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Accurate calculation of scaled optional error function, erfcx ()

The scaled additional error function, usually denoted by a symbol erfcx, is mathematically defined as erfcx (x): = e x 2 erfc (x). This is often found in diffusion problems in both physics and chemistry. Although some math environments, such as MATLAB and GNU Octave , do not have this function, it is not in the standard C math library, which provides just erf()that erfc().

Although you can implement your own erfcx(), based directly on a mathematical definition, this only works in a limited input domain, because in a positive half-plane it is erfc()not enough for moderate arguments, while exp()overflow, as indicated in this question , for example.

For use with C, you can adapt some erfcx()open source versions , such as, for example, the Faadeeva package , as indicated in the answers to this question . However, these implementations usually do not provide complete accuracy for a given floating point format. For example, tests with test vectors 2 32 show the maximum error erfcx(), as provided by the Faadeev package, equal to 8.41 ulps in the positive half-plane and 511.68 ulps in the negative half-plane.

A reasonable estimate for an exact implementation will be 4 ulps, which corresponds to the accuracy of the mathematical functions in the LA Intel Vector Math profile , which I found to be a reasonable estimate for non-trivial implementations of mathematical functions that require both good accuracy and good performance.

Like the erfcx()corresponding version, with one precision erfcxf()to be performed accurately, using only the standard C mathematical library and not requiring external libraries? We can assume that C floatnad doubletypes are mapped to IEEE 754-2008 binary32and binary64floating point types . We can assume hardware support for the smooth operation of multiple additions (FMA), since it is currently supported by all major processor architectures.

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erfcx(), , :

. M. Shepherd and J. G. Laframboise, " (1 + 2 x) exp (x 2) erfc x 0 ≤ x < ∞." , 36, № 153, 1981 ., . 249-253 ()

, , . , . K (x - K)/(x + K) . "" , .

erfcx . , , erfcf. FMA .

:

/*  
 * Based on: M. M. Shepherd and J. G. Laframboise, "Chebyshev Approximation of 
 * (1+2x)exp(x^2)erfc x in 0 <= x < INF", Mathematics of Computation, Vol. 36,
 * No. 153, January 1981, pp. 249-253.
 *
 */  
float my_erfcxf (float x)
{
    float a, d, e, m, p, q, r, s, t;

    a = fmaxf (x, 0.0f - x); // NaN-preserving absolute value computation

    /* Compute q = (a-2)/(a+2) accurately. [0,INF) -> [-1,1] */
    m = a - 2.0f;
    p = a + 2.0f;
#if FAST_RCP_SSE
    r = fast_recipf_sse (p);
#else
    r = 1.0f / p;
#endif
    q = m * r;
    t = fmaf (q + 1.0f, -2.0f, a); 
    e = fmaf (q, -a, t); 
    q = fmaf (r, e, q); 

    /* Approximate (1+2*a)*exp(a*a)*erfc(a) as p(q)+1 for q in [-1,1] */
    p =              0x1.f10000p-15f;  //  5.92470169e-5
    p = fmaf (p, q,  0x1.521cc6p-13f); //  1.61224554e-4
    p = fmaf (p, q, -0x1.6b4ffep-12f); // -3.46481771e-4
    p = fmaf (p, q, -0x1.6e2a7cp-10f); // -1.39681227e-3
    p = fmaf (p, q,  0x1.3c1d7ep-10f); //  1.20588380e-3
    p = fmaf (p, q,  0x1.1cc236p-07f); //  8.69014394e-3
    p = fmaf (p, q, -0x1.069940p-07f); // -8.01387429e-3
    p = fmaf (p, q, -0x1.bc1b6cp-05f); // -5.42122945e-2
    p = fmaf (p, q,  0x1.4ff8acp-03f); //  1.64048523e-1
    p = fmaf (p, q, -0x1.54081ap-03f); // -1.66031078e-1
    p = fmaf (p, q, -0x1.7bf5cep-04f); // -9.27637145e-2
    p = fmaf (p, q,  0x1.1ba03ap-02f); //  2.76978403e-1

    /* Divide (1+p) by (1+2*a) ==> exp(a*a)*erfc(a) */
    d = a + 0.5f;
#if FAST_RCP_SSE
    r = fast_recipf_sse (d);
#else
    r = 1.0f / d;
#endif
    r = r * 0.5f;
    q = fmaf (p, r, r); // q = (p+1)/(1+2*a)
    t = q + q;
    e = (p - q) + fmaf (t, -a, 1.0f); // residual: (p+1)-q*(1+2*a)
    r = fmaf (e, r, q);

    if (a > 0x1.fffffep127f) r = 0.0f; // 3.40282347e+38 // handle INF argument

    /* Handle negative arguments: erfcx(x) = 2*exp(x*x) - erfcx(|x|) */
    if (x < 0.0f) {
        s = x * x;
        d = fmaf (x, x, -s);
        e = expf (s);
        r = e - r;
        r = fmaf (e, d + d, r); 
        r = r + e;
        if (e > 0x1.fffffep127f) r = e; // 3.40282347e+38 // avoid creating NaN
    }
    return r;
}

expf(). Intel, 13.1.3.198 /fp:strict, 2,00450 ulps 2,38412 ulps . , , expf() 2.5 ulps.

, , , , , . , erfcxf() , . , , SSE ( < 2,0 ulps), .

/* Fast reciprocal approximation. HW approximation plus Newton iteration */
float fast_recipf_sse (float a)
{
    __m128 t;
    float e, r;
    t = _mm_set_ss (a);
    t = _mm_rcp_ss (t);
    _mm_store_ss (&r, t);
    e = fmaf (0.0f - a, r, 1.0f);
    r = fmaf (e, r, r);
    return r;
}

erfcx() erfcxf(), . , , . , . Intel /fp:strict 2 32 , 2,83788 ulps 2,77856 ulps .

double my_erfcx (double x)
{
    double a, d, e, m, p, q, r, s, t;

    a = fmax (x, 0.0 - x); // NaN preserving absolute value computation

    /* Compute q = (a-4)/(a+4) accurately. [0,INF) -> [-1,1] */
    m = a - 4.0;
    p = a + 4.0;
    r = 1.0 / p;
    q = m * r;
    t = fma (q + 1.0, -4.0, a); 
    e = fma (q, -a, t); 
    q = fma (r, e, q); 

    /* Approximate (1+2*a)*exp(a*a)*erfc(a) as p(q)+1 for q in [-1,1] */
    p =             0x1.edcad78fc8044p-31;  //  8.9820305531190140e-10
    p = fma (p, q,  0x1.b1548f14735d1p-30); //  1.5764464777959401e-09
    p = fma (p, q, -0x1.a1ad2e6c4a7a8p-27); // -1.2155985739342269e-08
    p = fma (p, q, -0x1.1985b48f08574p-26); // -1.6386753783877791e-08
    p = fma (p, q,  0x1.c6a8093ac4f83p-24); //  1.0585794011876720e-07
    p = fma (p, q,  0x1.31c2b2b44b731p-24); //  7.1190423171700940e-08
    p = fma (p, q, -0x1.b87373facb29fp-21); // -8.2040389712752056e-07
    p = fma (p, q,  0x1.3fef1358803b7p-22); //  2.9796165315625938e-07
    p = fma (p, q,  0x1.7eec072bb0be3p-18); //  5.7059822144459833e-06
    p = fma (p, q, -0x1.78a680a741c4ap-17); // -1.1225056665965572e-05
    p = fma (p, q, -0x1.9951f39295cf4p-16); // -2.4397380523258482e-05
    p = fma (p, q,  0x1.3be1255ce180bp-13); //  1.5062307184282616e-04
    p = fma (p, q, -0x1.a1df71176b791p-13); // -1.9925728768782324e-04
    p = fma (p, q, -0x1.8d4aaa0099bc8p-11); // -7.5777369791018515e-04
    p = fma (p, q,  0x1.49c673066c831p-8);  //  5.0319701025945277e-03
    p = fma (p, q, -0x1.0962386ea02b7p-6);  // -1.6197733983519948e-02
    p = fma (p, q,  0x1.3079edf465cc3p-5);  //  3.7167515521269866e-02
    p = fma (p, q, -0x1.0fb06dfedc4ccp-4);  // -6.6330365820039094e-02
    p = fma (p, q,  0x1.7fee004e266dfp-4);  //  9.3732834999538536e-02
    p = fma (p, q, -0x1.9ddb23c3e14d2p-4);  // -1.0103906603588378e-01
    p = fma (p, q,  0x1.16ecefcfa4865p-4);  //  6.8097054254651804e-02
    p = fma (p, q,  0x1.f7f5df66fc349p-7);  //  1.5379652102610957e-02
    p = fma (p, q, -0x1.1df1ad154a27fp-3);  // -1.3962111684056208e-01
    p = fma (p, q,  0x1.dd2c8b74febf6p-3);  //  2.3299511862555250e-01

    /* Divide (1+p) by (1+2*a) ==> exp(a*a)*erfc(a) */
    d = a + 0.5;
    r = 1.0 / d;
    r = r * 0.5;
    q = fma (p, r, r); // q = (p+1)/(1+2*a)
    t = q + q;
    e = (p - q) + fma (t, -a, 1.0); // residual: (p+1)-q*(1+2*a)
    r = fma (e, r, q);

    /* Handle argument of infinity */
    if (a > 0x1.fffffffffffffp1023) r = 0.0;

    /* Handle negative arguments: erfcx(x) = 2*exp(x*x) - erfcx(|x|) */
    if (x < 0.0) {
        s = x * x;
        d = fma (x, x, -s);
        e = exp (s);
        r = e - r;
        r = fma (e, d + d, r); 
        r = r + e;
        if (e > 0x1.fffffffffffffp1023) r = e; // avoid creating NaN
    }
    return r;
}
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Source: https://habr.com/ru/post/1656268/

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