浮点舍入错误

所以让我们尝试使用二进制表示法处理0.12345678。 所以0.1（二进制）是0.5（十进制），0.01（二进制） ）是0.25（十进制），并且 等等。 0.1 = 1/2 = 0.5 - 太大 0.01 = 1/4 = 0.25 - 太大了 0.001 = 1/8 = 0.125 - 太大了 0.0001 = 1/16 = 0.0625 - 太小了 0.00011 = 3/32 = 0.09375 - 太小了 0.000111 = 7/64 = 0.109375 - 太小了 0.0001111 = 15/128 = 0.1171875 - 太小了 0.00011111 = 31/256 = 0.12109375 - 太小 0.000111111 = 63/512 = 0.123046875 - 太小 0.0001111111 = 127/1024 = 0.1240234375 - 太大 0.00011111101 = 253/2048 = 0.12353515625 - 太大 0.000111111001 = 505/4096 = 0.123291015625 - 太小 0.0001111110011 = 1011/8192 = 0.1234130859375 - 太小了 0.00011111100111 = 2023/16384 = 0.12347412109375 - 太大了 0.000111111001101 = 4045/32768 = 0.123443603515625 - 太小了 0.0001111110011011 = 8091/65536 = 0.1234588623046875 - 太大 到目前为止，我们已经使用了16位。继续计算''，然后找出 如果要获得* b $ b 0.12345678的*精确*表示，你需要多少位数。您可能会对结果感到惊讶。Consider 1234.12345678It''s easy enough to deal with 1234. Here are the bits: 10011010010So let''s try to deal with 0.12345678, using binary notation.So 0.1 (binary) is 0.5 (decimal), 0.01 (binary) is 0.25 (decimal), andso on.0.1 = 1/2 = 0.5 - too large0.01 = 1/4 = 0.25 - too large0.001 = 1/8 = 0.125 - too large0.0001 = 1/16 = 0.0625 - too small0.00011 = 3/32 = 0.09375 - too small0.000111 = 7/64 = 0.109375 - too small0.0001111 = 15/128 = 0.1171875 - too small0.00011111 = 31/256 = 0.12109375 - too small0.000111111 = 63/512 = 0.123046875 - too small0.0001111111 = 127/1024 = 0.1240234375 - too large0.00011111101 = 253/2048 = 0.12353515625 - too large0.000111111001 = 505/4096 = 0.123291015625 - too small0.0001111110011 = 1011/8192 = 0.1234130859375 - too small0.00011111100111 = 2023/16384 = 0.12347412109375 - too large0.000111111001101 = 4045/32768 = 0.123443603515625 - too small0.0001111110011011 = 8091/65536 = 0.1234588623046875 - too largeSo far, we''ve used 16 bits on this. Keep on calculatin'', and find outhow many bits you need if you''re to get an *exact* representation of0.12345678. You might well be surprised by the result. 这取决于你想要达到的目标。 - Richard Heathfield Usenet是一个奇怪的地方 - dmr 29/7 / 1999http：//www.cpax.org.uk 电子邮件：rjh在上述域名， - www。That depends on what you want to achieve.--Richard Heathfield"Usenet is a strange place" - dmr 29/7/1999http://www.cpax.org.ukemail: rjh at the above domain, - www. 感谢您回复一个非常精细的例子，理查德。如果我告诉你，我会对你的二进制形式的非整数十进制数字的表示感兴趣，我会对你感到失望。 我知道带积分的二进制算术。我有时想知道并且 从不打扰我自己如何将小数表示为二进制文件。 我想了解你的例子。 我可以在表示中看到一个模式。 0.1是一半。 0.01是右移，你进一步减半。 0.001两个右移进一步将它减半等等。 你以0.011失去了我。我可以请你解释。Thank you for replying with a very elaborate example, Richard. I woulddisappoint you if I told you I am intrigued by the representation ofnon-integral decimal numbers in their binary form.I know binary arithmetic with integrals. I sometimes wondered andnever bothered myself as to how decimals were represented as binaries.I want to understand your example.I can see a pattern in the representation.0.1 is half.0.01 is a right shift and you further halve it.0.001 two right shifts further halving it and so on.You lost me at 0.011. Can I please request you to explain. 浮点数的表示是实现定义的。 IEEE 754很常见但不通用。The representation of floating-point numbers is implementation-defined.IEEE 754 is common but not universal. 您理解二进制整数。扩展概念。 二进制的13是1101（1 * 8 + 1 * 4 + 0 * 2 + 1 * 1）。 因此，每列代表一个乘数的一半，就像它左边的列 那样。 所以我们可以合理地想到二进制之外的列指向 代表一半，四分之一，八分之一等等。 所以0.11将是（半个+四分之一）=（四分之三）= 0.75 0.011将是（季度+第八）=（三分之八）= 0.375 等等。 当然，这并不一定是它们在内部存储的方式，但它确实可以让你很好地了解存储a所需的位数 特定精度的特定值。我建议您阅读 http：// docs .sun.com / source / 806-3568 / ncg_goldberg.html （标题：每个计算机科学家应该知道什么关于浮点数 Arithmetic） - Richard Heathfield Usenet是一个奇怪的地方 - dmr 29/7/1999 http://www.cpax.org.uk 电子邮件：rjh在上述域名中， - www。You understand binary integers. Extend the concept.13 in binary is 1101 (1 * eight + 1 * four + 0 * two + 1 * one).So each column represents a multiplier half as big as that of the columnto its left.So we might reasonably think of the columns beyond the binary point asrepresenting a half, a quarter, an eighth, etc.So 0.11 would be (half + quarter) = (three-quarters) = 0.750.011 would be (quarter + eighth) = (three-eighths) = 0.375and so on.That isn''t necessarily how they''re stored internally, of course, but itdoes give you a good idea of the number of bits you need for storing aparticular value to a particular precision. I recommend that you read http://docs.sun.com/source/806-3568/ncg_goldberg.html(Title: "What Every Computer Scientist Should Know About Floating-PointArithmetic")--Richard Heathfield"Usenet is a strange place" - dmr 29/7/1999 http://www.cpax.org.ukemail: rjh at the above domain, - www. 这篇关于浮点舍入错误的文章就介绍到这了，希望我们推荐的答案对大家有所帮助，也希望大家多多支持！