diff options
Diffstat (limited to 'manual')
| -rw-r--r-- | manual/math.texi | 78 |
1 files changed, 74 insertions, 4 deletions
diff --git a/manual/math.texi b/manual/math.texi index 57cf24f687..5e7c90e2e6 100644 --- a/manual/math.texi +++ b/manual/math.texi @@ -1227,10 +1227,80 @@ $${|d.d\dots d - (z/2^e)|}\over {2^{p-1}}$$ @noindent where @math{p} is the number of bits in the mantissa of the floating-point number representation. Ideally the error for all -functions is always less than 0.5ulps. Using rounding bits this is also -possible and normally implemented for the basic operations. To achieve -the same for the complex math functions requires a lot more work and -this has not yet been done. +functions is always less than 0.5ulps in round-to-nearest mode. Using +rounding bits this is also +possible and normally implemented for the basic operations. Except +for certain functions such as @code{sqrt}, @code{fma} and @code{rint} +whose results are fully specified by reference to corresponding IEEE +754 floating-point operations, and conversions between strings and +floating point, @theglibc{} does not aim for correctly rounded results +for functions in the math library, and does not aim for correctness in +whether ``inexact'' exceptions are raised. Instead, the goals for +accuracy of functions without fully specified results are as follows; +some functions have bugs meaning they do not meet these goals in all +cases. In future, @theglibc{} may provide some other correctly +rounding functions under the names such as @code{crsin} proposed for +an extension to ISO C. + +@itemize @bullet + +@item +Each function with a floating-point result behaves as if it computes +an infinite-precision result that is within a few ulp (in both real +and complex parts, for functions with complex results) of the +mathematically correct value of the function (interpreted together +with ISO C or POSIX semantics for the function in question) at the +exact value passed as the input. Exceptions are raised appropriately +for this value and in accordance with IEEE 754 / ISO C / POSIX +semantics, and it is then rounded according to the current rounding +direction to the result that is returned to the user. @code{errno} +may also be set (@pxref{Math Error Reporting}). + +@item +For the IBM @code{long double} format, as used on PowerPC GNU/Linux, +the accuracy goal is weaker for input values not exactly representable +in 106 bits of precision; it is as if the input value is some value +within 0.5ulp of the value actually passed, where ``ulp'' is +interpreted in terms of a fixed-precision 106-bit mantissa, but not +necessarily the exact value actually passed with discontiguous +mantissa bits. + +@item +Functions behave as if the infinite-precision result computed is zero, +infinity or NaN if and only if that is the mathematically correct +infinite-precision result. They behave as if the infinite-precision +result computed always has the same sign as the mathematically correct +result. + +@item +If the mathematical result is more than a few ulp above the overflow +threshold for the current rounding direction, the value returned is +the appropriate overflow value for the current rounding direction, +with the overflow exception raised. + +@item +If the mathematical result has magnitude well below half the least +subnormal magnitude, the returned value is either zero or the least +subnormal (in each case, with the correct sign), according to the +current rounding direction and with the underflow exception raised. + +@item +Where the mathematical result underflows and is not exactly +representable as a floating-point value, the underflow exception is +raised (so there may be spurious underflow exceptions in cases where +the underflowing result is exact, but not missing underflow exceptions +in cases where it is inexact). + +@item +@Theglibc{} does not aim for functions to satisfy other properties of +the underlying mathematical function, such as monotonicity, where not +implied by the above goals. + +@item +All the above applies to both real and complex parts, for complex +functions. + +@end itemize Therefore many of the functions in the math library have errors. The table lists the maximum error for each function which is exposed by one |