如果有人想复制问题中显示的相对误差函数，这是一种使用 Matlab 的方法（“快速”指数在 Matlab 中不是很快，但它是准确的）：
t = 1072632447+[0:ceil(1512775*pi)];
x = (t - 1072632447)/1512775;
ex = exp(x);
t = uint64(t);
import java.lang.Double;
et = arrayfun( @(n) java.lang.Double.longBitsToDouble(bitshift(n,32)), t );
plot(x, et./ex);
现在，误差的周期正好与二进制值tmp从尾数溢出到指数的时间一致。让我们通过丢弃成为指数的位（使其周期性）将我们的数据分成箱，并仅保留剩余的高八位（以使我们的查找表具有合理的大小）：
index = bitshift(bitand(t,uint64(2^20-2^12)),-12) + 1;
现在我们计算平均所需的调整：
relerrfix = ex./et;
adjust = NaN(1,256);
for i=1:256; adjust(i) = mean(relerrfix(index == i)); end;
et2 = et .* adjust(index);
相对误差降低到 +/- .0006。当然，其他表大小也是可能的（例如，具有 64 个条目的 6 位表给出 +/- .0025）并且误差在表大小上几乎是线性的。表条目之间的线性插值将进一步改善错误，但会以牺牲性能为代价。由于我们已经达到了准确度目标，让我们避免任何进一步的性能损失。
此时，获取 MatLab 计算的值并在 C# 中创建查找表是一些简单的编辑器技能。对于每个计算，我们添加一个位掩码、表查找和双精度乘法。
static double FastExp(double x)
{
var tmp = (long)(1512775 * x + 1072632447);
int index = (int)(tmp >> 12) & 0xFF;
return BitConverter.Int64BitsToDouble(tmp << 32) * ExpAdjustment[index];
}
加速与原始代码非常相似——对于我的计算机，编译为 x86 的速度大约快 30%，编译为 x64 的速度大约快 3 倍。使用 mono on ideone，这是一个巨大的净损失（但原来的也是如此）。
完整的源代码和测试用例：http: //ideone.com/UwNgx
using System;
using System.Diagnostics;
namespace fastexponent
{
class Program
{
static double[] ExpAdjustment = new double[256] {
1.040389835,
1.039159306,
1.037945888,
1.036749401,
1.035569671,
1.034406528,
1.033259801,
1.032129324,
1.031014933,
1.029916467,
1.028833767,
1.027766676,
1.02671504,
1.025678708,
1.02465753,
1.023651359,
1.022660049,
1.021683458,
1.020721446,
1.019773873,
1.018840604,
1.017921503,
1.017016438,
1.016125279,
1.015247897,
1.014384165,
1.013533958,
1.012697153,
1.011873629,
1.011063266,
1.010265947,
1.009481555,
1.008709975,
1.007951096,
1.007204805,
1.006470993,
1.005749552,
1.005040376,
1.004343358,
1.003658397,
1.002985389,
1.002324233,
1.001674831,
1.001037085,
1.000410897,
0.999796173,
0.999192819,
0.998600742,
0.998019851,
0.997450055,
0.996891266,
0.996343396,
0.995806358,
0.995280068,
0.99476444,
0.994259393,
0.993764844,
0.993280711,
0.992806917,
0.992343381,
0.991890026,
0.991446776,
0.991013555,
0.990590289,
0.990176903,
0.989773325,
0.989379484,
0.988995309,
0.988620729,
0.988255677,
0.987900083,
0.987553882,
0.987217006,
0.98688939,
0.98657097,
0.986261682,
0.985961463,
0.985670251,
0.985387985,
0.985114604,
0.984850048,
0.984594259,
0.984347178,
0.984108748,
0.983878911,
0.983657613,
0.983444797,
0.983240409,
0.983044394,
0.982856701,
0.982677276,
0.982506066,
0.982343022,
0.982188091,
0.982041225,
0.981902373,
0.981771487,
0.981648519,
0.981533421,
0.981426146,
0.981326648,
0.98123488,
0.981150798,
0.981074356,
0.981005511,
0.980944219,
0.980890437,
0.980844122,
0.980805232,
0.980773726,
0.980749562,
0.9807327,
0.9807231,
0.980720722,
0.980725528,
0.980737478,
0.980756534,
0.98078266,
0.980815817,
0.980855968,
0.980903079,
0.980955475,
0.981017942,
0.981085714,
0.981160303,
0.981241675,
0.981329796,
0.981424634,
0.981526154,
0.981634325,
0.981749114,
0.981870489,
0.981998419,
0.982132873,
0.98227382,
0.982421229,
0.982575072,
0.982735318,
0.982901937,
0.983074902,
0.983254183,
0.983439752,
0.983631582,
0.983829644,
0.984033912,
0.984244358,
0.984460956,
0.984683681,
0.984912505,
0.985147403,
0.985388349,
0.98563532,
0.98588829,
0.986147234,
0.986412128,
0.986682949,
0.986959673,
0.987242277,
0.987530737,
0.987825031,
0.988125136,
0.98843103,
0.988742691,
0.989060098,
0.989383229,
0.989712063,
0.990046579,
0.990386756,
0.990732574,
0.991084012,
0.991441052,
0.991803672,
0.992171854,
0.992545578,
0.992924825,
0.993309578,
0.993699816,
0.994095522,
0.994496677,
0.994903265,
0.995315266,
0.995732665,
0.996155442,
0.996583582,
0.997017068,
0.997455883,
0.99790001,
0.998349434,
0.998804138,
0.999264107,
0.999729325,
1.000199776,
1.000675446,
1.001156319,
1.001642381,
1.002133617,
1.002630011,
1.003131551,
1.003638222,
1.00415001,
1.004666901,
1.005188881,
1.005715938,
1.006248058,
1.006785227,
1.007327434,
1.007874665,
1.008426907,
1.008984149,
1.009546377,
1.010113581,
1.010685747,
1.011262865,
1.011844922,
1.012431907,
1.013023808,
1.013620615,
1.014222317,
1.014828902,
1.01544036,
1.016056681,
1.016677853,
1.017303866,
1.017934711,
1.018570378,
1.019210855,
1.019856135,
1.020506206,
1.02116106,
1.021820687,
1.022485078,
1.023154224,
1.023828116,
1.024506745,
1.025190103,
1.02587818,
1.026570969,
1.027268461,
1.027970647,
1.02867752,
1.029389072,
1.030114973,
1.030826088,
1.03155163,
1.032281819,
1.03301665,
1.033756114,
1.034500204,
1.035248913,
1.036002235,
1.036760162,
1.037522688,
1.038289806,
1.039061509,
1.039837792,
1.040618648
};
static double FastExp(double x)
{
var tmp = (long)(1512775 * x + 1072632447);
int index = (int)(tmp >> 12) & 0xFF;
return BitConverter.Int64BitsToDouble(tmp << 32) * ExpAdjustment[index];
}
static void Main(string[] args)
{
double[] x = new double[1000000];
double[] ex = new double[x.Length];
double[] fx = new double[x.Length];
Random r = new Random();
for (int i = 0; i < x.Length; ++i)
x[i] = r.NextDouble() * 40;
Stopwatch sw = new Stopwatch();
sw.Start();
for (int j = 0; j < x.Length; ++j)
ex[j] = Math.Exp(x[j]);
sw.Stop();
double builtin = sw.Elapsed.TotalMilliseconds;
sw.Reset();
sw.Start();
for (int k = 0; k < x.Length; ++k)
fx[k] = FastExp(x[k]);
sw.Stop();
double custom = sw.Elapsed.TotalMilliseconds;
double min = 1, max = 1;
for (int m = 0; m < x.Length; ++m) {
double ratio = fx[m] / ex[m];
if (min > ratio) min = ratio;
if (max < ratio) max = ratio;
}
Console.WriteLine("minimum ratio = " + min.ToString() + ", maximum ratio = " + max.ToString() + ", speedup = " + (builtin / custom).ToString());
}
}
}
