1. 目的
LeNet-5 中的 squashing function 中，使用到了 hyperbolic tangent（双曲正切三角函数）
f
(
a
)
=
A
tanh
(
S
a
)
f(a) = A \tanh(Sa)
f(a)=Atanh(Sa)
而 tanh 展开式出现了
exp
(
x
)
\exp(x)
exp(x):
tanh
(
x
)
=
e
x
−
e
−
x
e
x
+
e
−
x
\tanh(x) = \frac{e^x-e^{-x}}{e^x+e^{-x}}
tanh(x)=ex+e−xex−e−x
因此需要先计算 exp ( x ) \exp(x) exp(x).
2. exp ( x ) \exp(x) exp(x) 定义式
也叫做泰勒展开：
exp
(
x
)
=
∑
0
∞
x
n
n
!
\exp(x) = \sum_{0}^{\infty} \frac{x^n}{n!}
exp(x)=0∑∞n!xn
也可以显式的写出前面几项：
exp
(
x
)
=
1
+
x
+
x
2
2
!
+
x
3
3
!
+
.
.
.
+
x
n
n
!
\exp(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... + \frac{x^n}{n!}
exp(x)=1+x+2!x2+3!x3+...+n!xn
显然这是一个无限次累加的过程，具体实现时应当在一定迭代次数后停止，我选择的终止条件是 eps 小于 1e-5:
static double m_fabs(double n)
{
return n >= 0.0 ? n : -n;
}
double m_exp(double x)
{
double res = 1;
double eps = 1e-5;
double up = 1;
double down = 1;
for (int i = 1; ;i++)
{
up *= x;
down *= i;
double delta = up / down;
res += delta;
if (m_fabs(delta) < eps)
break;
}
return res;
}
3. exp ( x ) \exp(x) exp(x) 的快速计算
3.1 用极限公式近似
博客1 则使用极限进行近似:
e
x
=
lim
n
−
>
+
∞
(
1
+
x
n
)
n
e^x = \lim_{n->+\infty}(1 + \frac{x}{n})^n
ex=n−>+∞lim(1+nx)n
据说取 n = 256 n=256 n=256 时可以得到不错的效果, 对应的实现为：(来自博客2 )
inline double exp1(float x)
{
x = 1.0 + x / 256.0;
x *= x; x *= x; x *= x; x *= x;
x *= x; x *= x; x *= x; x *= x;
return x;
}
取 n = 1024 n=1024 n=1024 展开，对应的实现为
inline double exp2(double x)
{
x = 1.0 + x / 1024;
x *= x; x *= x; x *= x; x *= x;
x *= x; x *= x; x *= x; x *= x;
x *= x; x *= x;
return x;
}
3.2 转为 2 x ln 2 2^{\frac{x}{\ln 2}} 2ln2x近似
StackOverFlow 上的一个问答3 给出了另一种近似思路: 转换
e
x
e^x
ex 为
2
x
ln
2
2^{\frac{x}{\ln 2}}
2ln2x, 然后用 2 的幂次做快速计算。原理：
2
x
ln
2
=
(
2
1
ln
2
)
x
=
(
2
log
2
e
)
x
=
e
x
2^{\frac{x}{\ln 2}} = (2^\frac{1}{\ln 2}) ^ x = (2 ^ {\log_{2} e})^x = e^x
2ln2x=(2ln21)x=(2log2e)x=ex
具体计算，则是把上述公式倒过来用：
e
x
=
2
x
ln
2
e^x = 2^{\frac{x}{\ln 2}}
ex=2ln2x
而对于 f ( y ) = 2 y f(y) = 2^y f(y)=2y 的计算， 可以结合 IEEE-754 浮点数的二进制表示， 进行近似和加速, 例如参考4 给出的实现:
inline double fast_exp(double y)
{
double d;
*(reinterpret_cast<int*>(&d) + 0) = 0;
*(reinterpret_cast<int*>(&d) + 1) = static_cast<int>(1512775 * y + 1072632447);
return d;
}
C标准库中:
-
exp2f(x)表示 2 x 2^x 2x. -
logf(x)表示 ln ( x ) \ln(x) ln(x).
3.3 查表计算 exp ( x ) \exp(x) exp(x)
在前一小节用
e
x
=
2
x
ln
2
e^x = 2^{\frac{x}{\ln 2}}
ex=2ln2x 展开后， 计算
2
y
2^y
2y 的做法其实有多种， fast_exp() 函数里用到的公式是结合 IEEE-754 标准的二进制格式做的 hack 式实现， 有时候精度损失过大，例如计算
exp
(
40
)
\exp(40)
exp(40). 计算
2
y
2^y
2y 的过程可以考虑用多项式近似， 近似的过程中可以使用查表方式来加速， 计算多项式的过程用了 Horner’s method （其实就是秦九韶公式，高中课本提到过）：
a
0
+
a
1
x
+
a
2
x
2
+
a
3
x
3
+
.
.
.
+
a
n
x
n
=
a
0
+
x
(
a
1
+
x
(
a
2
+
x
(
a
3
+
.
.
.
+
x
(
a
n
−
1
+
x
a
n
)
)
)
)
a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ... + a_n x^n = a_0 + x(a_1 + x(a_2 + x(a_3 + ... + x(a_{n-1} + x a_n))))
a0+a1x+a2x2+a3x3+...+anxn=a0+x(a1+x(a2+x(a3+...+x(an−1+xan))))
github 仓库 Logarithmic and Exponential Function Approximation 5 给出了一份具体的实现，它假定了 exp ( x ) \exp(x) exp(x) 的 x ∈ [ − 710 , 710 ] x \in [-710, 710] x∈[−710,710] 的数据范围， 因此建立的表格是 1420 个元素, 对应的核心代码：
// Generated with:
// >>> from math import exp
// >>> [exp(i) for i in range(-710, 710)]
double EXP_TABLE[1420] = {
...
};
double __attribute__((noinline)) fast_exp(double x) {
double integer = trunc(x);
// X is now the fractional part of the number.
x = x - integer;
// Use a 4-part polynomial to approximate exp(x);
double c[] = { 0.28033708, 0.425302, 1.01273643, 1.00020947 };
// Use Horner's method to evaluate the polynomial.
double val = c[3] + x * (c[2] + x * (c[1] + x * (c[0])));
return val * EXP_TABLE[(unsigned)integer + 710];
}
4. 其他
参考6 是论文 A Fast, Compact Approximation of the Exponential Function 的主页。
参考7 是网友用 AVX2 指令做泰勒展开的加速实现。
参考8 则对利用IEEE-754二进制表示做快速计算的原理进行了初步理解。
5. 完整代码
简单尝试了几个数字， 如 x = 1 x=1 x=1, x = 20 x=20 x=20, x = 40 x=40 x=40, 感觉几个近似实现的误差都略大，打算先用 泰勒展开到精度小于 1 e − 9 1e-9 1e−9 的项。本文涉及的所有实现，代码如下：（没有考虑 SIMD/GPU 优化的方法）
#include <stdio.h>
#include <math.h>
#include <stdbool.h>
static double m_fabs(double n)
{
return n >= 0.0 ? n : -n;
}
double m_exp(double x)
{
double res = 1;
double eps = 1e-9;
double up = 1;
double down = 1;
for (int i = 1; ;i++)
{
up *= x;
down *= i;
double delta = up / down;
res += delta;
// printf("i=%d, delta=%lf\n", i, delta);
if (m_fabs(delta) < eps)
break;
}
return res;
}
double fast_exp(double x)
{
double d;
*((int*)&d + 0) = 0;
*((int*)&d + 1) = (int)(1512775 * x + 1072632447);
return d;
}
/// e^x = \lim_{n \rightarrow \infty}(1 + \frac{x){n)^n
double exp_by_limit(double x)
{
x = 1.0 + x / 1024;
x *= x; x *= x; x *= x; x *= x;
x *= x; x *= x; x *= x; x *= x;
x *= x; x *= x;
return x;
}
// Generated with:
// >>> from math import exp
// >>> [exp(i) for i in range(-710, 710)]
double EXP_TABLE[1420] = {
4.47628622567513e-309, 1.216780750623423e-308, 3.307553003638408e-308,
8.99086122645542e-308, 2.443969469407077e-307, 6.643397797997952e-307,
1.8058627513522668e-306, 4.9088439016919216e-306, 1.334362117671115e-305,
3.6271722970495225e-305, 9.85967654375977e-305, 2.680137958338607e-304,
7.285370309915161e-304, 1.9803689727037426e-303, 5.38320099214469e-303,
1.4633057435889614e-302, 3.977677412277625e-302, 1.0812448229266266e-301,
2.9391281542768673e-301, 7.98937865328318e-301, 2.171738281389827e-300,
5.9033967064708435e-300, 1.604709599338467e-299, 4.36205294383555e-299,
1.185728925200446e-298, 3.223145390850647e-298, 8.76141754643084e-298,
2.3816002108005187e-297, 6.473860575673281e-297, 1.7597777562830093e-296,
4.783571897030535e-296, 1.3003096562825466e-295, 3.5346081100426732e-295,
9.608060996252968e-295, 2.6117417612840555e-294, 7.09945017032607e-294,
1.929830639004783e-293, 5.245823558010209e-293, 1.4259626853041525e-292,
3.8761684555229417e-292, 1.0536518276694175e-291, 2.864122616676439e-291,
7.785492463390136e-291, 2.1163162688838255e-290, 5.752744056979149e-290,
1.5637579633862188e-289, 4.250734855980884e-289, 1.1554695316610313e-288,
3.140891831252265e-288, 8.537829190048485e-288, 2.3208225941796005e-287,
6.30864988483559e-287, 1.714868834405883e-286, 4.661496790756256e-286,
1.2671262019732887e-285, 3.444406129188316e-285, 9.362866590805559e-285,
2.5450910116073043e-284, 6.918274648626584e-284, 1.880582026165053e-283,
5.111951948651156e-283, 1.3895726089974245e-282, 3.7772499723621244e-282,
1.0267629961419394e-281, 2.791031194546799e-281, 7.586809378798905e-281,
2.0623086070371722e-280, 5.605936011183831e-280, 1.5238513990705191e-279,
4.142257567365285e-279, 1.1259823474166023e-278, 3.06073735414821e-278,
8.319946731466896e-278, 2.261596001389369e-277, 6.14765531389236e-277,
1.6711059727383288e-276, 4.542536999123976e-276, 1.2347895779821586e-275,
3.3565060717995146e-275, 9.123929462085072e-275, 2.4801411660927963e-274,
6.741722663803275e-274, 1.8325902209526951e-273, 4.981496696627458e-273,
1.3541111948971181e-272, 3.6808558548018004e-272, 1.000560358328482e-271,
2.7198050403207837e-271, 7.393196618055307e-271, 2.009679202108461e-270,
5.4628744561235025e-270, 1.4849632365233607e-269, 4.036548581771182e-269,
1.0972476659520735e-268, 2.982628391676622e-268, 8.107624558140589e-268,
2.2038808508361863e-267, 5.990769268916865e-267, 1.6284599242187592e-266,
4.426613020377647e-266, 1.2032781734912767e-265, 3.270849193582728e-265,
8.891089926545851e-265, 2.4168488182524857e-264, 6.569676224788449e-264,
1.785823150070186e-263, 4.8543706176772776e-263, 1.3195547438637656e-262,
3.5869216819018034e-262, 9.750264028019429e-262, 2.6503965530043108e-261,
7.204524788242109e-261, 1.9583928814561274e-260, 5.3234637826457406e-260,
1.4470674864825768e-259, 3.933537253059494e-259, 1.069246283655833e-258,
2.906512743009017e-258, 7.900720773506065e-258, 2.147638571035043e-257,
5.837886901742308e-257, 1.586902188160519e-256, 4.3136473816186357e-256,
1.172570929183388e-255, 3.187378249378541e-255, 8.66419237571129e-255,
2.3551716693169407e-254, 6.402020351605795e-254, 1.740249558719502e-253,
4.730488752451095e-253, 1.285880161551771e-252, 3.4953846767221607e-252,
9.501440650208043e-252, 2.582759346364262e-251, 7.020667798504735e-251,
1.908415370032299e-250, 5.1876108215107433e-250, 1.4101388229230154e-249,
3.8331547379562597e-249, 1.0419594869858195e-248, 2.832339539464062e-248,
7.69909710215122e-248, 2.0928315748319355e-247, 5.688906039890977e-247,
1.5464049912046552e-246, 4.2035645870299835e-246, 1.1426473231677555e-245,
3.10603745490428e-245, 8.443085172179486e-245, 2.2950684999667505e-244,
6.238642998528377e-244, 1.6958389897142937e-243, 4.6097683097327105e-243,
1.2530649429752794e-242, 3.406183664368772e-242, 9.258967159247676e-242,
2.5168482179282025e-241, 6.841502775783763e-241, 1.8597132674765122e-240,
5.055224781125599e-240, 1.374152566130957e-239, 3.735333950044147e-239,
1.0153690399631151e-238, 2.760059210511642e-238, 7.502618797404815e-238,
2.0394232342840765e-237, 5.543727118291579e-237, 1.5069412687587626e-236,
4.096291067421963e-236, 1.1134873572652229e-235, 3.02677244947294e-235,
8.227620548282767e-235, 2.236499142785329e-234, 6.079434979197592e-234,
1.6525617631251106e-233, 4.492128611109229e-233, 1.2210871574679187e-232,
3.319259031109752e-232, 9.022681508214216e-232, 2.4526191187752155e-231,
6.666909982697905e-231, 1.8122540257939923e-230, 4.926217186867559e-230,
1.3390846662104723e-229, 3.640009514928073e-229, 9.894571719847004e-229,
2.6896234506444874e-228, 7.311154551284224e-228, 1.987377856181155e-227,
5.402253112739128e-227, 1.4684846469095084e-226, 3.991755131065214e-226,
1.0850715436432725e-225, 2.9495302596635135e-225, 8.017654507333419e-225,
2.179424455414719e-224, 5.924289893653081e-224, 1.610388956444074e-223,
4.377491037053051e-223, 1.189925434026365e-222, 3.234552684535111e-222,
8.792425785565214e-222, 2.3900291240976666e-221, 6.496772737522576e-221,
1.7660059276035744e-220, 4.800501821955756e-220, 1.3049116870106872e-219,
3.547117726544988e-219, 9.642065659472201e-219, 2.6209851870952265e-218,
7.124576406741286e-218, 1.9366606581912876e-217, 5.264389475052911e-217,
1.4310094247967382e-216, 3.8898869157786035e-216, 1.057380891792158e-215,
2.874259263918443e-215, 7.813046727389575e-215, 2.123806294396449e-214,
5.773104057224809e-214, 1.5692923852557387e-213, 4.2657789743798256e-213,
1.1595589470279344e-212, 3.1520080147331386e-212, 8.568046109606362e-212,
2.329036404514219e-211, 6.3309773362105915e-211, 1.72093806494073e-210,
4.677994669831859e-210, 1.2716107904632215e-209, 3.4565965045886174e-209,
9.396003466738291e-209, 2.554098548377289e-208, 6.94275967214761e-208,
1.8872377456157127e-207, 5.130044069889206e-207, 1.3944905574373912e-206,
3.790618342239785e-206, 1.0303968958333958e-205, 2.800909158044528e-205,
7.6136604674769635e-205, 2.069607489679963e-204, 5.6257764312397845e-204,
1.5292445844012282e-203, 4.1569177650472634e-203, 1.1299674023126563e-202,
3.071569856457565e-202, 8.349392525651157e-202, 2.2696001981149314e-201,
6.169412976402867e-201, 1.6770203186015345e-200, 4.55861385801115e-200,
1.239159721319329e-199, 3.3683853530207066e-199, 9.156220696363793e-199,
2.4889188336286325e-198, 6.765582837962194e-198, 1.8390760887367006e-197,
4.999127113166508e-197, 1.3589036389877445e-196, 3.693883068487256e-196,
1.0041015221521447e-195, 2.7294309215942424e-195, 7.419362476203855e-195,
2.0167918197815843e-194, 5.482208555497131e-194, 1.4902187896230561e-193,
4.050834656260586e-193, 1.1011310236205293e-192, 2.993184452260193e-192,
8.136318905805023e-192, 2.2116807832197573e-191, 6.011971683378335e-191,
1.6342233380137666e-190, 4.4422796033665057e-190, 1.2075367922765428e-189,
3.282425319641051e-189, 8.922557099654142e-189, 2.4254024827378097e-188,
6.592927495525641e-188, 1.7921435007435354e-187, 4.871551112062132e-187,
1.3242248864327946e-186, 3.5996164455835086e-186, 9.784771973451989e-186,
2.659776785104989e-185, 7.230022902708112e-185, 1.9653239875774178e-184,
5.342304482466365e-184, 1.4521889196783625e-183, 3.9474587518512645e-183,
1.0730305393748917e-182, 2.916799416564376e-182, 7.928682851306888e-182,
2.1552394518322364e-181, 5.858548237893603e-181, 1.5925185216216938e-180,
4.328914158808713e-180, 1.1767208694848799e-179, 3.198658956689277e-179,
8.69485651740623e-179, 2.3635070472324053e-178, 6.424678257926741e-178,
1.7464086162218176e-177, 4.747230806540073e-177, 1.2904311236918859e-176,
3.50775547440964e-176, 9.535067964765462e-176, 2.5919001981743924e-175,
7.04551520987685e-175, 1.9151695967140057e-174, 5.20597071316492e-174,
1.4151295589086178e-173, 3.84672096489656e-173, 1.0456471698030763e-172,
2.842363700655332e-172, 7.726345597362994e-172, 2.1002384837706373e-171,
5.709040105864101e-171, 1.551877997771429e-170, 4.218441761327482e-170,
1.1466913584229263e-169, 3.1170302824520583e-169, 8.472966775545996e-169,
2.303191161910391e-168, 6.260722682888491e-168, 1.7018408701917146e-167,
4.626083112371067e-167, 1.2574997661299533e-166, 3.4182387635625514e-166,
9.291736316326398e-166, 2.5257557983503035e-165, 6.865716089780698e-165,
1.8662951286209762e-164, 5.0731161346720364e-164, 1.3790159402541388e-163,
3.7485539715481897e-163, 1.0189626143857429e-162, 2.7698275585638865e-162,
7.529171920409294e-162, 2.0466411214592676e-161, 5.5633473698397695e-161,
1.5122746060840868e-160, 4.110788581358434e-160, 1.1174281901343568e-159,
3.037484743850101e-159, 8.256739583429307e-159, 2.2444145171954394e-158,
6.100951197622044e-158, 1.6584104776811452e-157, 4.508027065606742e-157,
1.2254088054640357e-156, 3.3310064883265936e-156, 9.054614407697357e-156,
2.4612993808147185e-155, 6.6905053812661495e-155, 1.818667920110323e-154,
4.9436519592372975e-154, 1.3438239287020702e-153, 3.652892166039281e-153,
9.92959039626498e-153, 2.6991425138208544e-152, 7.337030047740496e-152,
1.994411545363099e-151, 5.421372662229435e-151, 1.473681879304291e-150,
4.00588267344223e-150, 1.0889118078156954e-149, 2.959969179979893e-149,
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8.04152429962318e+122, 2.185912937677754e+123, 5.941927417082968e+123,
1.6151833323879222e+124, 4.390523502060015e+124, 1.1934680253072109e+125,
3.2441824460394912e+125, 8.818602191274965e+125, 2.3971446088951858e+126,
6.516114630548348e+126, 1.77126359923757e+127, 4.814793655218451e+127,
1.3087966100760222e+128, 3.55767804231845e+128, 9.670771573941992e+128,
2.6287882636624796e+129, 7.145787367980123e+129, 1.9424263952412558e+130,
5.280062373303513e+130, 1.435269760248128e+131, 3.901467708219257e+131,
1.0605288775572162e+132, 2.882816376419849e+132, 7.836307370806225e+132,
2.130129192828224e+133, 5.790291477135095e+133, 1.5739644103777611e+134,
4.278478855371123e+134, 1.1630111326001581e+135, 3.161392028042583e+135,
8.593554502463442e+135, 2.3359703045918785e+136, 6.349825630792043e+136,
1.7260615626065507e+137, 4.691921780435012e+137, 1.2753965716307703e+138,
3.4668873247428877e+138, 9.423976816163585e+138, 2.56170249311968e+139,
6.963429336965459e+139, 1.8928563430431824e+140, 5.145317001177723e+140,
1.3986421705962793e+141, 3.801903596848382e+141, 1.0334645460866042e+142,
2.8092478959838913e+142, 7.636327507289818e+142, 2.075769029922787e+143,
5.642525234117172e+143, 1.533797381052233e+144, 4.169293549452358e+144,
1.133331489298786e+145, 3.080714392981317e+145, 8.374249953113352e+145,
2.2763571474522036e+146, 6.187780269002192e+146, 1.682013066372608e+147,
4.572185553551339e+147, 1.2428488906561565e+148, 3.378413554991113e+148,
9.183480175552067e+148, 2.4963287283217065e+149, 6.785725020057171e+149,
1.8445513014941297e+150, 5.014010284511975e+150, 1.362949304409567e+151,
3.7048803272874213e+151, 1.0070908870280797e+152, 2.7375568578151306e+152,
7.441451060972311e+152, 2.0227961196408315e+153, 5.498529934697141e+153,
1.4946554004725342e+154, 4.062894614912666e+154, 1.1044092602661211e+155,
3.0020956233632933e+155, 8.16054198028487e+155, 2.2182652975385555e+156,
6.0298702490003525e+156, 1.6390886725823477e+157, 4.4555049539136534e+157,
1.211131815283274e+158, 3.2921976053531405e+158, 8.949120926327824e+158,
2.4326232794719504e+159, 6.612555656075053e+159, 1.7974789879582895e+160,
4.8860544700039736e+160, 1.3281673078672893e+161, 3.6103330581290227e+161,
9.813902746597095e+161, 2.66769535023392e+162, 7.251547794405553e+162,
1.9711750597734883e+163, 5.358209345693946e+163, 1.4565123097479284e+164,
3.959210944514706e+164, 1.0762251165510499e+165, 2.9254831776519365e+165,
7.952287761273885e+165, 2.161655931614806e+166, 5.875990038289236e+166,
1.5972596945288e+167, 4.3418020029676826e+167, 1.1802241487434137e+168,
3.2081818570377667e+168, 8.720742444377757e+168, 2.370543571722357e+169,
6.443805514583285e+169, 1.7516079436415928e+170, 4.7613640437854577e+170,
1.2942729358900287e+171, 3.518198602696204e+171, 9.563455330619095e+171,
2.5996166842501676e+172, 7.066490793756186e+172, 1.9208713515640576e+173,
5.221469689764144e+173, 1.419342617553556e+174, 3.858173245653328e+174,
1.0487602224706297e+175, 2.8508258551525784e+175, 7.749348118162471e+175,
2.1064912172004347e+176, 5.726036797524517e+176, 1.556498177579872e+177,
4.231000712144986e+177, 1.1501052352020995e+178, 3.1263101616654833e+178,
8.498192102582143e+178, 2.3100481167203208e+179, 6.279361818546888e+179,
1.7069075125675549e+180, 4.639855674272614e+180, 1.2612435366047836e+181,
3.428415386814204e+181, 9.31939924638644e+181, 2.5332753623607178e+182,
6.886156383988143e+182, 1.8718513766522217e+183, 5.088219582729782e+183,
1.3831214830943832e+184, 3.759713994046786e+184, 1.0219962230220558e+185,
2.778073761794632e+185, 7.551587424805211e+185, 2.052734287286784e+186,
5.579910311786494e+186, 1.5167768804960472e+187, 4.123027032079202e+187,
1.1207549459546325e+188, 3.046527803744077e+188, 8.281321168812768e+188,
2.2510964848816967e+189, 6.119114668961948e+189, 1.663347821089645e+190,
4.521448156474929e+190, 1.229057036206545e+191, 3.340923407659982e+191,
9.0815713893156e+191, 2.4686270481430163e+192, 6.710424046209653e+192,
1.8240823746066321e+193, 4.958369972505633e+193, 1.3478246995039038e+194,
3.663767388609735e+194, 9.959152316158692e+194, 2.7071782767869983e+195,
7.35887351618917e+195, 2.000349215698554e+196, 5.437512923605682e+196,
1.4780692572248542e+197, 4.017808803118279e+197, 1.0921536659739205e+198,
2.968781464101838e+198, 8.069984706534065e+198, 2.193649278371395e+199,
5.962956971409261e+199, 1.6208997579264978e+200, 4.4060623577252635e+200,
1.1976919242062002e+201, 3.255664193661862e+201, 8.849812817195809e+201,
2.405628536624732e+202, 6.539176337129533e+202, 1.777532421030859e+203,
4.831834079584997e+203, 1.3134286776665033e+204, 3.5702693074778485e+204,
9.704998181222095e+204, 2.6380920201244107e+205, 7.1710776001069995e+205,
1.9493009930840557e+206, 5.298749467697559e+206, 1.4403494391599313e+207,
3.9152757070996186e+207, 1.0642822808016033e+208, 2.8930191842539453e+208,
7.86404147794091e+208, 2.137668104773499e+209, 5.810784364482288e+209,
1.5795349547066147e+210, 4.2936211647948715e+210, 1.167127239054906e+211,
3.172580765422527e+211, 8.623968643966744e+211, 2.3442377254095393e+212,
6.372298810568915e+212, 1.732170406228067e+213, 4.708527339044277e+213,
1.279910430452668e+214, 3.4791572651546824e+214, 9.457329972221242e+214,
2.5707688209230085e+215, 6.9880741710841e+215, 1.8995555035181914e+216,
5.1635272073628715e+216, 1.4035922178528375e+217, 3.8153592203558975e+217,
1.0371221637737106e+218, 2.8191903316782035e+218, 7.663353849568289e+218,
2.0831155514333153e+219, 5.662495150041624e+219, 1.5392257670095623e+220,
4.184049432358029e+220, 1.1373425541353215e+221, 3.091617597639242e+221,
8.403887936206959e+221, 2.2844135865397565e+222, 6.209679940975975e+222,
1.687966014410163e+223, 4.588367344027585e+223, 1.2472475573565076e+224,
3.3903703707521256e+224, 9.215982170561459e+224, 2.505163686563976e+225,
6.809740926502327e+225, 1.851079501702514e+226, 5.031755772510968e+226,
1.367773028166047e+227, 3.7179925679201674e+227, 1.0106551635723174e+228,
2.7472455659769343e+228, 7.467787700309786e+228, 2.0299551604542052e+229,
5.517990225249331e+229, 1.4999452558909891e+230, 4.077273932771829e+230,
1.1083179641103409e+231, 3.0127205819958637e+231, 8.189423612263916e+231,
2.2261161390770435e+232, 6.051211048892536e+232, 1.644889703437518e+233,
4.471273790673593e+233, 1.2154182295253221e+234, 3.3038492872965484e+234,
8.980793481625574e+234, 2.441232772624624e+235, 6.635958684864208e+235,
1.803840590747136e+236, 4.903347099264769e+236, 1.3328679318558793e+237,
3.6231106788996255e+237, 9.848635920948766e+237, 2.6771368059024047e+238,
7.277212331783397e+238, 1.9781514043324884e+239, 5.377173016337745e+239,
1.4616671698791204e+240, 3.9732233071375736e+240, 1.0800340716202018e+241,
2.935836991001829e+241, 7.980432343958154e+241, 2.1693064223828275e+242,
5.896786228322743e+242, 1.6029126850757262e+243, 4.357168424447843e+243,
1.1844011751712099e+244, 3.219536192073438e+244, 8.751606726979457e+244,
2.37893335357682e+245, 6.4666113061430065e+245, 1.757807200519635e+246,
4.778215371106989e+246, 1.298853601574382e+247, 3.5306501429882274e+247,
9.597302126331227e+247, 2.608817197223753e+248, 7.091500380984786e+248,
1.9276696622141338e+249, 5.239949414068466e+249, 1.4243659274306933e+250,
3.871828017611069e+250, 1.0524719743190776e+251, 2.860915442753964e+251,
7.776774460795963e+251, 2.1139464700806057e+252, 5.746302275955253e+252,
1.5620069057562017e+253, 4.245974987844624e+253, 1.1541756653549656e+254,
3.137374737984031e+254, 8.52826873932845e+254, 2.3182237942331857e+255,
6.301585614165449e+255, 1.7129485665464872e+256, 4.656276961528286e+256,
1.2657073052794837e+257, 3.440549168089086e+257, 9.352382283536447e+257,
2.5422410814139436e+258, 6.910527735169595e+258, 1.878476196757375e+259,
5.106227710838431e+259, 1.3880165998346134e+260, 3.7730203009299397e+260,
1.0256132522424933e+261, 2.787905866597553e+261, 7.578313856626495e+261,
2.059999284682719e+262, 5.599658622191666e+262, 1.522145027827762e+263,
4.1376191694234934e+263, 1.124721500132769e+264, 3.0573100158881035e+264,
8.310630260154467e+264, 2.2590635219219752e+265, 6.140771320975197e+265,
1.6692347094529326e+266, 4.537450378139021e+266, 1.2334068910429924e+267,
3.352747539018332e+267, 9.113712710724316e+267, 2.4773639651358133e+268,
6.734173448907929e+268, 1.83053813158578e+269, 4.975918539390998e+269,
1.3525948945519025e+270, 3.676734123126915e+270, 9.994399554971195e+270,
2.7167594696637367e+271, 7.384917898680968e+271, 2.007428812864643e+272,
5.456757263935073e+272, 1.4833004112866607e+273, 4.032028554146358e+273,
1.0960189950564043e+274, 2.9792885179077677e+274, 8.098545839965366e+274,
2.201412999372045e+275, 5.984060953126553e+275, 1.6266364149275224e+276,
4.421656208207252e+276, 1.2019307722462898e+277, 3.2671865772628366e+277,
8.881133903158874e+277, 2.414142490506832e+278, 6.562319663255584e+278,
1.7838234293167135e+279, 4.848934813091121e+279, 1.318077138980805e+280,
3.58290513539881e+280, 9.73934592264718e+280, 2.6474287042608523e+281,
7.19645733893315e+281, 1.956199921370272e+282, 5.317502699093823e+282,
1.4454470959728667e+283, 3.9291325749819406e+283, 1.0680489680179907e+284,
2.9032581016677402e+284, 7.891873741089921e+284, 2.1452336982897837e+285,
5.831349779859113e+285, 1.585125214197968e+286, 4.308817065586588e+286,
1.1712579131538248e+287, 3.1838091017649045e+287, 8.654490426610056e+287,
2.3525344061226884e+288, 6.394851526987996e+288, 1.7383008701505047e+289,
4.725191667724663e+289, 1.2844402646362043e+290, 3.491470631101721e+290,
9.490801171122244e+290, 2.579867236097942e+291, 7.012806227721897e+291,
1.9062783735320858e+292, 5.181801862756733e+292, 1.4085597842206858e+293,
3.828862465745284e+293, 1.04079272643043e+294, 2.829167955448184e+294,
7.690475842953428e+294, 2.090488073610356e+295, 5.682535743105387e+295,
1.5446733650052388e+296, 4.198857538998427e+296, 1.1413678148547691e+297,
3.1025593907077266e+297, 8.433630813475781e+297, 2.2924985388203488e+298,
6.231657119844268e+298, 1.6939400310060103e+299, 4.60460640478299e+299,
1.2516617917327736e+300, 3.4023695038436884e+300, 9.248599196001516e+300,
2.5140299133191857e+301, 6.833841829578011e+301, 1.8576308063905224e+302,
5.049564064997079e+302, 1.372613823952135e+303, 3.7311512151407716e+303,
1.0142320547350045e+304, 2.7569685642268427e+304, 7.49421754977065e+304,
2.037139538406043e+305, 5.5375193892845935e+305, 1.505253833063194e+306,
4.0917041416340054e+306, 1.1122405015634333e+307, 3.023383144276055e+307,
8.218407461554972e+307
};
double fast_exp_by_lut(double x)
{
double integer = (int)x;
// x is now the fractional part of the number.
x = x - integer;
// Use a 4-part polynomial to approximate exp(x);
double c[] = { 0.28033708, 0.425302, 1.01273643, 1.00020947 };
// Use Horner's method to evaluate the polynomial.
double val = c[3] + x * (c[2] + x * (c[1] + x * (c[0])));
return val * EXP_TABLE[(unsigned)integer + 710];
}
int main()
{
double x;
while (true)
{
printf("Please input double value x: ");
scanf("%lf", &x);
double y = m_exp(x);
double gt = exp(x);
double diff1 = m_fabs(y - gt);
double fy = fast_exp(x);
double diff2 = m_fabs(gt - fy);
double z = fast_exp_by_lut(x);
double uu = exp_by_limit(x);
printf(" exp(%lf) = %lf\n", x, y);
printf(" m_exp(%lf) = %lf\n", x, gt);
printf(" exp_by_limit(%lf) = %lf\n", x, uu);
printf(" fast_exp(%lf) = %lf\n", x, fy);
printf(" fast_exp_by_lut(%lf) = %lf\n", x, z);
printf(" abs(exp(%lf) - m_exp(%lf)) = %lf, %lf\n", x, x, diff1, gt-y);
//printf("diff1 = %lf\n", diff1);
printf(" abs(exp(%lf) - fast_exp(%lf) = %lf\n", x, x, diff2);
}
return 0;
}
6. References
-
exp近似计算，exp快速算法，C语言实现exp ↩︎
-
https://blog.csdn.net/just_sort/article/details/88128200 ↩︎
-
Fastest Implementation of the Natural Exponential Function Using SSE ↩︎
-
这个求指数函数exp()的快速近似方法的原理是什么? ↩︎
-
一种快速的幂运算方法（底数是自然数e，指数是浮点数） ↩︎
-
Logarithmic and Exponential Function Approximation ↩︎
-
A Fast, Compact Approximation of the Exponential Function ↩︎文章来源：https://www.toymoban.com/news/detail-500816.html
-
avx2实现exp函数的快速近似计算 ↩︎文章来源地址https://www.toymoban.com/news/detail-500816.html
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