Though different in detail, the MC68881 and SANE packages are both
implementations of the IEEE Standard 754. Following are the range and
precision of the data types supported by SANE:
Single Double Comp Extended
------ ------ ---- --------
Size (bytes:bits) 4:32 8:64 8:64 10:80
Range of binary exponents
Minimum -126 -1022 - -16383
Maximum 127 1023 - 16383
Significand precision
Bits 24 53 63 64
Decimal digits 7-8 15-16 18-19 19-20
Decimal range approximate
Maximum positive 3.4E+38 1.7E+308 9.2E18 1.1E+4932
Minimum positive norm 1.2E-38 2.3E+308 1.7E-4932
Minimum positive denorm 1.5E-45 5.0E-324 1.9E-4951
Maximum negative denorm -1.5E-45 -5.0E-324 -1.9E-4951
Maximum negative norm -1.2E-38 -2.3E-308 -1.7E-4932
Minimum negative -3.4E+38 -1.7E+308 -9.2E18 -1.1E+4932
If the compiler is making direct calls to the MC68881, accuracy may suffer in
some situations. For elemental functions, both SANE and the MC68881 have
errors in the least-significant bits of the fraction part of extended-format
results, but SANE's errors rarely exceed the last bit, whereas the MC68881
errors may extend to as many as the last five bits. The MC68881 is much more
likely to return an error in a double-precision result than SANE is.
Due to the architecture of the Macintosh, and how SANE is implemented on the
Macintosh, the extended-precision number format not only provides the most
accuracy, but also provides better performance than the computational or
double-number formats. The Macintosh converts double and computational numbers
into extended number before using them, and thus takes more time than just
using the extended-number format.
Single precision provides the most speed, but is hardly ever used because most
high-level language compilers today generate code using the extended number
format. You can use the single number format, but the code probably
would have to be written in assembly language.
For more complete and detailed information, we suggest the "Apple Numerics
Manual, Second Edition" from Addison-Wesley.