (magnitude) . . . . . . . . . . . . . . . 14 1.14 floor, floorf, ceil, ceilf—floor and ceiling. . . . . . . . . . . 15 1.15 fmod, fmodf—floating-point remainder (modulo). . . . . . . . . 16 1.16 frexp, frexpf—split floating-point number . . . . . . . . . . . . . 17

1.17 gamma, gammaf, lgamma, lgammaf, gamma_r, . . . . . . . . . . . . . 18

1.18 hypot, hypotf—distance from origin . . . . . . . . . . . . . . . . . . . 19

1.19 isnan,isnanf,isinf,isinff,finite,finitef—test for

exceptional numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.20 ldexp, ldexpf—load exponent . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.21 log, logf—natural logarithms . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.22 log10, log10f—base 10 logarithms . . . . . . . . . . . . . . . . . . . . . 23 1.23 pow, powf—x to the power y . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.24 remainder, remainderf—round and remainder. . . . . . . . . . 25 1.25 sqrt, sqrtf—positive square root . . . . . . . . . . . . . . . . . . . . . . 26 1.26 sin, sinf, cos, cosf—sine or cosine . . . . . . . . . . . . . . . . . . . . 27 1.27 sinh, sinhf—hyperbolic sine . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.28 tan, tanf—tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.29 tanh, tanhf—hyperbolic tangent . . . . . . . . . . . . . . . . . . . . . . . 30 1.30 cbrt, cbrtf—cube root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.31 copysign, copysignf—sign of y, magnitude of x . . . . . . . . 31 1.32 expm1, expm1f—exponential minus 1 . . . . . . . . . . . . . . . . . . . 31 1.33 ilogb, ilogbf—get exponent of floating point number … 32 1.34 infinity, infinityf—representation of infinity. . . . . . . . . 32 1.35 log1p, log1pf—log of 1 + x . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.36 matherr—modifiable math error handler . . . . . . . . . . . . . . . . 33 1.37 modf, modff—split fractional and integer parts . . . . . . . . . . 35 1.38 nan, nanf—representation of infinity. . . . . . . . . . . . . . . . . . . . 35 1.39 nextafter, nextafterf—get next number . . . . . . . . . . . . . . 35 1.40 scalbn, scalbnf—scale by integer . . . . . . . . . . . . . . . . . . . . . 36 ii Cygnus C Math Library

2 Reentrancy Properties of libm . . . . . . . . . . . . . 37

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39