| Long double format |
| ================== |
| |
| Each long double is made up of two IEEE doubles. The value of the |
| long double is the sum of the values of the two parts (except for |
| -0.0). The most significant part is required to be the value of the |
| long double rounded to the nearest double, as specified by IEEE. For |
| Inf values, the least significant part is required to be one of +0.0 |
| or -0.0. No other requirements are made; so, for example, 1.0 may be |
| represented as (1.0, +0.0) or (1.0, -0.0), and the low part of a NaN |
| is don't-care. |
| |
| Classification |
| -------------- |
| |
| A long double can represent any value of the form |
| s * 2^e * sum(k=0...105: f_k * 2^(-k)) |
| where 's' is +1 or -1, 'e' is between 1022 and -968 inclusive, f_0 is |
| 1, and f_k for k>0 is 0 or 1. These are the 'normal' long doubles. |
| |
| A long double can also represent any value of the form |
| s * 2^-968 * sum(k=0...105: f_k * 2^(-k)) |
| where 's' is +1 or -1, f_0 is 0, and f_k for k>0 is 0 or 1. These are |
| the 'subnormal' long doubles. |
| |
| There are four long doubles that represent zero, two that represent |
| +0.0 and two that represent -0.0. The sign of the high part is the |
| sign of the long double, and the sign of the low part is ignored. |
| |
| Likewise, there are four long doubles that represent infinities, two |
| for +Inf and two for -Inf. |
| |
| Each NaN, quiet or signalling, that can be represented as a 'double' |
| can be represented as a 'long double'. In fact, there are 2^64 |
| equivalent representations for each one. |
| |
| There are certain other valid long doubles where both parts are |
| nonzero but the low part represents a value which has a bit set below |
| 2^(e-105). These, together with the subnormal long doubles, make up |
| the denormal long doubles. |
| |
| Many possible long double bit patterns are not valid long doubles. |
| These do not represent any value. |
| |
| Limits |
| ------ |
| |
| The maximum representable long double is 2^1024-2^918. The smallest |
| *normal* positive long double is 2^-968. The smallest denormalised |
| positive long double is 2^-1074 (this is the same as for 'double'). |
| |
| Conversions |
| ----------- |
| |
| A double can be converted to a long double by adding a zero low part. |
| |
| A long double can be converted to a double by removing the low part. |
| |
| Comparisons |
| ----------- |
| |
| Two long doubles can be compared by comparing the high parts, and if |
| those compare equal, comparing the low parts. |
| |
| Arithmetic |
| ---------- |
| |
| The unary negate operation operates by negating the low and high parts. |
| |
| An absolute or absolute-negate operation must be done by comparing |
| against zero and negating if necessary. |
| |
| Addition and subtraction are performed using library routines. They |
| are not at present performed perfectly accurately, the result produced |
| will be within 1ulp of the range generated by adding or subtracting |
| 1ulp from the input values, where a 'ulp' is 2^(e-106) given the |
| exponent 'e'. In the presence of cancellation, this may be |
| arbitrarily inaccurate. Subtraction is done by negation and addition. |
| |
| Multiplication is also performed using a library routine. Its result |
| will be within 2ulp of the correct result. |
| |
| Division is also performed using a library routine. Its result will |
| be within 3ulp of the correct result. |
| |
| |
| Copyright (C) 2004-2025 Free Software Foundation, Inc. |
| |
| Copying and distribution of this file, with or without modification, |
| are permitted in any medium without royalty provided the copyright |
| notice and this notice are preserved. |
