darwin-ldouble-format revision 169689
1284345SsjgLong double format 2284345Ssjg================== 3284345Ssjg 4284345Ssjg Each long double is made up of two IEEE doubles. The value of the 5290816Ssjglong double is the sum of the values of the two parts (except for 6284345Ssjg-0.0). The most significant part is required to be the value of the 7284349Ssjglong double rounded to the nearest double, as specified by IEEE. For 8284349SsjgInf values, the least significant part is required to be one of +0.0 9284349Ssjgor -0.0. No other requirements are made; so, for example, 1.0 may be 10287898Sbdreweryrepresented as (1.0, +0.0) or (1.0, -0.0), and the low part of a NaN 11287869Sbdreweryis don't-care. 12284345Ssjg 13297438SbdreweryClassification 14284650Ssjg-------------- 15287898Sbdrewery 16287636SsjgA long double can represent any value of the form 17287636Ssjg s * 2^e * sum(k=0...105: f_k * 2^(-k)) 18287636Ssjgwhere 's' is +1 or -1, 'e' is between 1022 and -968 inclusive, f_0 is 19287867Sbdrewery1, and f_k for k>0 is 0 or 1. These are the 'normal' long doubles. 20284650Ssjg 21287867SbdreweryA long double can also represent any value of the form 22284650Ssjg s * 2^-968 * sum(k=0...105: f_k * 2^(-k)) 23287636Ssjgwhere 's' is +1 or -1, f_0 is 0, and f_k for k>0 is 0 or 1. These are 24287887Sbdrewerythe 'subnormal' long doubles. 25287887Sbdrewery 26284650SsjgThere are four long doubles that represent zero, two that represent 27284345Ssjg+0.0 and two that represent -0.0. The sign of the high part is the 28284345Ssjgsign of the long double, and the sign of the low part is ignored. 29284345Ssjg 30284345SsjgLikewise, there are four long doubles that represent infinities, two 31284345Ssjgfor +Inf and two for -Inf. 32287966Sbdrewery 33284345SsjgEach NaN, quiet or signalling, that can be represented as a 'double' 34287871Sbdrewerycan be represented as a 'long double'. In fact, there are 2^64 35284345Ssjgequivalent representations for each one. 36284345Ssjg 37284345SsjgThere are certain other valid long doubles where both parts are 38284345Ssjgnonzero but the low part represents a value which has a bit set below 39284345Ssjg2^(e-105). These, together with the subnormal long doubles, make up 40284345Ssjgthe denormal long doubles. 41284345Ssjg 42284345SsjgMany possible long double bit patterns are not valid long doubles. 43284345SsjgThese do not represent any value. 44284345Ssjg 45284345SsjgLimits 46287885Sbdrewery------ 47287885Sbdrewery 48287885SbdreweryThe maximum representable long double is 2^1024-2^918. The smallest 49284345Ssjg*normal* positive long double is 2^-968. The smallest denormalised 50284345Ssjgpositive long double is 2^-1074 (this is the same as for 'double'). 51292124Sbr 52284345SsjgConversions 53284345Ssjg----------- 54284345Ssjg 55284345SsjgA double can be converted to a long double by adding a zero low part. 56284345Ssjg 57284345SsjgA long double can be converted to a double by removing the low part. 58292124Sbr 59284345SsjgComparisons 60284345Ssjg----------- 61284345Ssjg 62284345SsjgTwo long doubles can be compared by comparing the high parts, and if 63284345Ssjgthose compare equal, comparing the low parts. 64284345Ssjg 65284345SsjgArithmetic 66284345Ssjg---------- 67284345Ssjg 68284345SsjgThe unary negate operation operates by negating the low and high parts. 69284345Ssjg 70284345SsjgAn absolute or absolute-negate operation must be done by comparing 71284345Ssjgagainst zero and negating if necessary. 72284345Ssjg 73284345SsjgAddition and subtraction are performed using library routines. They 74284345Ssjgare not at present performed perfectly accurately, the result produced 75284345Ssjgwill be within 1ulp of the range generated by adding or subtracting 76284345Ssjg1ulp from the input values, where a 'ulp' is 2^(e-106) given the 77284345Ssjgexponent 'e'. In the presence of cancellation, this may be 78284345Ssjgarbitrarily inaccurate. Subtraction is done by negation and addition. 79284345Ssjg 80284345SsjgMultiplication is also performed using a library routine. Its result 81284345Ssjgwill be within 2ulp of the correct result. 82284345Ssjg 83284345SsjgDivision is also performed using a library routine. Its result will 84284345Ssjgbe within 3ulp of the correct result. 85284345Ssjg