The C Standard, 7.12.1 [ISO/IEC 9899:2011], defines three types of errors that relate specifically to math functions in <math.h>
. Paragraph 2 states
A domain error occurs if an input argument is outside the domain over which the mathematical function is defined.
Paragraph 3 states
A pole error (also known as a singularity or infinitary) occurs if the mathematical function has an exact infinite result as the finite input argument(s) are approached in the limit.
Paragraph 4 states
A range error occurs if the mathematical result of the function cannot be represented in an object of the specified type, due to extreme magnitude.
An example of a domain error is the square root of a negative number, such as sqrt(1.0)
, which has no meaning in real arithmetic. Contrastingly, 10 raised to the 1millionth power, pow(10., 1e6)
, cannot be represented in many floatingpoint implementations because of the limited range of the type double
and consequently constitutes a range error. In both cases, the function will return some value, but the value returned is not the correct result of the computation. An example of a pole error is log(0.0)
, which results in negative infinity.
Programmers can prevent domain and pole errors by carefully boundschecking the arguments before calling mathematical functions and taking alternative action if the bounds are violated.
Range errors usually cannot be prevented because they are dependent on the implementation of floatingpoint numbers as well as on the function being applied. Instead of preventing range errors, programmers should attempt to detect them and take alternative action if a range error occurs.
The following table lists the double
forms of standard mathematical functions, along with checks that should be performed to ensure a proper input domain, and indicates whether they can also result in range or pole errors, as reported by the C Standard. Both float
and long double
forms of these functions also exist but are omitted from the table for brevity. If a function has a specific domain over which it is defined, the programmer must check its input values. The programmer must also check for range errors where they might occur. The standard math functions not listed in this table, such as fabs()
, have no domain restrictions and cannot result in range or pole errors.
Function  Domain  Range  Pole 


 No  No 
asin(x)  1 <= x && x <= 1  Yes  No 
atan(x)  None  Yes  No 

 No  No 

 Yes  No 
asinh(x)  None  Yes  No 

 Yes  Yes 
 None  Yes  No 
 None  Yes  No 
 None  Yes  No 

 No  Yes 

 No  Yes 

 Yes  No 
logb(x)  x != 0  Yes  Yes 
 None  Yes  No 
 None  Yes  No 

 Yes  Yes 

 No  No 
erf(x)  None  Yes  No 
 None  Yes  No 

 Yes  Yes 
 None  Yes  No 

 Yes  No 
 None  Yes  No 
 None  Yes  No 
 None  Yes  No 
The most reliable way to handle domain and pole errors is to prevent them by checking arguments beforehand, as in the following exemplar:
double safe_sqrt(double x) { if (x < 0) { fprintf(stderr, "sqrt requires a nonnegative argument"); /* Handle domain / pole error */ } return sqrt (x); } 
Programmers usually cannot prevent range errors, so the most reliable way to handle them is to detect when they have occurred and act accordingly.
The exact treatment of error conditions from math functions is tedious. The C Standard, 7.12.1 [ISO/IEC 9899:2011], defines the following behavior for floatingpoint overflow:
A floating result overflows if the magnitude of the mathematical result is finite but so large that the mathematical result cannot be represented without extraordinary roundoff error in an object of the specified type. If a floating result overflows and default rounding is in effect, then the function returns the value of the macro
HUGE_VAL
,HUGE_VALF
, orHUGE_VALL
according to the return type, with the same sign as the correct value of the function; if the integer expressionmath_errhandling & MATH_ERRNO
is nonzero, the integer expressionerrno
acquires the valueERANGE
; if the integer expressionmath_errhandling & MATH_ERREXCEPT
is nonzero, the "overflow" floatingpoint exception is raised.
It is preferable not to check for errors by comparing the returned value against HUGE_VAL
or 0
for several reasons:
HUGE_VAL
and 0
are possible, and programmers must know which are possible in each case.It can be unreliable to check for math errors using errno
because an implementation might not set errno
. For real functions, the programmer determines if the implementation sets errno
by checking whether math_errhandling & MATH_ERRNO
is nonzero. For complex functions, the C Standard, 7.3.2, paragraph 1, simply states that "an implementation may set errno
but is not required to" [ISO/IEC 9899:2011].
The obsolete System V Interface Definition (SVID3) [UNIX 1992] provides more control over the treatment of errors in the math library. The programmer can define a function named matherr()
that is invoked if errors occur in a math function. This function can print diagnostics, terminate the execution, or specify the desired return value. The matherr()
function has not been adopted by C or POSIX, so it is not generally portable.
The following errorhanding template uses C Standard functions for floatingpoint errors when the C macro math_errhandling
is defined and indicates that they should be used; otherwise, it examines errno
:
#include <math.h> #include <fenv.h> #include <errno.h> /* ... */ /* Use to call a math function and check errors */ { #pragma STDC FENV_ACCESS ON if (math_errhandling & MATH_ERREXCEPT) { feclearexcept(FE_ALL_EXCEPT); } errno = 0; /* Call the math function */ if ((math_errhandling & MATH_ERRNO) && errno != 0) { /* Handle range error */ } else if ((math_errhandling & MATH_ERREXCEPT) && fetestexcept(FE_INVALID  FE_DIVBYZERO  FE_OVERFLOW  FE_UNDERFLOW) != 0) { /* Handle range error */ } } 
See FLP03C. Detect and handle floatingpoint errors for more details on how to detect floatingpoint errors.
A subnormal number is a nonzero number that does not use all of its precision bits [IEEE 754 2006]. These numbers can be used to represent values that are closer to 0 than the smallest normal number (one that uses all of its precision bits). However, the asin()
, asinh()
, atan()
, atanh()
, and erf()
functions may produce range errors, specifically when passed a subnormal number. When evaluated with a subnormal number, these functions can produce an inexact, subnormal value, which is an underflow error. The C Standard, 7.12.1, paragraph 6 [ISO/IEC 9899:2011], defines the following behavior for floatingpoint underflow:
The result underflows if the magnitude of the mathematical result is so small that the mathematical result cannot be represented, without extraordinary roundoff error, in an object of the specified type. If the result underflows, the function returns an implementationdefined value whose magnitude is no greater than the smallest normalized positive number in the specified type; if the integer expression
math_errhandling & MATH_ERRNO
is nonzero, whethererrno
acquires the valueERANGE
is implementationdefined; if the integer expressionmath_errhandling & MATH_ERREXCEPT
is nonzero, whether the ‘‘underflow’’ floatingpoint exception is raised is implementationdefined.
Implementations that support floatingpoint arithmetic but do not support subnormal numbers, such as IBM S/360 hex floatingpoint or nonconforming IEEE754 implementations that skip subnormals (or support them by flushing them to zero), can return a range error when calling one of the following families of functions with the following arguments:
fmod((min+subnorm), min)
remainder((min+subnorm
), min)
remquo((min+subnorm
), min, quo)
where min
is the minimum value for the corresponding floating point type and subnorm
is a subnormal value.
If Annex F is supported and subnormal results are supported, the returned value is exact and a range error cannot occur. The C Standard, F.10.7.1 [ISO/IEC 9899:2011], specifies the following for the fmod()
, remainder()
, and remquo()
functions:
When subnormal results are supported, the returned value is exact and is independent of the current rounding direction mode.
Annex F, subclause F.10.7.2, paragraph 2, and subclause F.10.7.3, paragraph 2, of the C Standard identify when subnormal results are supported.
sqrt()
)This noncompliant code example determines the square root of x
:
#include <math.h> void func(double x) { double result; result = sqrt(x); } 
However, this code may produce a domain error if x
is negative.
sqrt()
)Because this function has domain errors but no range errors, bounds checking can be used to prevent domain errors:
#include <math.h> void func(double x) { double result; if (isless(x, 0.0)) { /* Handle domain error */ } result = sqrt(x); } 
sinh()
, Range Errors)This noncompliant code example determines the hyperbolic sine of x
:
#include <math.h> void func(double x) { double result; result = sinh(x); } 
This code may produce a range error if x
has a very large magnitude.
sinh()
, Range Errors)Because this function has no domain errors but may have range errors, the programmer must detect a range error and act accordingly:
#include <math.h> #include <fenv.h> #include <errno.h> void func(double x) { double result; { #pragma STDC FENV_ACCESS ON if (math_errhandling & MATH_ERREXCEPT) { feclearexcept(FE_ALL_EXCEPT); } errno = 0; result = sinh(x); if ((math_errhandling & MATH_ERRNO) && errno != 0) { /* Handle range error */ } else if ((math_errhandling & MATH_ERREXCEPT) && fetestexcept(FE_INVALID  FE_DIVBYZERO  FE_OVERFLOW  FE_UNDERFLOW) != 0) { /* Handle range error */ } } /* Use result... */ } 
pow()
)This noncompliant code example raises x
to the power of y
:
#include <math.h> void func(double x, double y) { double result; result = pow(x, y); } 
This code may produce a domain error if x
is negative and y
is not an integer value or if x
is 0 and y
is 0. A domain error or pole error may occur if x
is 0 and y
is negative, and a range error may occur if the result cannot be represented as a double
.
pow()
)Because the pow()
function can produce domain errors, pole errors, and range errors, the programmer must first check that x
and y
lie within the proper domain and do not generate a pole error and then detect whether a range error occurs and act accordingly:
#include <math.h> #include <fenv.h> #include <errno.h> void func(double x, double y) { double result; if (((x == 0.0f) && islessequal(y, 0.0))  isless(x, 0.0)) { /* Handle domain or pole error */ } { #pragma STDC FENV_ACCESS ON if (math_errhandling & MATH_ERREXCEPT) { feclearexcept(FE_ALL_EXCEPT); } errno = 0; result = pow(x, y); if ((math_errhandling & MATH_ERRNO) && errno != 0) { /* Handle range error */ } else if ((math_errhandling & MATH_ERREXCEPT) && fetestexcept(FE_INVALID  FE_DIVBYZERO  FE_OVERFLOW  FE_UNDERFLOW) != 0) { /* Handle range error */ } } /* Use result... */ } 
asin()
, Subnormal Number)This noncompliant code example determines the inverse sine of x
:
#include <math.h> void func(float x) { float result = asin(x); /* ... */ } 
asin()
, Subnormal Number)Because this function has no domain errors but may have range errors, the programmer must detect a range error and act accordingly:
#include <math.h> #include <fenv.h> #include <errno.h> void func(float x) { float result; { #pragma STDC FENV_ACCESS ON if (math_errhandling & MATH_ERREXCEPT) { feclearexcept(FE_ALL_EXCEPT); } errno = 0; result = asin(x); if ((math_errhandling & MATH_ERRNO) && errno != 0) { /* Handle range error */ } else if ((math_errhandling & MATH_ERREXCEPT) && fetestexcept(FE_INVALID  FE_DIVBYZERO  FE_OVERFLOW  FE_UNDERFLOW) != 0) { /* Handle range error */ } } /* Use result... */ } 
Failure to prevent or detect domain and range errors in math functions may cause unexpected results.
Rule  Severity  Likelihood  Remediation Cost  Priority  Level 

FLP32C  Medium  Probable  Medium  P8  L2 
Tool  Version  Checker  Description 

Astrée  stdliblimits  Partially checked  
CodeSonar  MATH.DOMAIN.ATAN MATH.DOMAIN.TOOHIGH MATH.DOMAIN.TOOLOW MATH.DOMAIN MATH.RANGE MATH.RANGE.GAMMA MATH.DOMAIN.LOG MATH.RANGE.LOG MATH.DOMAIN.FE_INVALID MATH.DOMAIN.POW MATH.RANGE.COSH.TOOHIGH MATH.RANGE.COSH.TOOLOW MATH.DOMAIN.SQRT  Arctangent Domain Error Argument Too High Argument Too Low Floating Point Domain Error Floating Point Range Error Gamma on Zero Logarithm on Negative Value Logarithm on Zero Raises FE_INVALID Undefined Power of Zero cosh on High Number cosh on Low Number sqrt on Negative Value  
Parasoft C/C++test  CERT_CFLP32a  Validate values passed to library functions  
Polyspace Bug Finder  CERTC: Rule FLP32C  Checks for invalid use of standard library floating point routine (rule partially covered)  
PRQA QAC 
 5025  
PRQA QAC++  5033  
RuleChecker  stdliblimits  Partially checked  
TrustInSoft Analyzer  outofrange argument  Partially verified. 
Search for vulnerabilities resulting from the violation of this rule on the CERT website.
Key here (explains table format and definitions)
Taxonomy  Taxonomy item  Relationship 

CERT C Secure Coding Standard  FLP03C. Detect and handle floatingpoint errors  Prior to 20180112: CERT: Unspecified Relationship 
CWE 2.11  CWE682, Incorrect Calculation  20170707: CERT: Rule subset of CWE 
Key here for mapping notes
Intersection( CWE391, FLP32C) =
CWE391  FLP32C
FLP32C – CWE391 =
Independent( INT34C, FLP32C, INT33C) CWE682 = Union( FLP32C, list) where list =
[ISO/IEC 9899:2011]  7.3.2, "Conventions" 
[IEEE 754 2006 ]  
[Plum 1985]  Rule 22 
[Plum 1989]  Topic 2.10, "conv—Conversions and Overflow" 
[UNIX 1992]  System V Interface Definition (SVID3) 