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2. Kahan summation algorithm - Wikipedia

   en.wikipedia.org/wiki/Kahan_summation_algorithm

   In principle, there is the () that grows linearly with , but in practice this term is effectively zero: since the final result is rounded to a precision , the term rounds to zero, unless is roughly / or larger. [2] In double precision, this corresponds to an of roughly , much larger than most sums.

3. Double-precision floating-point format - Wikipedia

   en.wikipedia.org/wiki/Double-precision_floating...

   Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide range of numeric values by using a floating radix point. Double precision may be chosen when the range or precision of single precision would be insufficient.

4. Floating-point error mitigation - Wikipedia

   en.wikipedia.org/wiki/Floating-point_error...

   Extension of precision is using of larger representations of real values than the one initially considered. The IEEE 754 standard defines precision as the number of digits available to represent real numbers. A programming language can include single precision (32 bits), double precision (64 bits), and quadruple precision (128 bits). While ...

5. Floating point operations per second - Wikipedia

   en.wikipedia.org/wiki/Floating_point_operations...

   This $30,000 cluster was built using only commercially available "gamer" grade hardware. [83] August 2012: 75.00¢ 102.72¢ Quad AMD Radeon 7970 System A quad AMD Radeon 7970 desktop computer reaching 16 TFLOPS of single-precision, 4 TFLOPS of double-precision computing performance. Total system cost was $3000; built using only commercially ...

6. Floating-point arithmetic - Wikipedia

   en.wikipedia.org/wiki/Floating-point_arithmetic

   For numbers with a base-2 exponent part of 0, i.e. numbers with an absolute value higher than or equal to 1 but lower than 2, an ULP is exactly 2 −23 or about 10 −7 in single precision, and exactly 2 −53 or about 10 −16 in double precision. The mandated behavior of IEEE-compliant hardware is that the result be within one-half of a ULP.

7. Distributive property - Wikipedia

   en.wikipedia.org/wiki/Distributive_property

   In approximate arithmetic, such as floating-point arithmetic, the distributive property of multiplication (and division) over addition may fail because of the limitations of arithmetic precision. For example, the identity 1 / 3 + 1 / 3 + 1 / 3 = ( 1 + 1 + 1 ) / 3 {\displaystyle 1/3+1/3+1/3=(1+1+1)/3} fails in decimal arithmetic , regardless of ...

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9. Extended precision - Wikipedia

   en.wikipedia.org/wiki/Extended_precision

   The largest possible exponent of a double-precision value is 1023 so the exponent of the largest possible product of two double-precision numbers is 2047 (an 11-bit value). Adding in a bias to account for negative exponents means that the exponent field must be at least 12 bits wide.