Essentially all architectures in current non-punch-card use, including embedded architectures and exotic signal processing architectures, offer one of two floating point systems: IEEE-754. IEEE-754 except for blah. That is, they mostly implement 754, but cheap out on some of the more expensive and/or fiddly bits. The most common cheap-outs: Flushing denormals to zero. This invalidates … Read more
Check out the article on “What every computer scientist should know about floating point arithmetic”
It doesn’t violate IEEE-754, because IEEE-754 defers to languages on this point: A language standard should also define, and require implementations to provide, attributes that allow and disallow value-changing optimizations, separately or collectively, for a block. These optimizations might include, but are not limited to: … ― Synthesis of a fusedMultiplyAdd operation from a multiplication … Read more
A 32 bit floating point number has 23 + 1 bits of mantissa and an 8 bit exponent (-126 to 127 is used though) so the largest number you can represent is: (1 + 1 / 2 + … 1 / (2 ^ 23)) * (2 ^ 127) = (2 ^ 23 + 2 ^ … Read more
The iPhone can do both single and double precision arithmetic in hardware. On the 1176 (original iPhone and iPhone3G), they operate at approximately the same speed, though you can fit more single-precision data in the caches. On the Cortex-A8 (iPhone3GS, iPhone4 and iPad), single-precision arithmetic is done on the NEON unit instead of VFP, and … Read more
SSE is fully specified*. Muller is an expert in floating point arithmetic; who are you going to trust, him or some guy on a gamedev forum? (*) there are actually a few exceptions for non-IEEE-754 operations like rsqrtss, where Intel never fully specified the behavior, but that doesn’t effect the IEEE-754 basic operations, and more … Read more
Formatting a floating point number is rather non-trivial. Search e.g. for the Dragon4 algorithm (here is one result). Very, very naively, you could try this: Handle NaN and Infinity. Print the sign (check < 0). Assume henceforth the number is positive real. If >= 1, truncate and use familiar integer formatting to print the integral … Read more
As for php, output is related to ini settings of precision: ini_set(‘precision’, 15); print 0.3 – 0.2; // 0.1 ini_set(‘precision’, 17); print 0.3 – 0.2; //0.099999999999999978 This may be also cause for other languages
The IEEE-754 standard defines a NaN as a number with all ones in the exponent, and a non-zero significand. The highest-order bit in the significand specifies whether the NaN is a signaling or quiet one. The remaining bits of the significand form what is referred to as the payload of the NaN. Whenever one of … Read more
It’s a nonsense from the mathematical perspective. Yes. No. Sort of. The thing is: Floating-point numbers are approximations. You want to use a wide range of exponents and a limited number of digits and get results which are not completely wrong. 🙂 The idea behind IEEE-754 is that every operation could trigger “traps” which indicate … Read more