我应该使用什么参考来使用 erf / erfc 功能答案

【问题标题】：what reference should I use to use erf / erfc function我应该使用什么参考来使用 erf / erfc 功能
【发布时间】：2022-03-31 22:31:33
【问题描述】：

我正在尝试在 c# 中绘制一个函数。我必须使用erf / erfc函数，但我在Math下没有找到它。所以想问一下，怎么用/哪里找erf/erfc函数。非常感谢。

【问题讨论】：

我从未听说过“erf”或“erfc”函数，可能是因为我缺乏数学英语知识。你能描述一下这些功能到底是什么吗？
是错误函数。在这里你可以找到我想要绘制的函数：en.wikipedia.org/wiki/…
谢谢。我想就是这样

标签： c# function math

【解决方案1】：

Math 类不包含任何erf 函数。因此，你必须实施你的。

请查看here 以获取自定义解决方案。另一种方法是使用某人的其他库实现的错误函数，例如one。

【讨论】：

【解决方案2】：

如果您需要比接受答案中链接的 John Cooks 代码更高的精度，请查看以下内容：https://math.stackexchange.com/q/1889960

/// <summary>
/// Returns the value of the gaussian error function at <paramref name="x"/>.
/// </summary>
public static double Erf(double x)
{
    /*
    Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
    *
    * Developed at SunPro, a Sun Microsystems, Inc. business.
    * Permission to use, copy, modify, and distribute this
    * software is freely granted, provided that this notice
    * is preserved.
    */

    #region Constants

    const double tiny = 1e-300;
    const double erx = 8.45062911510467529297e-01;

    // Coefficients for approximation to erf on [0, 0.84375]
    const double efx = 1.28379167095512586316e-01; /* 0x3FC06EBA; 0x8214DB69 */
    const double efx8 = 1.02703333676410069053e+00; /* 0x3FF06EBA; 0x8214DB69 */
    const double pp0 = 1.28379167095512558561e-01; /* 0x3FC06EBA; 0x8214DB68 */
    const double pp1 = -3.25042107247001499370e-01; /* 0xBFD4CD7D; 0x691CB913 */
    const double pp2 = -2.84817495755985104766e-02; /* 0xBF9D2A51; 0xDBD7194F */
    const double pp3 = -5.77027029648944159157e-03; /* 0xBF77A291; 0x236668E4 */
    const double pp4 = -2.37630166566501626084e-05; /* 0xBEF8EAD6; 0x120016AC */
    const double qq1 = 3.97917223959155352819e-01; /* 0x3FD97779; 0xCDDADC09 */
    const double qq2 = 6.50222499887672944485e-02; /* 0x3FB0A54C; 0x5536CEBA */
    const double qq3 = 5.08130628187576562776e-03; /* 0x3F74D022; 0xC4D36B0F */
    const double qq4 = 1.32494738004321644526e-04; /* 0x3F215DC9; 0x221C1A10 */
    const double qq5 = -3.96022827877536812320e-06; /* 0xBED09C43; 0x42A26120 */

    // Coefficients for approximation to erf in [0.84375, 1.25]
    const double pa0 = -2.36211856075265944077e-03; /* 0xBF6359B8; 0xBEF77538 */
    const double pa1 = 4.14856118683748331666e-01; /* 0x3FDA8D00; 0xAD92B34D */
    const double pa2 = -3.72207876035701323847e-01; /* 0xBFD7D240; 0xFBB8C3F1 */
    const double pa3 = 3.18346619901161753674e-01; /* 0x3FD45FCA; 0x805120E4 */
    const double pa4 = -1.10894694282396677476e-01; /* 0xBFBC6398; 0x3D3E28EC */
    const double pa5 = 3.54783043256182359371e-02; /* 0x3FA22A36; 0x599795EB */
    const double pa6 = -2.16637559486879084300e-03; /* 0xBF61BF38; 0x0A96073F */
    const double qa1 = 1.06420880400844228286e-01; /* 0x3FBB3E66; 0x18EEE323 */
    const double qa2 = 5.40397917702171048937e-01; /* 0x3FE14AF0; 0x92EB6F33 */
    const double qa3 = 7.18286544141962662868e-02; /* 0x3FB2635C; 0xD99FE9A7 */
    const double qa4 = 1.26171219808761642112e-01; /* 0x3FC02660; 0xE763351F */
    const double qa5 = 1.36370839120290507362e-02; /* 0x3F8BEDC2; 0x6B51DD1C */
    const double qa6 = 1.19844998467991074170e-02; /* 0x3F888B54; 0x5735151D */

    // Coefficients for approximation to erfc in [1.25, 1/0.35]
    const double ra0 = -9.86494403484714822705e-03; /* 0xBF843412; 0x600D6435 */
    const double ra1 = -6.93858572707181764372e-01; /* 0xBFE63416; 0xE4BA7360 */
    const double ra2 = -1.05586262253232909814e+01; /* 0xC0251E04; 0x41B0E726 */
    const double ra3 = -6.23753324503260060396e+01; /* 0xC04F300A; 0xE4CBA38D */
    const double ra4 = -1.62396669462573470355e+02; /* 0xC0644CB1; 0x84282266 */
    const double ra5 = -1.84605092906711035994e+02; /* 0xC067135C; 0xEBCCABB2 */
    const double ra6 = -8.12874355063065934246e+01; /* 0xC0545265; 0x57E4D2F2 */
    const double ra7 = -9.81432934416914548592e+00; /* 0xC023A0EF; 0xC69AC25C */
    const double sa1 = 1.96512716674392571292e+01; /* 0x4033A6B9; 0xBD707687 */
    const double sa2 = 1.37657754143519042600e+02; /* 0x4061350C; 0x526AE721 */
    const double sa3 = 4.34565877475229228821e+02; /* 0x407B290D; 0xD58A1A71 */
    const double sa4 = 6.45387271733267880336e+02; /* 0x40842B19; 0x21EC2868 */
    const double sa5 = 4.29008140027567833386e+02; /* 0x407AD021; 0x57700314 */
    const double sa6 = 1.08635005541779435134e+02; /* 0x405B28A3; 0xEE48AE2C */
    const double sa7 = 6.57024977031928170135e+00; /* 0x401A47EF; 0x8E484A93 */
    const double sa8 = -6.04244152148580987438e-02; /* 0xBFAEEFF2; 0xEE749A62 */

    // Coefficients for approximation to erfc in [1/0.35, 28]
    const double rb0 = -9.86494292470009928597e-03; /* 0xBF843412; 0x39E86F4A */
    const double rb1 = -7.99283237680523006574e-01; /* 0xBFE993BA; 0x70C285DE */
    const double rb2 = -1.77579549177547519889e+01; /* 0xC031C209; 0x555F995A */
    const double rb3 = -1.60636384855821916062e+02; /* 0xC064145D; 0x43C5ED98 */
    const double rb4 = -6.37566443368389627722e+02; /* 0xC083EC88; 0x1375F228 */
    const double rb5 = -1.02509513161107724954e+03; /* 0xC0900461; 0x6A2E5992 */
    const double rb6 = -4.83519191608651397019e+02; /* 0xC07E384E; 0x9BDC383F */
    const double sb1 = 3.03380607434824582924e+01; /* 0x403E568B; 0x261D5190 */
    const double sb2 = 3.25792512996573918826e+02; /* 0x40745CAE; 0x221B9F0A */
    const double sb3 = 1.53672958608443695994e+03; /* 0x409802EB; 0x189D5118 */
    const double sb4 = 3.19985821950859553908e+03; /* 0x40A8FFB7; 0x688C246A */
    const double sb5 = 2.55305040643316442583e+03; /* 0x40A3F219; 0xCEDF3BE6 */
    const double sb6 = 4.74528541206955367215e+02; /* 0x407DA874; 0xE79FE763 */
    const double sb7 = -2.24409524465858183362e+01; /* 0xC03670E2; 0x42712D62 */

    #endregion

    if (double.IsNaN(x))
        return double.NaN;

    if (double.IsNegativeInfinity(x))
        return -1.0;

    if (double.IsPositiveInfinity(x))
        return 1.0;

    int n0, hx, ix, i;
    double R, S, P, Q, s, y, z, r;
    unsafe
    {
        double one = 1.0;
        n0 = ((*(int*)&one) >> 29) ^ 1;
        hx = *(n0 + (int*)&x);
    }
    ix = hx & 0x7FFFFFFF;

    if (ix < 0x3FEB0000) // |x| < 0.84375
    {
        if (ix < 0x3E300000) // |x| < 2**-28
        {
            if (ix < 0x00800000)
                return 0.125 * (8.0 * x + efx8 * x); // avoid underflow
            return x + efx * x;
        }
        z = x * x;
        r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4)));
        s = 1.0 + z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5))));
        y = r / s;
        return x + x * y;
    }
    if (ix < 0x3FF40000) // 0.84375 <= |x| < 1.25
    {
        s = Math.Abs(x) - 1.0;
        P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 + s * (pa5 + s * pa6)))));
        Q = 1.0 + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 + s * (qa5 + s * qa6)))));
        if (hx >= 0)
            return erx + P / Q;
        else
            return -erx - P / Q;
    }
    if (ix >= 0x40180000) // inf > |x| >= 6
    {
        if (hx >= 0)
            return 1.0 - tiny;
        else
            return tiny - 1.0;
    }
    x = Math.Abs(x);
    s = 1.0 / (x * x);
    if (ix < 0x4006DB6E) // |x| < 1/0.35
    {
        R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 + s * (ra5 + s * (ra6 + s * ra7))))));
        S = 1.0 + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 + s * (sa5 + s * (sa6 + s * (sa7 + s * sa8)))))));
    }
    else // |x| >= 1/0.35
    {
        R = rb0 + s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 + s * (rb5 + s * rb6)))));
        S = 1.0 + s * (sb1 + s * (sb2 + s * (sb3 + s * (sb4 + s * (sb5 + s * (sb6 + s * sb7))))));
    }
    z = x;
    unsafe { *(1 - n0 + (int*)&z) = 0; }
    r = Math.Exp(-z * z - 0.5625) * Math.Exp((z - x) * (z + x) + R / S);
    if (hx >= 0)
        return 1.0 - r / x;
    else
        return r / x - 1.0;
}

/// <summary>
/// Returns the value of the complementary error function at <paramref name="x"/>.
/// </summary>
public static double Erfc(double x)
{
    /*
    Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
    *
    * Developed at SunPro, a Sun Microsystems, Inc. business.
    * Permission to use, copy, modify, and distribute this
    * software is freely granted, provided that this notice
    * is preserved.
    */

    #region Constants

    const double tiny = 1e-300;
    const double erx = 8.45062911510467529297e-01;

    // Coefficients for approximation to erf on [0, 0.84375]
    const double efx = 1.28379167095512586316e-01; /* 0x3FC06EBA; 0x8214DB69 */
    const double efx8 = 1.02703333676410069053e+00; /* 0x3FF06EBA; 0x8214DB69 */
    const double pp0 = 1.28379167095512558561e-01; /* 0x3FC06EBA; 0x8214DB68 */
    const double pp1 = -3.25042107247001499370e-01; /* 0xBFD4CD7D; 0x691CB913 */
    const double pp2 = -2.84817495755985104766e-02; /* 0xBF9D2A51; 0xDBD7194F */
    const double pp3 = -5.77027029648944159157e-03; /* 0xBF77A291; 0x236668E4 */
    const double pp4 = -2.37630166566501626084e-05; /* 0xBEF8EAD6; 0x120016AC */
    const double qq1 = 3.97917223959155352819e-01; /* 0x3FD97779; 0xCDDADC09 */
    const double qq2 = 6.50222499887672944485e-02; /* 0x3FB0A54C; 0x5536CEBA */
    const double qq3 = 5.08130628187576562776e-03; /* 0x3F74D022; 0xC4D36B0F */
    const double qq4 = 1.32494738004321644526e-04; /* 0x3F215DC9; 0x221C1A10 */
    const double qq5 = -3.96022827877536812320e-06; /* 0xBED09C43; 0x42A26120 */

    // Coefficients for approximation to erf in [0.84375, 1.25]
    const double pa0 = -2.36211856075265944077e-03; /* 0xBF6359B8; 0xBEF77538 */
    const double pa1 = 4.14856118683748331666e-01; /* 0x3FDA8D00; 0xAD92B34D */
    const double pa2 = -3.72207876035701323847e-01; /* 0xBFD7D240; 0xFBB8C3F1 */
    const double pa3 = 3.18346619901161753674e-01; /* 0x3FD45FCA; 0x805120E4 */
    const double pa4 = -1.10894694282396677476e-01; /* 0xBFBC6398; 0x3D3E28EC */
    const double pa5 = 3.54783043256182359371e-02; /* 0x3FA22A36; 0x599795EB */
    const double pa6 = -2.16637559486879084300e-03; /* 0xBF61BF38; 0x0A96073F */
    const double qa1 = 1.06420880400844228286e-01; /* 0x3FBB3E66; 0x18EEE323 */
    const double qa2 = 5.40397917702171048937e-01; /* 0x3FE14AF0; 0x92EB6F33 */
    const double qa3 = 7.18286544141962662868e-02; /* 0x3FB2635C; 0xD99FE9A7 */
    const double qa4 = 1.26171219808761642112e-01; /* 0x3FC02660; 0xE763351F */
    const double qa5 = 1.36370839120290507362e-02; /* 0x3F8BEDC2; 0x6B51DD1C */
    const double qa6 = 1.19844998467991074170e-02; /* 0x3F888B54; 0x5735151D */

    // Coefficients for approximation to erfc in [1.25, 1/0.35]
    const double ra0 = -9.86494403484714822705e-03; /* 0xBF843412; 0x600D6435 */
    const double ra1 = -6.93858572707181764372e-01; /* 0xBFE63416; 0xE4BA7360 */
    const double ra2 = -1.05586262253232909814e+01; /* 0xC0251E04; 0x41B0E726 */
    const double ra3 = -6.23753324503260060396e+01; /* 0xC04F300A; 0xE4CBA38D */
    const double ra4 = -1.62396669462573470355e+02; /* 0xC0644CB1; 0x84282266 */
    const double ra5 = -1.84605092906711035994e+02; /* 0xC067135C; 0xEBCCABB2 */
    const double ra6 = -8.12874355063065934246e+01; /* 0xC0545265; 0x57E4D2F2 */
    const double ra7 = -9.81432934416914548592e+00; /* 0xC023A0EF; 0xC69AC25C */
    const double sa1 = 1.96512716674392571292e+01; /* 0x4033A6B9; 0xBD707687 */
    const double sa2 = 1.37657754143519042600e+02; /* 0x4061350C; 0x526AE721 */
    const double sa3 = 4.34565877475229228821e+02; /* 0x407B290D; 0xD58A1A71 */
    const double sa4 = 6.45387271733267880336e+02; /* 0x40842B19; 0x21EC2868 */
    const double sa5 = 4.29008140027567833386e+02; /* 0x407AD021; 0x57700314 */
    const double sa6 = 1.08635005541779435134e+02; /* 0x405B28A3; 0xEE48AE2C */
    const double sa7 = 6.57024977031928170135e+00; /* 0x401A47EF; 0x8E484A93 */
    const double sa8 = -6.04244152148580987438e-02; /* 0xBFAEEFF2; 0xEE749A62 */

    // Coefficients for approximation to erfc in [1/0.35, 28]
    const double rb0 = -9.86494292470009928597e-03; /* 0xBF843412; 0x39E86F4A */
    const double rb1 = -7.99283237680523006574e-01; /* 0xBFE993BA; 0x70C285DE */
    const double rb2 = -1.77579549177547519889e+01; /* 0xC031C209; 0x555F995A */
    const double rb3 = -1.60636384855821916062e+02; /* 0xC064145D; 0x43C5ED98 */
    const double rb4 = -6.37566443368389627722e+02; /* 0xC083EC88; 0x1375F228 */
    const double rb5 = -1.02509513161107724954e+03; /* 0xC0900461; 0x6A2E5992 */
    const double rb6 = -4.83519191608651397019e+02; /* 0xC07E384E; 0x9BDC383F */
    const double sb1 = 3.03380607434824582924e+01; /* 0x403E568B; 0x261D5190 */
    const double sb2 = 3.25792512996573918826e+02; /* 0x40745CAE; 0x221B9F0A */
    const double sb3 = 1.53672958608443695994e+03; /* 0x409802EB; 0x189D5118 */
    const double sb4 = 3.19985821950859553908e+03; /* 0x40A8FFB7; 0x688C246A */
    const double sb5 = 2.55305040643316442583e+03; /* 0x40A3F219; 0xCEDF3BE6 */
    const double sb6 = 4.74528541206955367215e+02; /* 0x407DA874; 0xE79FE763 */
    const double sb7 = -2.24409524465858183362e+01; /* 0xC03670E2; 0x42712D62 */

    #endregion

    if (double.IsNaN(x))
        return double.NaN;

    if (double.IsNegativeInfinity(x))
        return 2.0;

    if (double.IsPositiveInfinity(x))
        return 0.0;

    int n0, hx, ix;
    double R, S, P, Q, s, y, z, r;
    unsafe
    {
        double one = 1.0;
        n0 = ((*(int*)&one) >> 29) ^ 1;
        hx = *(n0 + (int*)&x);
    }
    ix = hx & 0x7FFFFFFF;

    if (ix < 0x3FEB0000) // |x| < 0.84375
    {
        if (ix < 0x3C700000) // |x| < 2**-56
            return 1.0 - x;
        z = x * x;
        r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4)));
        s = 1.0 + z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5))));
        y = r / s;
        if (hx < 0x3FD00000) // x < 1/4
            return 1.0 - (x + x * y);
        else
        {
            r = x * y;
            r += (x - 0.5);
            return 0.5 - r;
        }
    }
    if (ix < 0x3FF40000) // 0.84375 <= |x| < 1.25
    {
        s = Math.Abs(x) - 1.0;
        P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 + s * (pa5 + s * pa6)))));
        Q = 1.0 + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 + s * (qa5 + s * qa6)))));
        if (hx >= 0)
        {
            z = 1.0 - erx;
            return z - P / Q;
        }
        else
        {
            z = erx + P / Q;
            return 1.0 + z;
        }
    }
    if (ix < 0x403C0000) // |x| < 28
    {
        x = Math.Abs(x);
        s = 1.0 / (x * x);
        if (ix < 0x4006DB6D) // |x| < 1/.35 ~ 2.857143
        {
            R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 + s * (ra5 + s * (ra6 + s * ra7))))));
            S = 1.0 + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 + s * (sa5 + s * (sa6 + s * (sa7 + s * sa8)))))));
        }
        else // |x| >= 1/.35 ~ 2.857143
        {
            if (hx < 0 && ix >= 0x40180000)
                return 2.0 - tiny; // x < -6
            R = rb0 + s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 + s * (rb5 + s * rb6)))));
            S = 1.0 + s * (sb1 + s * (sb2 + s * (sb3 + s * (sb4 + s * (sb5 + s * (sb6 + s * sb7))))));
        }
        z = x;
        unsafe { *(1 - n0 + (int*)&z) = 0; }
        r = Math.Exp(-z * z - 0.5625) *
        Math.Exp((z - x) * (z + x) + R / S);
        if (hx > 0)
            return r / x;
        else
            return 2.0 - r / x;
    }
    else
    {
        if (hx > 0)
            return tiny * tiny;
        else
            return 2.0 - tiny;
    }
}

【讨论】：

【解决方案3】：

每当数学函数或技术不可用时，例如在这种情况下，我寻找解决方案的第一个地方是书 Numerical Recipies in C。这本书现在已经是第三版了。如果这还不够，还有诸如Abramowitz and Stegun 之类的书籍，这是一本已有 40 年历史的数学函数手册，至今仍被视为重要参考。

【讨论】：

确实，第 6.2 章数字食谱，第 2 版，2 涵盖了 错误函数 并给出了实现。

【解决方案4】：

数学类中没有 erf 函数。看看这个。 functions in math class 还有这个Erf func

【讨论】：

【解决方案5】：

如果您想向后兼容 ANSI C 中的某些内容，您可以尝试使用 Universal C Runtime Library(UCRT) 中的那个

[DllImport("ucrtbase.dll")]
static extern double erfc(double  x);

【讨论】：