精确计算java中float和double的精度

[本文相关的代码放在github上。地址为：https://github.com/VigourJiang/StructuredFloat]

Java中double类型的格式基本遵循IEEE 754标准。

虽然数学意义上的小数是连续的。但double只能表示当中的一些离散点。把这些离散点组成的集合记为S。S的大小还是有限的。假设要保存的小数P刚好在集合S内，那么double类型就能精确的表示P；否则double类型只能从集合S中找一个与P近期的离散点P'取代P。

以上表述对于float也成立。IEEE 754中float和double的表示策略全然同样。差别只体如今各个字段（指数字段、小数字段）的bit数量不同。

也就是说。float和double都是不精确的，因此偶尔会有一些奇怪的事情发生：

       double d = 1e16d;

       double d2 = d +1.0d - d;

       // 下面代码输出0.0，而不是1.0
       System.out.println(d2);

上述结果或许并不有趣。由于在学C语言的时候。你的老师就可能提醒过你。但有趣的是，float和double到底有多不精确？给定一个double d = XXX，与d最接近的另外两个double的数值是多少？

我用Java写了一个解析原始数据类型float和double的类StructuredFloat，给定一个float或者double数值P。它能够计算出与P的绝对值近期的、能够被float/double表示的浮点数。以下是使用方法：

	    double cur = 3.23d; // any valid double value
	    StructuredFloat sd = com.vigour.StructuredFloat.StructedFloatBuilder.buildDouble(cur);
	    StructuredFloat smaller = sd.absSmaller();
	    StructuredFloat bigger = sd.absBigger();

	    if(smaller != null){
	      double smaller_double = smaller.getBigDecimal().doubleValue();
	      // now you get the nearest double value whose absolute value is smaller
	    }
	    if(bigger!= null){
	      double bigger_double = bigger.getBigDecimal().doubleValue();
	      // now you get the nearest double value whose absolute value is bigger
	    }

以下是一些有趣的输出，能够看到，在1e16d附近double的精度就小于1了，在1e7f（一千万）附近。float的精度就等于1了，float果然难堪大用。

-------------Some Interesting Resule for Double--------------------
// 0.1d附近的double
Nearest smaller double:   0.09999999999999999167332731531132594682276248931884765625
Current double:           0.1000000000000000055511151231257827021181583404541015625
Nearest bigger double :   0.10000000000000001942890293094023945741355419158935546875

// 1.0d附近的double
Nearest smaller double:   0.99999999999999988897769753748434595763683319091796875
Current double:           1
Nearest bigger double :   1.0000000000000002220446049250313080847263336181640625

// 10.0d附近的double
Nearest smaller double:    9.9999999999999982236431605997495353221893310546875000
Current double:           10.00
Nearest bigger double :   10.0000000000000017763568394002504646778106689453125000

// 1e14d附近的double
Nearest smaller double:    99999999999999.9843750000000000000000000000000000000000000000000000
Current double:           100000000000000.00000000000000000000000000000000
Nearest bigger double :   100000000000000.0156250000000000000000000000000000000000000000000000

// 1e15d附近的double
Nearest smaller double:    999999999999999.8750000000000000000000000000000000000000000000000000
Current double:           1000000000000000.0000000000000000000000000000000000
Nearest bigger double :   1000000000000000.1250000000000000000000000000000000000000000000000000

// 1e16d附近的double
Nearest smaller double:    9999999999999998.0000000000000000000000000000000000000000000000000000
Current double:           10000000000000000.0000000000000000000000000000000000000
Nearest bigger double :   10000000000000002.0000000000000000000000000000000000000000000000000000

// 1e17d附近的double
Nearest smaller double:    99999999999999984.0000000000000000000000000000000000000000000000000000
Current double:           100000000000000000.000000000000000000000000000000000000000
Nearest bigger double :   100000000000000016.0000000000000000000000000000000000000000000000000000

// 1e304d附近的double
Nearest smaller double:    9999999999999998174371273630364736815867488735718786093662414371947263704524926751224722911637244940234972882804879769415602664816552507597839565690480126952738889402600333599657997758603312171995012866291845554976690497648524473448849371595248581587050582985041870802940253992811266476846330599148879872.0000000000000000000000000000000000000000000000000000
Current double:            9999999999999999392535525055364621860040287220117324953190771571323204563013233902843309257440507748436856118056162172578717193742636030530235798840866882774987301441682011041067710253162440905843719802548551599076639682550821832659549112269607949805346034918662572406407604380845959862074904348138143744.000000000000000000000000000000000000000000000000
Nearest bigger double :   10000000000000000610699776480364506904213085704515863812719128770699145421501541054461895603243770556638739353307444575741831722668719553462632031991253638597235713480763688482477422747721569639692426738805257643176588867453119191870248852943967318023641486852283274009874954768880653247303478097127407616.0000000000000000000000000000000000000000000000000000

-------------Some Interesting Resule for Float--------------------

// 0.1f附近的float
Nearest smaller float:    0.0999999940395355224609375
Current float:            0.100000001490116119384765625
Nearest bigger float:     0.10000000894069671630859375

// 1.0f附近的float
Nearest smaller float:    0.999999940395355224609375
Current float:            1
Nearest bigger float:     1.00000011920928955078125

// 10.0f附近的float
Nearest smaller float:     9.99999904632568359375000
Current float:            10.00
Nearest bigger float:     10.00000095367431640625000

// 1e5f附近的float
Nearest smaller float:     99999.99218750000000000000000
Current float:            100000.00000000000
Nearest bigger float:     100000.00781250000000000000000

// 1e6f附近的float
Nearest smaller float:     999999.93750000000000000000000
Current float:            1000000.0000000000000
Nearest bigger float:     1000000.06250000000000000000000

// 1e7f附近的float
Nearest smaller float:     9999999.00000000000000000000000
Current float:            10000000.0000000000000000
Nearest bigger float:     10000001.00000000000000000000000

// 1e8f附近的float
Nearest smaller float:     99999992.00000000000000000000000
Current float:            100000000.000000000000000000
Nearest bigger float:     100000008.00000000000000000000000

// 1e38f附近的float
Nearest smaller float:     99999986661652122824821048795547566080.00000000000000000000
Current float:             99999996802856924650656260769173209088.00000000000000000000000
Nearest bigger float:     100000006944061726476491472742798852096.0000000000000000000000