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:

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:

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

Astrée
stdlib-limits
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_C-FLP32-a
Validate values passed to library functions
Polyspace Bug Finder

CERT-C: Rule FLP32-CChecks for invalid use of standard library floating point routine (rule partially covered)


PRQA QA-C

 

5025
PRQA QA-C++

5033
RuleChecker

stdlib-limits
Partially checked
TrustInSoft Analyzer

out-of-range argumentPartially 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 StandardFLP03-C. Detect and handle floating-point errorsPrior to 2018-01-12: CERT: Unspecified Relationship
CWE 2.11CWE-682, Incorrect Calculation2017-07-07: CERT: Rule subset of CWE

CERT-CWE Mapping Notes

Key here for mapping notes

CWE-391 and FLP32-C

Intersection( CWE-391, FLP32-C) =



CWE-391 - FLP32-C



FLP32-C – CWE-391 =



CWE-682 and FLP32-C

Independent( INT34-C, FLP32-C, INT33-C) CWE-682 = Union( FLP32-C, list) where list =



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)