优化单精度浮点计算：减小误差的策略

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"深入理解单精度浮点运算" 在计算机科学中，浮点运算是一种用于表示和处理实数的方法，特别是在需要快速计算和有限存储空间的场景下。单精度浮点运算是一种这样的标准，它在IEEE 754标准中定义，使用32位二进制来表示一个浮点数。这个格式包括符号位、指数部分和尾数部分，旨在提供足够的精度同时保持计算效率。单精度浮点数的结构如下： - 符号位（Sign）：1位，0表示正数，1表示负数。 - 指数值（Exponent）：8位，偏移量为127，因此指数范围是-126到127。 - 尾数（Fraction）：23位，不包含前导1，因此可以表示的有效数字范围是从1（不显示）到2（显示为1.0），总共24个二进制位。在进行单精度浮点运算时，有几个关键知识点需要理解： 1. **精度与误差**：由于浮点数的二进制表示不是完全精确的，特别是对于十进制下的有理数，可能会存在舍入误差。例如，1/3的二进制表示是无限循环的，因此在存储为单精度浮点数时会引入误差。为了减小误差，可以使用更高精度的浮点格式，如双精度，或者使用特定算法优化计算过程。 2. **浮点运算规则**：加法、减法、乘法和除法都有特定的规则。这些运算通常涉及到对齐尾数、调整指数以及处理溢出和下溢等特殊情况。例如，两个浮点数相加时，需要先将它们的指数对齐，然后相加尾数，最后处理结果的指数和符号。 3. **舍入策略**：当计算结果超出了单精度浮点数的表示范围，需要进行舍入。IEEE 754标准定义了多种舍入模式，如最近偶数、向零舍入、向上舍入和向下舍入，每种方式都可能影响结果的精确度。 4. **溢出与下溢**：指数过大或过小可能导致溢出或下溢。溢出通常会导致无穷大或NaN（非数字），而下溢则可能引发“渐近零”行为，其中非常小的数值被表示为0。 5. **比较浮点数**：由于浮点数的不精确性，直接比较两个浮点数是否相等可能产生错误的结果。通常建议使用一个微小的容差值（epsilon）来判断两个浮点数是否足够接近。 6. **性能考虑**：单精度浮点运算通常比双精度更快，因为涉及的数据量更小。在计算密集型应用中，如图形渲染或物理模拟，使用单精度可以提高速度。深入理解单精度浮点运算涉及到其内部结构、精度限制、运算规则以及处理误差的方法。这不仅有助于编写高效且精确的代码，也对理解和调试与浮点计算相关的程序问题至关重要。

ulps, even though 11.2 - 11.1 is exactly equal to .1

. The subtraction did not

introduce any error, but rather exposed the error introduced in the earlier

multiplications.

Benign cancellation occurs when subtracting exactly known quantities. If x and y

have no rounding error, then by Theorem 2 if the subtraction is done with a

guard digit, the difference x-y has a very small relative error (less than 2ε).

A formula that exhibits catastrophic cancellation can sometimes be rearranged

to eliminate the problem. Again consider the quadratic formula

(4)

When , then does not involve a cancellation and .

But the other addition (subtraction) in one of the formulas will have a

catastrophic cancellation. To avoid this, multiply the numerator and

denominator of r

(and similarly for r

) to obtain

(5)

If and , then computing r

using formula (4) will involve a

cancellation. Therefore, use (5) for computing r

and (4) for r

. On the other

hand, if b < 0, use (4) for computing r

and (5) for r

The expression x

- y

is another formula that exhibits catastrophic cancellation.

It is more accurate to evaluate it as (x - y)(x + y).

Unlike the quadratic formula,

this improved form still has a subtraction, but it is a benign cancellation of

quantities without rounding error, not a catastrophic one. By Theorem 2, the

6. 700, not 70. Since .1 - .0292 = .0708, the error in terms of ulp(0.0292) is 708 ulps. -- Ed.

b− b

4ac−+

b− b

4ac−−

ac» b

4ac−

4ac− b≈

b− b

4ac−−

b− b

4ac−−

b− b

4ac−+

ac» b 0>

10 What Every Computer Scientist Should Know About Floating-Point Arithmetic

relative error in x - y is at most 2ε. The same is true of x + y. Multiplying two

quantities with a small relative error results in a product with a small relative

error (see “Rounding Error” on page 50).

In order to avoid confusion between exact and computed values, the following

notation is used. Whereas x - y denotes the exact difference of x and y, x y

denotes the computed difference (i.e., with rounding error). Similarly ⊕, ⊗, and

denote computed addition, multiplication, and division, respectively. All

caps indicate the computed value of a function, as in LN(x) or SQRT(x). Lower

case functions and traditional mathematical notation denote their exact values

as in ln(x) and .

Although (xy)⊗ (x ⊕ y) is an excellent approximation to x

- y

, the

ﬂoating-point numbers x and y might themselves be approximations to some

true quantities and . For example, and might be exactly known decimal

numbers that cannot be expressed exactly in binary. In this case, even though

xy is a good approximation to x - y, it can have a huge relative error

compared to the true expression , and so the advantage of (x + y)(x - y)

over x

- y

is not as dramatic. Since computing (x + y)(x - y) is about the same

amount of work as computing x

- y

, it is clearly the preferred form in this

case. In general, however, replacing a catastrophic cancellation by a benign one

is not worthwhile if the expense is large because the input is often (but not

always) an approximation. But eliminating a cancellation entirely (as in the

quadratic formula) is worthwhile even if the data are not exact. Throughout

this paper, it will be assumed that the ﬂoating-point inputs to an algorithm are

exact and that the results are computed as accurately as possible.

The expression x

- y

is more accurate when rewritten as (x - y)(x + y) because

a catastrophic cancellation is replaced with a benign one. We next present more

interesting examples of formulas exhibiting catastrophic cancellation that can

be rewritten to exhibit only benign cancellation.

The area of a triangle can be expressed directly in terms of the lengths of its

sides a, b, and c as

(6)

7. Although the expression (x - y)(x + y) does not cause a catastrophic cancellation, it is slightly less accurate

than x

- y

if or . In this case, (x - y)(x + y) has three rounding errors, but x

- y

has only two

since the rounding error committed when computing the smaller of x

and y

does not affect the ﬁnal

subtraction.

xy» xy«

−

Assa−()sb−()sc−()where s, abc++()2⁄==

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