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33 .Nd IEEE standard 754 for floating-point arithmetic
35 The IEEE Standard 754 for Binary Floating-Point Arithmetic
36 defines representations of floating-point numbers and abstract
39 .Ss IEEE STANDARD 754 Floating-Point Arithmetic
43 .Bd -ragged -offset indent -compact
49 Zero is represented ambiguously as +0 or \-0.
50 .Bd -ragged -offset indent -compact
53 with like signs; but x\-x yields +0 for every
57 .Fn copysign x \(+-0 .
60 finite x = y then \*(If = 1/(x\-y) \(!= \-1/(y\-x) = \-\*(If.
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69 (finite)/\(+-\*(If\0=\0\(+-0
70 (nonzero)/0 = \(+-\*(If.
72 \*(If\-\*(If, \*(If\(**0 and \*(If/\*(If
73 are, like 0/0 and sqrt(\-3),
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81 Some \*(Nas, called Signaling \*(Nas, trap any floating-point operation
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94 Every algebraic operation (+, \-, \(**, /,
119 +\*(If or towards \-\*(If
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125 IEEE 754 recognizes five kinds of floating-point exceptions,
127 .Bl -column -offset indent "Invalid Operation" "Gradual Underflow"
130 Overflow	\(+-\*(If
131 Divide by Zero	\(+-\*(If
147 Single-precision:
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157 If x and x' are consecutive positive single-precision
161 .Bl -column "XXX" -compact
162 5.9e\-08 < 0.5**24 < (x'\-x)/x \(<= 0.5**23 < 1.2e\-07.
165 .Bl -column "XXX" -compact
167 	Underflow threshold = 0.5**126 = 1.2e\-38
172 .Bl -column "XXX" -compact
173 0.5**149 = 1.4e\-45.
177 Double-precision:
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192 If x and x' are consecutive positive double-precision
196 .Bl -column "XXX" -compact
197 1.1e\-16 < 0.5**53 < (x'\-x)/x \(<= 0.5**52 < 2.3e\-16.
200 .Bl -column "XXX" -compact
202 	Underflow threshold = 0.5**1022 = 2.2e\-308
207 .Bl -column "XXX" -compact
208 0.5**1074 = 4.9e\-324.
212 Extended-precision:
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223 If x and x' are consecutive positive extended-precision
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228 1.0e\-19 < 0.5**63 < (x'\-x)/x \(<= 0.5**62 < 2.2e\-19.
231 .Bl -column "XXX" -compact
233 	Underflow threshold = 0.5**16382 = 3.4e\-4932
238 .Bl -column "XXX" -compact
239 0.5**16445 = 5.7e\-4953.
243 Quad-extended-precision:
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254 If x and x' are consecutive positive quad-extended-precision
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259 9.6e\-35 < 0.5**113 < (x'\-x)/x \(<= 0.5**112 < 2.0e\-34.
262 .Bl -column "XXX" -compact
264 	Underflow threshold = 0.5**16382 = 3.4e\-4932
269 .Bl -column "XXX" -compact
270 0.5**16494 = 6.5e\-4966.
274 For each kind of floating-point exception, IEEE 754
282 .Bl -enum
299 IEEE 754; if x \(!= y then x\-y is correct to
319 .Bl -enum
324 traditionally associated with error-handling
329 .Bl -dash
332 the offending operation's result and resume
337 In a subprogram that lacks an error-handling
341 an error-handling statement is encountered or the
357 to be able to resume execution after each one as if
366 .Bl -enum
386 .Bl -tag -width indent -offset indent -compact
403 .Bl -tag -width indent -offset indent -compact
413 Divide-by-Zero is signaled only when
432 the title "A Proposed Radix- and Word-length-independent
433 Standard for Floating-point Arithmetic" by
443 .St -ieee754