Java中的半精度浮点
是否有可以在IEEE 754半精度数字上执行计算或者将它们转换为双精度和从双精度转换的Java库？
这些方法中的任何一种都是合适的：
- 将数字保持为半精度格式，并使用整数算术和位进行计算（ MicroFloat用于单精度和双精度）
- 以单精度或双精度执行所有计算，转换为半精度进行传输（在这种情况下，我需要的是经过良好测试的转换函数。）
编辑 ：转换需要100％准确 – 输入文件中有很多NaN，无穷大和次正规。
相关问题，但对于JavaScript： 在Javascript中解压缩半精度浮点数
您可以使用Float.intBitsToFloat()和Float.floatToIntBits()将它们转换为原始浮点值和从原始浮点值转换它们。 如果你可以使用截断精度（而不是舍入），那么转换应该可以通过几个位移来实现。
我现在已经付出了更多的努力，结果并没有我在开始时预期的那么简单。 这个版本现在在我能想象的每个方面都经过测试和validation，我非常有信心它可以为所有可能的输入值生成精确的结果。 它支持任一方向的精确舍入和次正规转换。
// ignores the higher 16 bits public static float toFloat( int hbits ) { int mant = hbits & 0x03ff; // 10 bits mantissa int exp = hbits & 0x7c00; // 5 bits exponent if( exp == 0x7c00 ) // NaN/Inf exp = 0x3fc00; // -> NaN/Inf else if( exp != 0 ) // normalized value { exp += 0x1c000; // exp - 15 + 127 if( mant == 0 && exp > 0x1c400 ) // smooth transition return Float.intBitsToFloat( ( hbits & 0x8000 ) << 16 | exp << 13 | 0x3ff ); } else if( mant != 0 ) // && exp==0 -> subnormal { exp = 0x1c400; // make it normal do { mant <<= 1; // mantissa * 2 exp -= 0x400; // decrease exp by 1 } while( ( mant & 0x400 ) == 0 ); // while not normal mant &= 0x3ff; // discard subnormal bit } // else +/-0 -> +/-0 return Float.intBitsToFloat( // combine all parts ( hbits & 0x8000 ) << 16 // sign << ( 31 - 15 ) | ( exp | mant ) << 13 ); // value << ( 23 - 10 ) }
// returns all higher 16 bits as 0 for all results public static int fromFloat( float fval ) { int fbits = Float.floatToIntBits( fval ); int sign = fbits >>> 16 & 0x8000; // sign only int val = ( fbits & 0x7fffffff ) + 0x1000; // rounded value if( val >= 0x47800000 ) // might be or become NaN/Inf { // avoid Inf due to rounding if( ( fbits & 0x7fffffff ) >= 0x47800000 ) { // is or must become NaN/Inf if( val < 0x7f800000 ) // was value but too large return sign | 0x7c00; // make it +/-Inf return sign | 0x7c00 | // remains +/-Inf or NaN ( fbits & 0x007fffff ) >>> 13; // keep NaN (and Inf) bits } return sign | 0x7bff; // unrounded not quite Inf } if( val >= 0x38800000 ) // remains normalized value return sign | val - 0x38000000 >>> 13; // exp - 127 + 15 if( val < 0x33000000 ) // too small for subnormal return sign; // becomes +/-0 val = ( fbits & 0x7fffffff ) >>> 23; // tmp exp for subnormal calc return sign | ( ( fbits & 0x7fffff | 0x800000 ) // add subnormal bit + ( 0x800000 >>> val - 102 ) // round depending on cut off >>> 126 - val ); // div by 2^(1-(exp-127+15)) and >> 13 | exp=0 }
与本书相比，我实现了两个小扩展，因为16位浮点数的一般精度相当低，这可能使浮点格式的固有exception在视觉上可感知，而较大的浮点类型由于其足够的精度而通常不会被注意到。
第一个是toFloat()函数中的这两行：
if( mant == 0 && exp > 0x1c400 ) // smooth transition return Float.intBitsToFloat( ( hbits & 0x8000 ) << 16 | exp << 13 | 0x3ff );
类型大小的正常范围内的浮点数采用指数，因此精度为值的大小。 但这并非顺利采用，而是按步骤进行：切换到下一个更高的指数会导致精度降低一半。 对于尾数的所有值，精度现在保持相同，直到下一个跳到下一个更高的指数。 上面的扩展代码通过返回该特定半浮点值的覆盖32位浮点范围的地理中心中的值，使这些转换更平滑。 每个正常的半浮点值都精确映射到8192个32位浮点值。 返回的值应该恰好位于这些值的中间。 但是在半浮点指数的转变处，较低的4096值具有两倍于上4096值的精度，因此覆盖的数量空间仅为另一侧的一半。 所有这些8192 32位浮点值映射到相同的半浮点值，因此将半浮点数转换为32位并返回将导致相同的半浮点值，无论选择了哪个8192 中间 32位值。 现在，扩展现在在转换时产生更平滑的半步长因子sqrt（2），如右图所示，而左图像应该将锐步步骤可视化为2而没有抗锯齿。 您可以安全地从代码中删除这两行以获得标准行为。
covered number space on either side of the returned value: 6.0E-8 ####### ########## 4.5E-8 | # 3.0E-8 ######### ########
第二个扩展名是fromFloat()函数：
{ // avoid Inf due to rounding if( ( fbits & 0x7fffffff ) >= 0x47800000 ) ... return sign | 0x7bff; // unrounded not quite Inf }
此扩展稍微扩展了半浮点格式的数字范围，方法是保存一些32位值，从而升级为Infinity。 受影响的值是那些在没有舍入的情况下小于无穷大的值，并且由于舍入而仅变为无穷大。 如果您不想要此扩展名，可以安全地删除上面显示的行。
我试图尽可能地优化fromFloat()函数中正常值的路径，这使得它由于使用了预先计算和未移位的常量而变得不那么可读。 我没有在'toFloat（）'中投入太多精力，因为它无论如何都不会超过查找表的性能。 因此，如果速度真的很重要，可以使用toFloat()函数只填充带有0x10000元素的静态查找表，而不是使用此表进行实际转换。 使用当前的x64服务器虚拟机大约快3倍，使用x86客户端虚拟机大约快5倍。
我把代码放在公共领域。
x4u的代码将值1正确编码为0x3c00（参考： https ：//en.wikipedia.org/wiki/Half-precision_floating-point_format）。 但是具有平滑性改进的解码器将其解码为1.000122。 维基百科条目表示整数值0..2048可以准确表示。 不太好…
从toFloat代码中删除"| 0x3ff"可确保toFloat(fromFloat(k)) == k表示-2048..2048范围内的整数k，可能代价是平滑度稍差。
在我看到这里发布的解决方案之前，我已经掀起了一些简单的事情：
public static float toFloat(int nHalf) { int S = (nHalf >>> 15) & 0x1; int E = (nHalf >>> 10) & 0x1F; int T = (nHalf ) & 0x3FF; E = E == 0x1F ? 0xFF // it's 2^w-1; it's all 1's, so keep it all 1's for the 32-bit float : E - 15 + 127; // adjust the exponent from the 16-bit bias to the 32-bit bias // sign S is now bit 31 // exp E is from bit 30 to bit 23 // scale T by 13 binary digits (it grew from 10 to 23 bits) return Float.intBitsToFloat(S << 31 | E << 23 | T << 13); }
不过，我确实喜欢其他发布解决方案中的方法。 以供参考：
// notes from the IEEE-754 specification: // left to right bits of a binary floating point number: // size bit ids name description // ---------- ------------ ---- --------------------------- // 1 bit S sign // w bits E[0]..E[w-1] E biased exponent // t=p-1 bits d[1]..d[p-1] T trailing significant field // The range of the encoding's biased exponent E shall include: // ― every integer between 1 and 2^w − 2, inclusive, to encode normal numbers // ― the reserved value 0 to encode ±0 and subnormal numbers // ― the reserved value 2w − 1 to encode +/-infinity and NaN // The representation r of the floating-point datum, and value v of the floating-point datum // represented, are inferred from the constituent fields as follows: // a) If E == 2^w−1 and T != 0, then r is qNaN or sNaN and v is NaN regardless of S // b) If E == 2^w−1 and T == 0, then r=v=(−1)^S * (+infinity) // c) If 1 <= E <= 2^w−2, then r is (S, (E−bias), (1 + 2^(1−p) * T)) // the value of the corresponding floating-point number is // v = (−1)^S * 2^(E−bias) * (1 + 2^(1−p) * T) // thus normal numbers have an implicit leading significand bit of 1 // d) If E == 0 and T != 0, then r is (S, emin, (0 + 2^(1−p) * T)) // the value of the corresponding floating-point number is // v = (−1)^S * 2^emin * (0 + 2^(1−p) * T) // thus subnormal numbers have an implicit leading significand bit of 0 // e) If E == 0 and T ==0, then r is (S, emin, 0) and v = (−1)^S * (+0) // parameter bin16 bin32 // -------------------------------------------- ----- ----- // k, storage width in bits 16 32 // p, precision in bits 11 24 // emax, maxiumum exponent e 15 127 // bias, Ee 15 127 // sign bit 1 1 // w, exponent field width in bits 5 8 // t, trailing significant field width in bits 10 23
我创建了一个名为HalfPrecisionFloat的java类，它使用x4u的解决方案。 该类具有便捷方法和错误检查。 它更进一步，并且具有从2字节半精度值返回Double和Float的方法。
希望这会对某人有所帮助。
==>
import java.nio.ByteBuffer; /** * Accepts various forms of a floating point half-precision (2 byte) number * and contains methods to convert to a * full-precision floating point number Float and Double instance. * * This implemention was inspired by x4u who is a user contributing * to stackoverflow.com. * (https://stackoverflow.com/users/237321/x4u). * * @author dougestep */ public class HalfPrecisionFloat { private short halfPrecision; private Float fullPrecision; /** * Creates an instance of the class from the supplied the supplied * byte array. The byte array must be exactly two bytes in length. * * @param bytes the two-byte byte array. */ public HalfPrecisionFloat(byte[] bytes) { if (bytes.length != 2) { throw new IllegalArgumentException("The supplied byte array " + "must be exactly two bytes in length"); } final ByteBuffer buffer = ByteBuffer.wrap(bytes); this.halfPrecision = buffer.getShort(); } /** * Creates an instance of this class from the supplied short number. * * @param number the number defined as a short. */ public HalfPrecisionFloat(final short number) { this.halfPrecision = number; this.fullPrecision = toFullPrecision(); } /** * Creates an instance of this class from the supplied * full-precision floating point number. * * @param number the float number. */ public HalfPrecisionFloat(final float number) { if (number > Short.MAX_VALUE) { throw new IllegalArgumentException("The supplied float is too " + "large for a two byte representation"); } if (number < Short.MIN_VALUE) { throw new IllegalArgumentException("The supplied float is too " + "small for a two byte representation"); } final int val = fromFullPrecision(number); this.halfPrecision = (short) val; this.fullPrecision = number; } /** * Returns the half-precision float as a number defined as a short. * * @return the short. */ public short getHalfPrecisionAsShort() { return halfPrecision; } /** * Returns a full-precision floating pointing number from the * half-precision value assigned on this instance. * * @return the full-precision floating pointing number. */ public float getFullFloat() { if (fullPrecision == null) { fullPrecision = toFullPrecision(); } return fullPrecision; } /** * Returns a full-precision double floating point number from the * half-precision value assigned on this instance. * * @return the full-precision double floating pointing number. */ public double getFullDouble() { return new Double(getFullFloat()); } /** * Returns the full-precision float number from the half-precision * value assigned on this instance. * * @return the full-precision floating pointing number. */ private float toFullPrecision() { int mantisa = halfPrecision & 0x03ff; int exponent = halfPrecision & 0x7c00; if (exponent == 0x7c00) { exponent = 0x3fc00; } else if (exponent != 0) { exponent += 0x1c000; if (mantisa == 0 && exponent > 0x1c400) { return Float.intBitsToFloat( (halfPrecision & 0x8000) << 16 | exponent << 13 | 0x3ff); } } else if (mantisa != 0) { exponent = 0x1c400; do { mantisa <<= 1; exponent -= 0x400; } while ((mantisa & 0x400) == 0); mantisa &= 0x3ff; } return Float.intBitsToFloat( (halfPrecision & 0x8000) << 16 | (exponent | mantisa) << 13); } /** * Returns the integer representation of the supplied * full-precision floating pointing number. * * @param number the full-precision floating pointing number. * @return the integer representation. */ private int fromFullPrecision(final float number) { int fbits = Float.floatToIntBits(number); int sign = fbits >>> 16 & 0x8000; int val = (fbits & 0x7fffffff) + 0x1000; if (val >= 0x47800000) { if ((fbits & 0x7fffffff) >= 0x47800000) { if (val < 0x7f800000) { return sign | 0x7c00; } return sign | 0x7c00 | (fbits & 0x007fffff) >>> 13; } return sign | 0x7bff; } if (val >= 0x38800000) { return sign | val - 0x38000000 >>> 13; } if (val < 0x33000000) { return sign; } val = (fbits & 0x7fffffff) >>> 23; return sign | ((fbits & 0x7fffff | 0x800000) + (0x800000 >>> val - 102) >>> 126 - val); }
这是unit testing
import org.junit.Assert; import org.junit.Test; import java.nio.ByteBuffer; public class TestHalfPrecision { private byte[] simulateBytes(final float fullPrecision) { HalfPrecisionFloat halfFloat = new HalfPrecisionFloat(fullPrecision); short halfShort = halfFloat.getHalfPrecisionAsShort(); ByteBuffer buffer = ByteBuffer.allocate(2); buffer.putShort(halfShort); return buffer.array(); } @Test public void testHalfPrecisionToFloatApproach() { final float startingValue = 1.2f; final float closestValue = 1.2001953f; final short shortRepresentation = (short) 15565; byte[] bytes = simulateBytes(startingValue); HalfPrecisionFloat halfFloat = new HalfPrecisionFloat(bytes); final float retFloat = halfFloat.getFullFloat(); Assert.assertEquals(new Float(closestValue), new Float(retFloat)); HalfPrecisionFloat otherWay = new HalfPrecisionFloat(retFloat); final short shrtValue = otherWay.getHalfPrecisionAsShort(); Assert.assertEquals(new Short(shortRepresentation), new Short(shrtValue)); HalfPrecisionFloat backAgain = new HalfPrecisionFloat(shrtValue); final float backFlt = backAgain.getFullFloat(); Assert.assertEquals(new Float(closestValue), new Float(backFlt)); HalfPrecisionFloat dbl = new HalfPrecisionFloat(startingValue); final double retDbl = dbl.getFullDouble(); Assert.assertEquals(new Double(startingValue), new Double(retDbl)); } @Test(expected = IllegalArgumentException.class) public void testInvalidByteArray() { ByteBuffer buffer = ByteBuffer.allocate(4); buffer.putFloat(Float.MAX_VALUE); byte[] bytes = buffer.array(); new HalfPrecisionFloat(bytes); } @Test(expected = IllegalArgumentException.class) public void testInvalidMaxFloat() { new HalfPrecisionFloat(Float.MAX_VALUE); } @Test(expected = IllegalArgumentException.class) public void testInvalidMinFloat() { new HalfPrecisionFloat(-35000); } @Test public void testCreateWithShort() { HalfPrecisionFloat sut = new HalfPrecisionFloat(Short.MAX_VALUE); Assert.assertEquals(Short.MAX_VALUE, sut.getHalfPrecisionAsShort()); } }
我对小的正浮点数感兴趣，所以我用12位尾数，无符号位和4位指数构建了这个变量，偏差为15 ，这样它可以表示0到1.00之间的数字（独占）非常好。 它在尾数额外有2位分辨率，但相同的指数低。
public static float toFloat(int hbits) { int mant = hbits & 0x0fff; // 12 bits mantissa int exp = (hbits & 0xf000) >>> 12; // 4 bits exponent if (exp == 0xf) { exp = 0xff; } else { if (exp != 0) { // normal value exp += 127 - 15; } else { // subnormal value if (mant != 0) { // not zero exp += 127 - 15; // make it noral exp++; do { mant <<= 1; exp--; } while ((mant & 0x1000) == 0); mant &= 0x0fff; } } } return Float.intBitsToFloat(exp << 23 | mant << 11); } public static int fromFloat(float fval) { int fbits = Float.floatToIntBits( fval ); int val = ( fbits & 0x7fffffff ) + 0x400; // rounded value if( val < 0x32000000 ) // too small for subnormal or negative return 0; // becomes 0 if( val >= 0x47800000 ) // might be or become NaN/Inf { // avoid Inf due to rounding if( ( fbits & 0x7fffffff ) >= 0x47800000 ) { // is or must become NaN/Inf if( val < 0x7f800000 ) // was value but too large return 0xf000; // make it +/-Inf return 0xf000 | // remains +/-Inf or NaN ( fbits & 0x007fffff ) >>> 11; // keep NaN (and Inf) bits } return 0x7fff; // unrounded not quite Inf } if( val >= 0x38800000 ) // remains normalized value return val - 0x38000000 >>> 11; // exp - 127 + 15 val = ( fbits & 0x7fffffff ) >>> 23; // tmp exp for subnormal calc return ( ( fbits & 0x7f_ffff | 0x80_0000 ) // add subnormal bit + ( 0x800000 >>> val - 100 ) // round depending on cut off >>> 124 - val ); // div by 2^(1-(exp-127+15)) and >> 11 | exp=0 }
测试给出：
Smallest subnormal float : 0.0000000149 Largest subnormal float : 0.0000610203 Smallest normal float : 0.0000610352 Smallest normal float + ups: 0.0000610501 E=1, M=fff (max) : 0.0001220554 Largest normal float : 0.0078115463
法线：
0.9990000129 => 3f7fbe77 => eff8 => 0.9990234375 | error: 0.002% 0.8991000056 => 3f662b6b => ecc5 => 0.8990478516 | error: 0.006% 0.8091899753 => 3f4f2713 => e9e5 => 0.8092041016 | error: 0.002% 0.7282709479 => 3f3a6ff7 => e74e => 0.7282714844 | error: 0.000% 0.6554438472 => 3f27cb2b => e4f9 => 0.6553955078 | error: 0.007% 0.5898994207 => 3f1703a6 => e2e0 => 0.5898437500 | error: 0.009% 0.5309094787 => 3f07e9af => e0fd => 0.5308837891 | error: 0.005% 0.4778185189 => 3ef4a4a1 => de95 => 0.4778442383 | error: 0.005% 0.4300366640 => 3edc2dc4 => db86 => 0.4300537109 | error: 0.004% 0.3870329857 => 3ec62930 => d8c5 => 0.3870239258 | error: 0.002% 0.3483296633 => 3eb25844 => d64b => 0.3483276367 | error: 0.001% 0.3134966791 => 3ea082a3 => d410 => 0.3134765625 | error: 0.006% 0.2821469903 => 3e907592 => d20f => 0.2821655273 | error: 0.007% 0.2539322972 => 3e82036a => d040 => 0.2539062500 | error: 0.010% 0.2285390645 => 3e6a0625 => cd41 => 0.2285461426 | error: 0.003% 0.2056851536 => 3e529f21 => ca54 => 0.2056884766 | error: 0.002% 0.1851166338 => 3e3d8f37 => c7b2 => 0.1851196289 | error: 0.002% 0.1666049659 => 3e2a9a7e => c553 => 0.1665954590 | error: 0.006% 0.1499444693 => 3e198b0b => c331 => 0.1499328613 | error: 0.008% 0.1349500120 => 3e0a3056 => c146 => 0.1349487305 | error: 0.001% 0.1214550063 => 3df8bd67 => bf18 => 0.1214599609 | error: 0.004% 0.1093095019 => 3ddfdda9 => bbfc => 0.1093139648 | error: 0.004% 0.0983785465 => 3dc97ab1 => b92f => 0.0983734131 | error: 0.005% 0.0885406882 => 3db554d2 => b6ab => 0.0885467529 | error: 0.007% 0.0796866193 => 3da332bd => b466 => 0.0796813965 | error: 0.007% 0.0717179552 => 3d92e0dd => b25c => 0.0717163086 | error: 0.002% 0.0645461604 => 3d8430c7 => b086 => 0.0645446777 | error: 0.002% 0.0580915436 => 3d6df166 => adbe => 0.0580902100 | error: 0.002% 0.0522823893 => 3d56260f => aac5 => 0.0522842407 | error: 0.004% 0.0470541492 => 3d40bbda => a817 => 0.0470504761 | error: 0.008% 0.0423487313 => 3d2d75dd => a5af => 0.0423507690 | error: 0.005% 0.0381138586 => 3d1c1d47 => a384 => 0.0381164551 | error: 0.007% 0.0343024731 => 3d0c80c0 => a190 => 0.0343017578 | error: 0.002% 0.0308722258 => 3cfce7c0 => 9f9d => 0.0308723450 | error: 0.000% 0.0277850032 => 3ce39d60 => 9c74 => 0.0277862549 | error: 0.005% 0.0250065029 => 3cccda70 => 999b => 0.0250053406 | error: 0.005% 0.0225058515 => 3cb85e31 => 970c => 0.0225067139 | error: 0.004% 0.0202552658 => 3ca5ee5f => 94be => 0.0202560425 | error: 0.004% 0.0182297379 => 3c955688 => 92ab => 0.0182304382 | error: 0.004% 0.0164067633 => 3c86677a => 90cd => 0.0164070129 | error: 0.002% 0.0147660868 => 3c71ed75 => 8e3e => 0.0147666931 | error: 0.004% 0.0132894777 => 3c59bc1c => 8b38 => 0.0132904053 | error: 0.007% 0.0119605297 => 3c43f619 => 887f => 0.0119609833 | error: 0.004% 0.0107644768 => 3c305d7d => 860c => 0.0107650757 | error: 0.006% 0.0096880291 => 3c1eba8a => 83d7 => 0.0096874237 | error: 0.006% 0.0087192263 => 3c0edb16 => 81db => 0.0087184906 | error: 0.008% 0.0078473035 => 3c0091fa => 8012 => 0.0078468323 | error: 0.006% 0.0070625730 => 3be76d28 => 7cee => 0.0070629120 | error: 0.005% 0.0063563157 => 3bd048a4 => 7a09 => 0.0063562393 | error: 0.001% 0.0057206838 => 3bbb7493 => 776f => 0.0057210922 | error: 0.007% 0.0051486152 => 3ba8b5b7 => 7517 => 0.0051488876 | error: 0.005% 0.0046337536 => 3b97d6be => 72fb => 0.0046339035 | error: 0.003% 0.0041703782 => 3b88a7ab => 7115 => 0.0041704178 | error: 0.001% 0.0037533403 => 3b75fa9a => 6ebf => 0.0037531853 | error: 0.004% 0.0033780062 => 3b5d618a => 6bac => 0.0033779144 | error: 0.003% 0.0030402055 => 3b473e2f => 68e8 => 0.0030403137 | error: 0.004% 0.0027361847 => 3b335190 => 666a => 0.0027360916 | error: 0.003% 0.0024625661 => 3b216301 => 642c => 0.0024623871 | error: 0.007% 0.0022163095 => 3b113f81 => 6228 => 0.0022163391 | error: 0.001% 0.0019946785 => 3b02b927 => 6057 => 0.0019946098 | error: 0.003% 0.0017952106 => 3aeb4d46 => 5d6a => 0.0017952919 | error: 0.005% 0.0016156895 => 3ad3c58b => 5a79 => 0.0016157627 | error: 0.005% 0.0014541205 => 3abe9830 => 57d3 => 0.0014541149 | error: 0.000% 0.0013087085 => 3aab88f8 => 5571 => 0.0013086796 | error: 0.002% 0.0011778376 => 3a9a61ac => 534c => 0.0011777878 | error: 0.004% 0.0010600538 => 3a8af181 => 515e => 0.0010600090 | error: 0.004% 0.0009540484 => 3a7a191b => 4f43 => 0.0009540319 | error: 0.002% 0.0008586436 => 3a611698 => 4c23 => 0.0008586645 | error: 0.002% 0.0007727792 => 3a4a9455 => 4953 => 0.0007728338 | error: 0.007% 0.0006955012 => 3a36524c => 46ca => 0.0006954670 | error: 0.005% 0.0006259511 => 3a2416de => 4483 => 0.0006259680 | error: 0.003% 0.0005633560 => 3a13ae2e => 4276 => 0.0005633831 | error: 0.005% 0.0005070204 => 3a04e990 => 409d => 0.0005069971 | error: 0.005% 0.0004563183 => 39ef3e03 => 3de8 => 0.0004563332 | error: 0.003% 0.0004106865 => 39d75169 => 3aea => 0.0004106760 | error: 0.003% 0.0003696179 => 39c1c945 => 3839 => 0.0003696084 | error: 0.003% 0.0003326561 => 39ae6857 => 35cd => 0.0003326535 | error: 0.001% 0.0002993904 => 399cf781 => 339f => 0.0002993941 | error: 0.001% 0.0002694514 => 398d4527 => 31a9 => 0.0002694726 | error: 0.008% 0.0002425062 => 397e4946 => 2fc9 => 0.0002425015 | error: 0.002% 0.0002182556 => 3964db8b => 2c9b => 0.0002182424 | error: 0.006% 0.0001964300 => 394df8ca => 29bf => 0.0001964271 | error: 0.001% 0.0001767870 => 39395fe9 => 272c => 0.0001767874 | error: 0.000% 0.0001591083 => 3926d651 => 24db => 0.0001591146 | error: 0.004% 0.0001431975 => 39162749 => 22c5 => 0.0001432002 | error: 0.002% 0.0001288777 => 3907235b => 20e4 => 0.0001288652 | error: 0.010% 0.0001159900 => 38f33fa3 => 1e68 => 0.0001159906 | error: 0.001% 0.0001043910 => 38daec79 => 1b5e => 0.0001043975 | error: 0.006% 0.0000939519 => 38c50806 => 18a1 => 0.0000939518 | error: 0.000% 0.0000845567 => 38b15405 => 162b => 0.0000845641 | error: 0.009% 0.0000761010 => 389f986b => 13f3 => 0.0000761002 | error: 0.001% 0.0000684909 => 388fa2c6 => 11f4 => 0.0000684857 | error: 0.008% 0.0000616418 => 388145b2 => 1029 => 0.0000616461 | error: 0.007%
对于次正常测试：
0.0000554776 => 3868b0a6 => 0e8b => 0.0000554770 | error: 0.001% 0.0000499299 => 38516bc8 => 0d17 => 0.0000499338 | error: 0.008% 0.0000449369 => 383c7a9a => 0bc8 => 0.0000449419 | error: 0.011% 0.0000404432 => 3829a18a => 0a9a => 0.0000404418 | error: 0.004% 0.0000363989 => 3818aafc => 098b => 0.0000364035 | error: 0.013% 0.0000327590 => 380966af => 0896 => 0.0000327528 | error: 0.019% 0.0000294831 => 37f7526e => 07bb => 0.0000294894 | error: 0.021% 0.0000265348 => 37de96fc => 06f5 => 0.0000265390 | error: 0.016% 0.0000238813 => 37c854af => 0643 => 0.0000238866 | error: 0.022% 0.0000214932 => 37b44c37 => 05a2 => 0.0000214875 | error: 0.026% 0.0000193438 => 37a24498 => 0512 => 0.0000193417 | error: 0.011% 0.0000174095 => 37920a89 => 0490 => 0.0000174046 | error: 0.028% 0.0000156685 => 37836fe1 => 041b => 0.0000156611 | error: 0.047% 0.0000141017 => 376c962e => 03b2 => 0.0000140965 | error: 0.037% 0.0000126915 => 3754ed8f => 0354 => 0.0000126958 | error: 0.034% 0.0000114223 => 373fa29a => 02ff => 0.0000114292 | error: 0.060% 0.0000102801 => 372c78be => 02b2 => 0.0000102818 | error: 0.016% 0.0000092521 => 371b3978 => 026d => 0.0000092536 | error: 0.016% 0.0000083269 => 370bb3b9 => 022f => 0.0000083297 | error: 0.034% 0.0000074942 => 36fb76b3 => 01f7 => 0.0000074953 | error: 0.014% 0.0000067448 => 36e2513a => 01c5 => 0.0000067502 | error: 0.081% 0.0000060703 => 36cbaf81 => 0197 => 0.0000060648 | error: 0.091% 0.0000054633 => 36b75127 => 016f => 0.0000054687 | error: 0.100% 0.0000049169 => 36a4fc3c => 014a => 0.0000049174 | error: 0.009% 0.0000044253 => 36947c9c => 0129 => 0.0000044256 | error: 0.009% 0.0000039827 => 3685a359 => 010b => 0.0000039786 | error: 0.103% 0.0000035845 => 36708c6d => 00f1 => 0.0000035912 | error: 0.188% 0.0000032260 => 36587e62 => 00d8 => 0.0000032187 | error: 0.228% 0.0000029034 => 3642d825 => 00c3 => 0.0000029057 | error: 0.080% 0.0000026131 => 362f5c21 => 00af => 0.0000026077 | error: 0.205% 0.0000023518 => 361dd2ea => 009e => 0.0000023544 | error: 0.112% 0.0000021166 => 360e0a9f => 008e => 0.0000021160 | error: 0.029% 0.0000019049 => 35ffacb7 => 0080 => 0.0000019073 | error: 0.127% 0.0000017144 => 35e61b71 => 0073 => 0.0000017136 | error: 0.047% 0.0000015430 => 35cf18b2 => 0068 => 0.0000015497 | error: 0.436% 0.0000013887 => 35ba6306 => 005d => 0.0000013858 | error: 0.208% 0.0000012498 => 35a7bf85 => 0054 => 0.0000012517 | error: 0.150% 0.0000011248 => 3596f92b => 004b => 0.0000011176 | error: 0.645% 0.0000010124 => 3587e040 => 0044 => 0.0000010133 | error: 0.091% 0.0000009111 => 357493a6 => 003d => 0.0000009090 | error: 0.236% 0.0000008200 => 355c1e7b => 0037 => 0.0000008196 | error: 0.054% 0.0000007380 => 35461b6e => 0032 => 0.0000007451 | error: 0.955% 0.0000006642 => 35324be3 => 002d => 0.0000006706 | error: 0.955% 0.0000005978 => 3520777f => 0028 => 0.0000005960 | error: 0.291% 0.0000005380 => 35106b8c => 0024 => 0.0000005364 | error: 0.291% 0.0000004842 => 3501fa64 => 0020 => 0.0000004768 | error: 1.522% 0.0000004358 => 34e9f5e7 => 001d => 0.0000004321 | error: 0.838% 0.0000003922 => 34d29083 => 001a => 0.0000003874 | error: 1.218% 0.0000003530 => 34bd820f => 0018 => 0.0000003576 | error: 1.315% 0.0000003177 => 34aa8ea7 => 0015 => 0.0000003129 | error: 1.499% 0.0000002859 => 34998063 => 0013 => 0.0000002831 | error: 0.978% 0.0000002573 => 348a26bf => 0011 => 0.0000002533 | error: 1.557% 0.0000002316 => 3478ac24 => 0010 => 0.0000002384 | error: 2.947% 0.0000002084 => 345fce20 => 000e => 0.0000002086 | error: 0.087% 0.0000001876 => 34496cb6 => 000d => 0.0000001937 | error: 3.264% 0.0000001688 => 3435483d => 000b => 0.0000001639 | error: 2.914% 0.0000001519 => 3423276a => 000a => 0.0000001490 | error: 1.933% 0.0000001368 => 3412d6ac => 0009 => 0.0000001341 | error: 1.933% 0.0000001231 => 3404279b => 0008 => 0.0000001192 | error: 3.144% 0.0000001108 => 33ede0e3 => 0007 => 0.0000001043 | error: 5.834% 0.0000000997 => 33d61732 => 0007 => 0.0000001043 | error: 4.629% 0.0000000897 => 33c0ae79 => 0006 => 0.0000000894 | error: 0.354% 0.0000000808 => 33ad69d3 => 0005 => 0.0000000745 | error: 7.735% 0.0000000727 => 339c1271 => 0005 => 0.0000000745 | error: 2.517% 0.0000000654 => 338c76ff => 0004 => 0.0000000596 | error: 8.874% 0.0000000589 => 337cd631 => 0004 => 0.0000000596 | error: 1.251% 0.0000000530 => 33638d92 => 0004 => 0.0000000596 | error: 12.501% 0.0000000477 => 334ccc36 => 0003 => 0.0000000447 | error: 6.249% 0.0000000429 => 33385163 => 0003 => 0.0000000447 | error: 4.168% 0.0000000386 => 3325e2d9 => 0003 => 0.0000000447 | error: 15.742% 0.0000000348 => 33154c29 => 0002 => 0.0000000298 | error: 14.265% 0.0000000313 => 33065e25 => 0002 => 0.0000000298 | error: 4.739% 0.0000000282 => 32f1dca9 => 0002 => 0.0000000298 | error: 5.846% 0.0000000253 => 32d9acfe => 0002 => 0.0000000298 | error: 17.606% 0.0000000228 => 32c3e87e => 0002 => 0.0000000298 | error: 30.673% 0.0000000205 => 32b0513e => 0001 => 0.0000000149 | error: 27.404% 0.0000000185 => 329eaf84 => 0001 => 0.0000000149 | error: 19.337% 0.0000000166 => 328ed12a => 0001 => 0.0000000149 | error: 10.375% 0.0000000150 => 3280890c => 0001 => 0.0000000149 | error: 0.416% 0.0000000135 => 32675d15 => 0001 => 0.0000000149 | error: 10.648% 0.0000000121 => 32503a2c => 0001 => 0.0000000149 | error: 22.943% 0.0000000109 => 323b678e => 0001 => 0.0000000149 | error: 36.603% 0.0000000098 => 3228aa00 => 0001 => 0.0000000149 | error: 51.781% 0.0000000088 => 3217cc33 => 0001 => 0.0000000149 | error: 68.646% 0.0000000080 => 32089e2e => 0001 => 0.0000000149 | error: 87.384% 0.0000000072 => 31f5e986 => 0000 => 0.0000000000 | error: 100.000%