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 notes:ieee_754-1985 [2013/02/20 16:19]andy notes:ieee_754-1985 [2013/02/24 00:18] (current)andy [NaN] Both sides previous revision Previous revision 2013/02/24 00:18 andy [NaN] 2013/02/24 00:12 andy [Zero] 2013/02/24 00:11 andy [Normalised Values] 2013/02/24 00:09 andy [IEEE 754-1985] 2013/02/20 16:19 andy 2013/02/20 12:59 andy 2013/02/14 15:42 andy created Next revision Previous revision 2013/02/24 00:18 andy [NaN] 2013/02/24 00:12 andy [Zero] 2013/02/24 00:11 andy [Normalised Values] 2013/02/24 00:09 andy [IEEE 754-1985] 2013/02/20 16:19 andy 2013/02/20 12:59 andy 2013/02/14 15:42 andy created Line 5: Line 5: Briefly, floating point numbers are a way to represent a wide numeric range in a limited set of bits by storing a limited number of significant digits. As the absolute size of the number increases, the absolute precision decreases. Briefly, floating point numbers are a way to represent a wide numeric range in a limited set of bits by storing a limited number of significant digits. As the absolute size of the number increases, the absolute precision decreases. - The principle is similar to exponential notation for numbers (e.g. **3.523 x 106​**) except that IEEE floating point values use powers of 2 instead of 10. + The principle is similar to exponential notation for numbers (e.g. $3.523 \times ​10^6$) except that IEEE floating point values use powers of 2 instead of 10. Line 45: Line 45: The most common form of IEEE floating point numbers is the normalised form --- this is where the exponent has a value in the valid range (once the bias has been subtracted from the unsigned value stored). As explained above, the significand stores only the digits after the leading **1**, which is implicit. The most common form of IEEE floating point numbers is the normalised form --- this is where the exponent has a value in the valid range (once the bias has been subtracted from the unsigned value stored). As explained above, the significand stores only the digits after the leading **1**, which is implicit. - If the standard C library is available, the ''​[[man>​frexp|frexp()]]''​ function normalises a floating point value such that the fractional part will be in the range **0.5 ≤ x < 1.0**. Multiplying this value by **2** and reducing the exponent by **1** yields a value in the desired range **1.0 ≤ x < 2.0**. At this point the leading digit can then be discarded as the implicit leading **1** (see the [[#​Significand]] section for details). + If the standard C library is available, the ''​[[man>​frexp|frexp()]]''​ function normalises a floating point value such that the fractional part will be in the range $0.5 \le \times ​< 1.0$. Multiplying this value by **2** and reducing the exponent by **1** yields a value in the desired range $1.0 \le \times ​< 2.0$. At this point the leading digit can then be discarded as the implicit leading **1** (see the [[#​Significand]] section for details). If ''​frexp()''​ is available then it should be used, as it will likely use the underlying hardware representation to avoid expensive loops. However, a naive implementation can quite simply mimic its functionality --- the following version demonstrates the principle, but a production version would also need to check for special values (zero, NaN, infinities) as well as catching under- and overflows: If ''​frexp()''​ is available then it should be used, as it will likely use the underlying hardware representation to avoid expensive loops. However, a naive implementation can quite simply mimic its functionality --- the following version demonstrates the principle, but a production version would also need to check for special values (zero, NaN, infinities) as well as catching under- and overflows: Line 90: Line 90: ^ Significand | Zero | ^ Significand | Zero | - A value of exactly zero is represented by a exponent and significand of zero. The sign bit may be set or unset and IEEE 754 has the concept of both a positive and negative zero. For standard comparisons,​ however, these will both compare equal with zero, so the comparison ​**-0.0 < 0.0** yields **false**. + A value of exactly zero is represented by a exponent and significand of zero. The sign bit may be set or unset and IEEE 754 has the concept of both a positive and negative zero. For standard comparisons,​ however, these will both compare equal with zero, so the comparison ​$-0.0 < 0.0$ yields **false**. To determine the sign of a floating point value including zero, the ''​[[man>​copysign|copysign()]]''​ function can be used with a non-zero value, or the ''​[[man>​signbit|signbit()]]''​ macro can be used more directly on some platforms (not available on WinCE, for example). To determine the sign of a floating point value including zero, the ''​[[man>​copysign|copysign()]]''​ function can be used with a non-zero value, or the ''​[[man>​signbit|signbit()]]''​ macro can be used more directly on some platforms (not available on WinCE, for example). Line 118: Line 118: * Operations which are provided an existing NaN value as an argument. * Operations which are provided an existing NaN value as an argument. * Operations whose results are mathematically indeterminate - some examples are listed below: * Operations whose results are mathematically indeterminate - some examples are listed below: - * **0.0 / 0.0** and **±∞ ​/ ±∞** + * $0.0 / 0.0$ and $\pm\infty ​/ \pm\infty$ - * **0.0 x ±∞** + * $0.0 \times \pm\infty$ - * **∞ -- ∞** and equivalents + * $\infty ​- \infty$ ​and equivalents * Operations which yield complex results - some examples are listed below: * Operations which yield complex results - some examples are listed below: - * **√--n** + * $\sqrt{-n}$ - * **log(--n)** + * $\log{-n}$ - * **sin⁻¹(x)** or **cos⁻¹(x)** where **x < --1** or **x > 1** + * $\sin^{-1}{x}$ or $\cos^{-1}{x}$ where $x < -1$ or $x > 1$ ===== Limits ===== ===== Limits =====