Catalog

1.IEEE Floating point standard

1.1 Basic concepts

1.2 Normalized value

1.3 Denormalization value

1.4 Special values

1.5 Normalized value is the first positive integer that cannot be accurately expressed

2. Rounding and operation

2.1 Rounding mode ： Round to even

2.2 Multiplication

2.3 Addition operation

1.IEEE Floating point standard

1.1 Basic concepts

●IEEE 754 standard :

On 1985 establish , Before, every computer manufacturer designed its own rules , Support all mainstream CPU

● For the accuracy of numerical operation ：

Support rounding , Overflow and other operations , Defined on a small and consistent set of rules —— Quite elegant , Easy to understand .

● Mathematical form ：

(-1)^s * M * 2^E

s: Sign bit .

M: mantissa , It's a binary decimal , The scope is [1.0, 2.0).

E: Order code , Express 2 The power of . 

1.2 Normalized value

● Judge the condition : exp ≠ 000…0 and exp ≠ 111…1

● The level code field is interpreted as offset （ biased） A signed integer in the form of : E = Exp – Bias

●Exp: An unsigned number exp

●Bias = 2^(k-1) - 1, among k Is the order code number

● mantissa ： M = 1.xxx…x(2 Base number )

xxx…x: frac The bit of "s" means

minimum value 000…0 (M = 1.0)

Maximum ：111…1 (M = 2.0 – 0.0…01)

1.3 Denormalization value

● Judge the condition : exp = 000…0

● Order code value : E = –Bias + 1 ( Notice that it's not E = 0 – Bias)

● mantissa : M = 0.xxx…x2

●xxx…x: frac value

● Example

exp = 000…0, frac = 000…0（+0 And -0 There are differences ）

exp = 000…0, frac ≠ 000…0

1.4 Special values

● Judge the condition : exp = 111…1

● example : exp = 111…1, frac = 000…0

Express ∞ ( infinite ),s＝1 Negative time is infinite ,s＝0 Timing is infinite

Can express the result of overflow

E.g., 1.0/0.0 = −1.0/−0.0 = +∞, 1.0/−0.0 = −∞

● example : exp = 111…1, frac ≠ 000…0

It's not a number (NaN)

Some unspeakable numbers

E.g., sqrt(–1), ∞ − ∞, ∞ * 0

1.5 Normalized value is the first positive integer that cannot be accurately expressed

IEEE Able to express float The growth of the number is shown in the figure ：

Set order code yes M position , Mantissa has N position ：

When (2^M-1-Bias) < (N+1) when , Integers in the range can be accurately expressed ;

When (2^M-1-Bias) = (N+1) when , The first integer that cannot be accurately expressed is 2^(N+1)+1. 

2. Rounding and operation

2.1 Rounding mode ： Round to even

Operation idea ： First calculate the exact result , Then get the approximate result by rounding .

2.2 Multiplication

{(-1)^s1 * M1 * 2^E1} * {(-1)^s2 * M2 * 2^E2}

● Precise results ：(-1)^s * M * 2^E

▲s = s1 + s2

▲M = M1 * M2

▲E = E1 + E2

● adjustment

▲ If M ≥ 2, M Moves to the right one , E＝E+1

▲ If E Out of range , overflow .

▲ take M Round to frac Range of digits .

2.3 Addition operation

1） Antithetic order ： Small steps to large ones

When the order codes of two numbers are the same , Indicates that the positions of the decimal points of two numbers are aligned , At this point, you can add . Before adding , It is necessary to align the order , Small steps align with large ones . The order code of small order plus K Equal to the order code of large order , At the same time, the mantissa of the small order should be shifted to the right K position .

2） Add mantissa .

3） Normalize the result and round .

4） Judge whether it overflows according to the order code .