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I've been exploring the evolution of floating-point arithmetic formats from single to octuple precision. Here's what I THINK I have learned about the key specifications and capabilities for each precision level I've learned about:

Single Precision (32-bit):
Bits (Significand/Sign/Exponent): 24 (1 implicit + 23 explicit)/23/1/8
Decimal Precision: ~7-8 digits
Bias: 127
Range: 2^-126 to 2^127

Double Precision (64-bit):
Bits (Significand/Sign/Exponent): 53 (1 implicit + 52 explicit)/52/1/11
Decimal Precision: ~15-17 digits
Bias: 1023
Range: 2^-1022 to 2^1023

Quadruple Precision (128-bit):
Bits (Significand/Sign/Exponent): 113 (1 implicit + 112 explicit)/112/1/15
Decimal Precision: ~33-36 digits
Bias: 16383
Range: 2^-16382 to 2^16383

Octuple Precision (256-bit):
Bits (Significand/Sign/Exponent): 237 (1 implicit + 236 explicit)/236/1/19
Decimal Precision: ~69-72 digits
Bias: 262143
Range: 2^-262142 to 2^262143

Lastly, I learned about arbitrary precision floating point, where you can basically choose the precision you want. So, this led me to the obvious question of how precise have people gone?

Question: Given these advancements, what is the highest precision floating-point arithmetic that has ever been utilized, in practical or theoretical applications? How many bits were used for the exponent, significand, and sign? (Is there ever any situation in which you would use more than one sign bit?) What specific application or use case required such high precision?

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  • $\begingroup$ "Extended" and "arbitrary" precision are two different things. True arbitrary-precision libraries are just that: arbitrary. You can have the equivalent of thousands of bits of precision, virtually as many as you want. The "size" (memory footprint) of an arbitrary-precision number typically varies, at runtime, as needed. (A lot like variable-length strings.) One popular arbitrary-precision library is GMP. $\endgroup$ Commented May 14 at 15:48
  • $\begingroup$ Also notable are the Unix/Linux arbitrary-precision calculators dc and bc. $\endgroup$ Commented May 14 at 15:51
  • $\begingroup$ @SteveSummit how did I not make that clear in the question? I said, " arbitrary precision floating point, where you can basically choose the precision you want." That was the ENTIRE point of the question, I want to know HOW arbitrary people have gone. $\endgroup$ Commented May 16 at 16:28
  • $\begingroup$ @SteveSummit those calculators are fascinating. It makes PERFECT sense to me that one of the first Unix programs was an arbitrary precision calculator, considering precise math is one of the ultimate quests of man. It makes me wonder how they achieve this with finite bits, maybe that's another question. $\endgroup$ Commented May 16 at 16:32
  • $\begingroup$ Sorry, perhaps I read your question too hastily. The specific examples you cited (quadruple, octuple) are examples of what I believe are called extended precision (not "arbitrary"), which is why I felt the need to make that point. $\endgroup$ Commented May 16 at 20:38

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