Long double format
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Long double format
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Each long double is made up of two IEEE doubles. The value of the
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Each long double is made up of two IEEE doubles. The value of the
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long double is the sum of the values of the two parts (except for
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long double is the sum of the values of the two parts (except for
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-0.0). The most significant part is required to be the value of the
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-0.0). The most significant part is required to be the value of the
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long double rounded to the nearest double, as specified by IEEE. For
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long double rounded to the nearest double, as specified by IEEE. For
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Inf values, the least significant part is required to be one of +0.0
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Inf values, the least significant part is required to be one of +0.0
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or -0.0. No other requirements are made; so, for example, 1.0 may be
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or -0.0. No other requirements are made; so, for example, 1.0 may be
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represented as (1.0, +0.0) or (1.0, -0.0), and the low part of a NaN
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represented as (1.0, +0.0) or (1.0, -0.0), and the low part of a NaN
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is don't-care.
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is don't-care.
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Classification
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Classification
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A long double can represent any value of the form
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A long double can represent any value of the form
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s * 2^e * sum(k=0...105: f_k * 2^(-k))
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s * 2^e * sum(k=0...105: f_k * 2^(-k))
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where 's' is +1 or -1, 'e' is between 1022 and -968 inclusive, f_0 is
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where 's' is +1 or -1, 'e' is between 1022 and -968 inclusive, f_0 is
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1, and f_k for k>0 is 0 or 1. These are the 'normal' long doubles.
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1, and f_k for k>0 is 0 or 1. These are the 'normal' long doubles.
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A long double can also represent any value of the form
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A long double can also represent any value of the form
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s * 2^-968 * sum(k=0...105: f_k * 2^(-k))
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s * 2^-968 * sum(k=0...105: f_k * 2^(-k))
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where 's' is +1 or -1, f_0 is 0, and f_k for k>0 is 0 or 1. These are
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where 's' is +1 or -1, f_0 is 0, and f_k for k>0 is 0 or 1. These are
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the 'subnormal' long doubles.
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the 'subnormal' long doubles.
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There are four long doubles that represent zero, two that represent
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There are four long doubles that represent zero, two that represent
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+0.0 and two that represent -0.0. The sign of the high part is the
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+0.0 and two that represent -0.0. The sign of the high part is the
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sign of the long double, and the sign of the low part is ignored.
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sign of the long double, and the sign of the low part is ignored.
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Likewise, there are four long doubles that represent infinities, two
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Likewise, there are four long doubles that represent infinities, two
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for +Inf and two for -Inf.
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for +Inf and two for -Inf.
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Each NaN, quiet or signalling, that can be represented as a 'double'
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Each NaN, quiet or signalling, that can be represented as a 'double'
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can be represented as a 'long double'. In fact, there are 2^64
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can be represented as a 'long double'. In fact, there are 2^64
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equivalent representations for each one.
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equivalent representations for each one.
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There are certain other valid long doubles where both parts are
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There are certain other valid long doubles where both parts are
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nonzero but the low part represents a value which has a bit set below
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nonzero but the low part represents a value which has a bit set below
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2^(e-105). These, together with the subnormal long doubles, make up
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2^(e-105). These, together with the subnormal long doubles, make up
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the denormal long doubles.
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the denormal long doubles.
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Many possible long double bit patterns are not valid long doubles.
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Many possible long double bit patterns are not valid long doubles.
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These do not represent any value.
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These do not represent any value.
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Limits
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Limits
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The maximum representable long double is 2^1024-2^918. The smallest
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The maximum representable long double is 2^1024-2^918. The smallest
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*normal* positive long double is 2^-968. The smallest denormalised
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*normal* positive long double is 2^-968. The smallest denormalised
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positive long double is 2^-1074 (this is the same as for 'double').
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positive long double is 2^-1074 (this is the same as for 'double').
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Conversions
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Conversions
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-----------
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A double can be converted to a long double by adding a zero low part.
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A double can be converted to a long double by adding a zero low part.
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A long double can be converted to a double by removing the low part.
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A long double can be converted to a double by removing the low part.
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Comparisons
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Comparisons
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Two long doubles can be compared by comparing the high parts, and if
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Two long doubles can be compared by comparing the high parts, and if
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those compare equal, comparing the low parts.
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those compare equal, comparing the low parts.
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Arithmetic
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Arithmetic
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----------
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----------
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The unary negate operation operates by negating the low and high parts.
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The unary negate operation operates by negating the low and high parts.
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An absolute or absolute-negate operation must be done by comparing
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An absolute or absolute-negate operation must be done by comparing
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against zero and negating if necessary.
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against zero and negating if necessary.
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Addition and subtraction are performed using library routines. They
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Addition and subtraction are performed using library routines. They
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are not at present performed perfectly accurately, the result produced
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are not at present performed perfectly accurately, the result produced
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will be within 1ulp of the range generated by adding or subtracting
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will be within 1ulp of the range generated by adding or subtracting
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1ulp from the input values, where a 'ulp' is 2^(e-106) given the
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1ulp from the input values, where a 'ulp' is 2^(e-106) given the
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exponent 'e'. In the presence of cancellation, this may be
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exponent 'e'. In the presence of cancellation, this may be
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arbitrarily inaccurate. Subtraction is done by negation and addition.
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arbitrarily inaccurate. Subtraction is done by negation and addition.
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Multiplication is also performed using a library routine. Its result
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Multiplication is also performed using a library routine. Its result
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will be within 2ulp of the correct result.
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will be within 2ulp of the correct result.
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Division is also performed using a library routine. Its result will
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Division is also performed using a library routine. Its result will
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be within 3ulp of the correct result.
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be within 3ulp of the correct result.
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