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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.

Programmers can prevent domain and pole errors by carefully bounds-checking 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 floating-point 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 

acos(x)

-1 <= x && x <= 1

No

No
asin(x)-1 <= x && x <= 1YesNo
atan(x)NoneYesNo

atan2(y, x)

x != 0 && y != 0

No

No

acosh(x)

x >= 1

Yes

No
asinh(x)NoneYesNo

atanh(x)

-1 < x && x < 1

Yes

Yes

cosh(x), sinh(x)

None

Yes

No

exp(x), exp2(x), expm1(x)

None

Yes

No

ldexp(x, exp)

None

Yes

No

log(x), log10(x), log2(x)

x >= 0

No

Yes

log1p(x)

x >= -1

No

Yes

ilogb(x)

x != 0 && !isinf(x) && !isnan(x)

Yes

No
logb(x)x != 0Yes Yes

scalbn(x, n), scalbln(x, n)

None

Yes

No

hypot(x, y)

None

Yes

No

pow(x,y)

x > 0 || (x == 0 && y > 0) ||
(x < 0 && y is an integer)

Yes

Yes

sqrt(x)

x >= 0

No

No
erf(x)NoneYesNo

erfc(x)

None

Yes

No

lgamma(x), tgamma(x)

x != 0 && ! (x < 0 && x is an integer)

Yes

Yes

lrint(x), lround(x)

None

Yes

No

fmod(x, y), remainder(x, y),
remquo(x, y, quo)

y != 0

Yes

No

nextafter(x, y),
nexttoward(x, y)

None

Yes

No

fdim(x,y)

None

Yes

No 

fma(x,y,z)

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 floating-point 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, or HUGE_VALL according to the return type, with the same sign as the correct value of the function; if the integer expression math_errhandling & MATH_ERRNO is nonzero, the integer expression errno acquires the value ERANGE; if the integer expression math_errhandling & MATH_ERREXCEPT is nonzero, the "overflow" floating-point 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 and 0 are possible, and programmers must know which are possible in each case.
  • Different versions of the library have varied in their error-return 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 error-handing template uses C Standard functions for floating-point 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 FLP03-C. Detect and handle floating-point errors for more details on how to detect floating-point 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 floating-point 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 implementation-defined 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, whether errno  acquires the value ERANGE  is implementation-defined; if the integer expression math_errhandling & MATH_ERREXCEPT is nonzero, whether the ‘‘underflow’’ floating-point exception is raised is implementation-defined.

Implementations that support floating-point arithmetic but do not support subnormal numbers, such as IBM S/360 hex floating-point or nonconforming IEEE-754 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.

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

FLP32-C

Medium

Probable

Medium

P8

L2

Automated Detection

Tool

Version

Checker

Description

Polyspace Bug FinderR2016aInvalid use of standard library floating point routine

Wrong arguments to standard library function

 PRQA QA-C 9.1 5025 

Related Vulnerabilities

Search for vulnerabilities resulting from the violation of this rule on the CERT website.

Related Guidelines

Bibliography

[ISO/IEC 9899:2011]

7.3.2, "Conventions"
7.12.1, "Treatment of Error Conditions"
F.10.7, "Remainder Functions" 

[IEEE 754 2006 ] 
[Plum 1985]Rule 2-2
[Plum 1989]Topic 2.10, "conv—Conversions and Overflow"
[UNIX 1992]System V Interface Definition (SVID3)

 


 

21 Comments

  1. 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 "Non-Compliant
    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 non-complex ones, that is) either to set errno or to set
    the exception flags, or both.  So I think the recommendation (for
    non-complex 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:

    #include <math.h>
    #if defined(math_errhandling) && (math_errhandling & MATH_ERREXCEPT)
    #include <fenv.h>
    #endif

    [...]

    #if defined(math_errhandling) && (math_errhandling & MATH_ERREXCEPT)
        feclearexcept(FE_ALL_EXCEPT);
    #endif
    errno = 0;

    /* call the function */

    #if !defined(math_errhandling) || (math_errhandling & MATH_ERRNO)
    if (errno != 0){
        /* handle error */
    }
    #endif
    #if defined(math_errhandling) && (math_errhandling & MATH_ERREXCEPT)
    if (fetestexcept(FE_INVALID | FE_DIVBYZERO | FE_OVERFLOW) != 0){
        /* handle error */
    }
    #endif

    Other functions besides pow() where detection should be used because
    prevention would be too complicated include erfc(), lgamma() and tgamma().

  2. There is essentially no reason for a program to invoke pow() with a negative base.

  3. I've addressed these comments. I included Geoff's code sample under 'Compliant Example: Error Checking'

    1. Your changes helped a lot, but there were still some problems relating to pow().  I have attempted to fix them.

  4. 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 (smile)

    1. "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.

  5. 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?

    1. The difference between x > 0.0 and isgreater(x,0) is that isgreater(x,0) will not raise a floating-point exception if x 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).

  6. For consistency with the way pow() was treated, shouldn't the new tgamma() 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 it

    1. In that case, we should add the FE_UNDERFLOW flag to the Error Checking section

    2. agreed; the pow() section and the tgamma() section both follow the same outline.

  7. 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 exportable-java guideline.

  8. 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?

  9. I think row 6 in the table should be asinh() and not asin()

    1. Agreed. I've fixed it, thanks!

  10. 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 > -1.0

     

     

    1. 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.

      1. I looked at the wording in the standard, and I agree that log(), log10(), and log2() seem to be correct, while log1p() was incorrect with its domain. I've corrected.

  11. I noticed that in the documentation for cos(), sin(), and tan():

    Errors are reported as specified in math_errhandling

    If the implementation supports IEEE floating-point arithmetic (IEC 60559),

    • if the argument is ±0, the result is 1.0
    • if the argument is ±∞, NaN is returned and FE_INVALID is raised
    • if the argument is NaN, NaN is returned

    I think it makes sense to add cos(), sin(), and tan() to the list as well.

    1. tan() could certainly be in that list with a pole check, since tan() 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() and sin() 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.