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The C Standard, 7.12.1 [ISO/IEC 9899:20112024], defines three types of errors that relate specifically to math functions in <math.h>.  Paragraph 2 states

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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 arange error occurs if and only if the mathematical result of the function cannot be represented in an object of the specified type, due to extreme magnitude.result overflows or underflows

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 1-millionth power, pow(10., 1e6), cannot be represented in many floating-point 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.

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The exact treatment of error conditions from math functions is tedious. The C Standard, 7.12.1 paragraph 5 [ISO/IEC 9899:20112024], 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 typea finite result value with ordinary accuracy would have magnitude (absolute value) too large for the representation with full precision in the specified type. A result that is exactly an infinity does not overflow. 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 however, for the types with reduced-precision representations of numbers beyond the overflow threshold, the function may return a representation of the result with less than full precision for the type. If a floating resultoverflowsanddefaultroundingisineffectandtheintegerexpressionmath_errhandling & MATH_ERRNO is nonzero, then the integer expression errno acquires the value ERANGE; if . If a floating result overflows, and the integer expression math_errhandling & MATH_ERREXCEPT is nonzero, the the "overflow" floating-point exception is raised (regardless of whether default rounding is in effect).

It is preferable not to check for errors by comparing the returned value against HUGE_VAL or 0 for several reasons:

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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  

The C Standard, 7.3.2, paragraph 1 , simply states that "an [ISO/IEC 9899:2024],  states:

 an implementation may set errno but is not required to

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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.

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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,  77.12.1, paragraph 6 [ISO/IEC 9899:20112024], 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.

underflows if a nonzero result value with ordinary accuracy would have magnitude (absolute value) less than the minimum normalized number in the type; however a zero result that is specified to be an exact zero does not underflow. Also, a result with ordinary accuracy and the magnitude of the minimum normalized number may underflow.269) 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

acquires the value ERANGE

 is

is implementation-defined; if the integer expression math_errhandling & MATH_ERREXCEPT

is

s nonzero, whether the

‘‘underflow’’

"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:

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If Annex F is supported and subnormal results subnormal results are supported, the returned value is exact and a range error cannot occur. The C Standard, F.10.7.1 paragraph 2 [ISO/IEC 9899:20112024], specifies the following for the fmod(), remainder(), and remquo() functions:

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Bibliography

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