Python余弦函数精度 [英] Python cosine function precision

问题描述

从数学中我们知道90度角的余弦为0，但Python表示它的余弦要大得多。

From mathematics we know that the cosine of a 90 degree angle is 0 but Python says it's a bit more than that.

import math
math.cos(math.radians(90))
6.123233995736766e-17

Python和数字 0之间有什么关系？

What's the matter between Python and the number "0"?

推荐答案

在我之后重复：

计算机无法处理实数。

Computers cannot process real numbers.

Python使用double精度IEEE浮点数，精度四舍五入至53个二进制数，并且对范围有限制。由于π/ 2是一个无理数，因此计算机会将其四舍五入为最接近的可表示数字（或四舍五入为可表示的数字-有些运算具有精确的舍入，有些运算的误差大于1/2 ULP）。

Python uses double precision IEEE floats, which round to 53 binary digits of precision and have limits on range. Since π/2 is an irrational number, the computer rounds it to the nearest representable number (or to a close representable number — some operations have exact rounding, some have error greater than 1/2 ULP).

因此，您从未要求计算机计算 cos（π/ 2），您实际上是要求计算机计算 cos （π/ 2 +ε），其中ε是计算π/ 2的舍入误差。然后再次将结果四舍五入。

Therefore, you never asked the computer to compute cos(π/2), you really asked it to compute cos(π/2+ε), where ε is the roundoff error for computing π/2. The result is then rounded again.

可能性1：该程序执行符号计算，而不是数字计算。这适用于Mathematica和Maxima之类的程序，不适用于Excel。

Possibility 1: The program does symbolic computations, not numeric ones. This applies to programs like Mathematica and Maxima, not Excel.

可能性2：该程序正在隐藏数据（最有可能）。 Excel只会显示您要求的数字，例如，

Possibility 2: The program is hiding the data (most likely). Excel will only show you the digits you ask for, e.g.,

>>> '%.10f' % math.cos(math.radians(90))
'0.0000000000'

Python具有用于打印浮点数的微调功能，因此它们在往返文本和返回时均能幸免。这意味着默认情况下，Python打印的位数比 printf 多。

Python has a finely tuned function for printing out floats so that they survive a round trip to text and back. This means that Python prints more digits by default than, for example, printf.

可能性3： 您使用的程序有两个舍入错误被取消。

Possibility 3: The program you are using had two round-off errors that canceled.

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