FLP32-C. Prevent or detect domain and range errors in math functions

Subclause 7.12.1 of the C Standard [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. On the other hand, 10 raised to the 1-millionth power, pow(10., 1e6), cannot be represented in many implementations' floating-point representation 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.

Domain and pole errors can be prevented by carefully bounds-checking the arguments before calling 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 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. 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.

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 template:

if (/* Arguments that will cause a domain or pole error */) {
  /* Handle domain or pole error */
} else {
  /* Perform computation */
}

Range Checking

Range errors usually cannot be prevented, so the most reliable way to handle range errors is to detect when they have occurred and act accordingly.

The exact treatment of error conditions from math functions is quite complicated. Subclause 7.12.1, paragraph 5, of the C Standard [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.
• There are multiple different result possibilities aside from HUGE_VAL and 0, and you must know which are possible in each case.
• Different versions of the library have differed in their error-return behavior.

It is also difficult to check for math errors using errno because an implementation might not set it. For real functions, the programmer determine if the implementation sets errno by checking whether math_errhandling & MATH_ERRNO is nonzero. For complex functions, the C Standard, subclause 7.3.2, paragraph 1, simply states that "an implementation may set errno but is not required to" [ISO/IEC 9899:2011].

The System V Interface Definition (SVID3) [UNIX 1992] provides more control over the treatment of errors in the math library. The user can provide 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, 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>
#if defined(math_errhandling) \
  && (math_errhandling & MATH_ERREXCEPT)
#include <fenv.h>

#pragma STDC FENV_ACCESS ON
#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 range error */
}
#endif
#if defined(math_errhandling) \
  && (math_errhandling & MATH_ERREXCEPT)
if (fetestexcept(FE_INVALID
               | FE_DIVBYZERO
               | FE_OVERFLOW
               | FE_UNDERFLOW) != 0) {
  /* Handle range error */
}
#endif

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 ]. They 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, certain functions may produce range errors specifically when applied with a subnormal number. These functions are: asin(), asinh(), atan(), atanh(), and erf(). When evaluated with a subnormal number, these functions can produce an inexact, subnormal value, which is an underflow error. Subclause 7.12.1, paragraph 6, of the C Standard [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 does not support subnormals, such as IBM S/360 hex floating point or sloppy 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 family 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.

These error cases are not an issue if Annex F is fully support.  In this case, if subnormal results are supported, the returned value is exact and there cannot be a range error.  The C Standard [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.

Subclause F.10.7.2, paragraph 2, and subclause F.10.7.3, paragraph 2 of the C Standard [ISO/IEC 9899:2011] cover for 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 <errno.h>
#include <math.h>

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

#pragma STD FENV_ACCESS ON
#endif

void func(double x) { 
  double result;
  errno = 0;
 
  #if defined(math_errhandling) \
    && (math_errhandling & MATH_ERREXCEPT)
    feclearexcept(FE_ALL_EXCEPT);
  #endif
 
  result = sinh(x); 
 
  #if !defined(math_errhandling) \
    || (math_errhandling & MATH_ERRNO)
  if (errno != 0) {
    /* Handle range error */
  }
  #endif
  #if defined(math_errhandling) \
    && (math_errhandling & MATH_ERREXCEPT)
  if (fetestexcept(FE_INVALID
                 | FE_DIVBYZERO
                 | FE_OVERFLOW
                 | FE_UNDERFLOW) != 0) {
    /* Handle range error */
  }
  #endif
}

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);
}

However, this code may produce a domain error if x is negative and y is not an integer 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, then detect whether a range error occurs and act accordingly:

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

#pragma STDC FENV_ACCESS ON
#endif

void func(double x, double y) {
  #if defined(math_errhandling) \
    && (math_errhandling & MATH_ERREXCEPT)
    feclearexcept(FE_ALL_EXCEPT);
  #endif
  errno = 0;

  double result;

  if (((x == 0.0f) && islessequal(y, 0.0)) || isless(x, 0.0)) {
    /* Handle domain or pole error */
  }

  result = pow(x, y);

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

Noncompliant Code Example (asin(), Subnormal Number)

This noncompliant code example determines the arcsin of x:

#include <math.h>
 
void func(float x) {
  float result = asin(x);
  /* ... */
}

Compliant Soluction (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 <errno.h>
#include <math.h>

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

#pragma STDC FENV_ACCESS ON
#endif

void func(float x) { 
  float result;
  errno = 0;
  result = asin(x);
 
  #if defined(math_errhandling) \
    && (math_errhandling & MATH_ERREXCEPT)
    feclearexcept(FE_ALL_EXCEPT);
  #endif
 
  #if !defined(math_errhandling) \
    || (math_errhandling & MATH_ERRNO)
  if (errno != 0) {
    /* Handle range error */
  }
  #endif
  #if defined(math_errhandling) \
    && (math_errhandling & MATH_ERREXCEPT)
  if (fetestexcept(FE_INVALID
                 | FE_DIVBYZERO
                 | FE_OVERFLOW
                 | FE_UNDERFLOW) != 0) {
    /* Handle range error */
  }
  #endif
}

Risk Assessment

Failure to prevent or detect domain and range errors in math functions may cause unexpected results.

Automated Detection

Related Vulnerabilities

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

Related Guidelines

Bibliography