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 
Domain and Pole Checking
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); }
Range Checking
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:
 These are, in general, valid (albeit unlikely) data values.
 Making such tests requires detailed knowledge of the various error returns for each math function.
 Multiple results aside from
HUGE_VAL
and0
are possible, and programmers must know which are possible in each case.  Different versions of the library have varied in their errorreturn behavior.
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.
Subnormal Numbers
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+
), min)subnorm
remquo
((min+
), min, quo)subnorm
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.
Noncompliant Code Example (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.
Compliant Solution (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); }
Noncompliant Code Example (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.
Compliant Solution (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... */ }
Noncompliant Code Example (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
.
Compliant Solution (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... */ }
Noncompliant Code Example (asin()
, Subnormal Number)
This noncompliant code example determines the inverse sine of x
:
#include <math.h> void func(float x) { float result = asin(x); /* ... */ }
Compliant Solution (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... */ }
Risk Assessment
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 
Automated Detection
Tool  Version  Checker  Description 

Astrée  23.04  stdliblimits  Partially checked 
Axivion Bauhaus Suite  7.2.0  CertCFLP32  Partially implemented 
CodeSonar  8.0p0  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 
Helix QAC  2023.3  C5025 C++5033  
Parasoft C/C++test  2023.1  CERT_CFLP32a  Validate values passed to library functions 
PClint Plus  1.4  2423  Partially supported: reports domain errors for functions with the Semantics *dom_1, *dom_lt0, or *dom_lt1, including standard library math functions 
Polyspace Bug Finder  R2023b  CERTC: Rule FLP32C  Checks for invalid use of standard library floating point routine (rule fully covered) 
RuleChecker  23.04  stdliblimits  Partially checked 
TrustInSoft Analyzer  1.38  outofrange argument  Partially verified. 
Related Vulnerabilities
Search for vulnerabilities resulting from the violation of this rule on the CERT website.
Related Guidelines
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 
CERTCWE Mapping Notes
Key here for mapping notes
CWE391 and FLP32C
Intersection( CWE391, FLP32C) =
 Failure to detect range errors in floatingpoint calculations
CWE391  FLP32C
 Failure to detect errors in functions besides floatingpoint calculations
FLP32C – CWE391 =
 Failure to detect domain errors in floatingpoint calculations
CWE682 and FLP32C
Independent( INT34C, FLP32C, INT33C) CWE682 = Union( FLP32C, list) where list =
 Incorrect calculations that do not involve floatingpoint range errors
Bibliography
[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) 
25 Comments
Geoff Clare
The treatment of pow() in this rule is somewhat lacking.
The specified bounds check (x != 0  y > 0) is insufficient: C99 says
a domain error also occurs "if x is finite and negative and y is finite
and not an integer value."
The CCE for pow() says "This code tests x and y to ensure that there
will be no range or domain errors" but it does not detect the range
error given in the introductory text at the top of the page: pow(10., 1e6)
Given that the page begins with "Prevent or detect domain errors and
range errors ...", and given the complexity involved in preventing pow()
errors, I think that the document should recommend detection for this
function instead of prevention.
On the subject of detection, the subsection entitled "NonCompliant
Coding Example (Error Checking)" only talks about return values and errno.
There is no mention of error checking by examining the exception flags.
C90 required the maths functions to set errno on error. C99 requires
them (the noncomplex ones, that is) either to set errno or to set
the exception flags, or both. So I think the recommendation (for
noncomplex maths functions) should be:
1. If there is a simple bounds check that can be done to prevent domain
and range errors, then do it. (This applies to all the current
examples except pow().)
2. Otherwise, detect errors as follows:
Other functions besides pow() where detection should be used because
prevention would be too complicated include erfc(), lgamma() and tgamma().
Douglas A. Gwyn
There is essentially no reason for a program to invoke pow() with a negative base.
David Svoboda
I've addressed these comments. I included Geoff's code sample under 'Compliant Example: Error Checking'
Geoff Clare
Your changes helped a lot, but there were still some problems relating to pow(). I have attempted to fix them.
Alex Volkovitsky
Shaun,
regarding the second NCCE under pow(), what does "result cannot be represented as a double" mean? It means the result is either a NaN or Infty... we can check for those two after computing the pow() to ensure no range errors happened, ahh... the beauty of floating point
Geoff Clare
"result cannot be represented as a double" means the true (mathematical) result is outside the range of values that can be represented by double. It does not mean the result is NaN or Infinity. E.g. for pow(10.,1e6) the true result is ten to the power of one million, which is larger than DBL_MAX and therefore cannot be represented as a double. The true result is not infinity.
Also note that some implementations do not support Inifinity and/or NaN, and so applications cannot reply on them being returned.
David Svoboda
This should be checkable by Rose. But there is a snag. The isLess() etc. functions, which are being used to do range checking, are not defined. If we could simply check for "x > 0.0", then we can do it. Is that what isGreater(x, 0) really means?
Geoff Clare
The difference between
x > 0.0
andisgreater(x,0)
is thatisgreater(x,0)
will not raise a floatingpoint exception ifx
is a NaN.If you want to switch to using the operators, you would have to add explicit
isnan()
checks (in the cases where there isn't one already).Geoff Clare
For consistency with the way
pow()
was treated, shouldn't the newtgamma()
examples have a NCCE that does no checking, then a NCCE that does only the domain checks, and then just give a reference to the Error Checking and Detection section instead of having a CS that duplicates code from itAlex Volkovitsky
In that case, we should add the
FE_UNDERFLOW
flag to the Error Checking sectionDavid Svoboda
agreed; the
pow()
section and thetgamma()
section both follow the same outline.Jonathan Paulson
Java universally deals with this issue by returning NaN; it might be worth a guideline to check if the result a math operation is Nan? Removing the exportablejava guideline.
Geoff Clare
The new safe_sqrt() exemplar seems to me to be not very exemplary, as it handles the domain error in a particularly unhelpful way. Is there a reason not to use the usual convention of a /* Handle error */ comment here?
David Svoboda
Agreed...fixed.
Jeremy Hall
I think row 6 in the table should be asinh() and not asin()
Aaron Ballman
Agreed. I've fixed it, thanks!
Jeremy Hall
For
log(x)
,log10(x)
,log2(x)
I think the domain should be x > 0 rather than x >= 0 because they produce a pole error for x == 0. Log1p looks correct, the domain is given as x > 1.0Geoff Clare
No, you have it backwards. If x == 0 was outside the domain of the function, log(0) would produce a domain error, not a pole error. The mistake is for log1p(x) which should give the domain as x >= 1.0.
Aaron Ballman
I looked at the wording in the standard, and I agree that
log()
,log10()
, andlog2()
seem to be correct, whilelog1p()
was incorrect with its domain. I've corrected.Eddie O'Hagan
I noticed that in the documentation for cos(), sin(), and tan():
I think it makes sense to add cos(), sin(), and tan() to the list as well.
Aaron Ballman
tan()
could certainly be in that list with a pole check, sincetan()
returns an infinity as the function approaches pi / 2. However, I don't believe any IEEE floating point representation can exactly represent pi / 2, so I don't believe a pole error can technically occur in practice. I'm uncertain of how we might want to represent that in the list.We would have to do a lot more research before adding
cos()
andsin()
to that list, because that's a difference in specification between POSIX and C. The C standard does not define any domain error when given infinity or NaN, while POSIX does. I suspect similar distinctions occur for other functions. We would need to make mention of where POSIX and C differ.Wolfgang Stanglmeier
The domain of atan2( y, x ) should be
x != 0  y != 0
.x or y may be zero, only the case x=0 ∧ y=0 is undefined.
Joseph C. Sible
Wait, isn't even that case defined too? In the C17 standard, under F.10.1.4, it says "atan2(±0, −0) returns ±π" and "atan2(±0, +0) returns ±0".
David Svoboda
Correct. The C23 draft standard (sF.10.1.4) says:
atan2(±0,−0)returns ±π.404
Footnote 404:
atan2(0, 0) does not raise the "invalid" floatingpoint exception, nor does atan2(y, 0) raise the "dividebyzero" floating point exception.
Whether passing zeroes in constitutes a domain error depends on how you define atan2(y,x). Every definition I've seen references the "intermediate value" of y/x, which is undefined for x=y=0. But ISO C is clear on atan2(0,0) for both positive and negative zero :) So I took out the domain error.
Wolfgang Stanglmeier
Thanks for the clarifications and the reference to the standards. I noticed that the domain was obviously too narrow, but didn't realize that there is no domain restriction at all.