epsilon 用于各种浮点值

Question

有一个最接近于零的FLT_MIN 常数。如何最接近some number值？

举个例子：

float nearest_to_1000 = 1000.0f + epsilon;
// epsilon must be the smallest value satisfying condition:
// nearest_to_1000 > 1000.0f

我更喜欢不使用特殊函数的数字公式。

Answer 1

C 在<math.h> 标头中为此提供了一个函数。 nextafterf(x, INFINITY) 是x 之后的下一个可表示值，朝向INFINITY。

但是，如果您想自己动手：

假设 IEEE 754，以下返回您寻找的 epsilon，用于单精度（浮点）。请参阅底部有关使用库例程的说明。

#include <float.h>
#include <math.h>


/*  Return the ULP of q.

    This was inspired by Algorithm 3.5 in Siegfried M. Rump, Takeshi Ogita, and
    Shin'ichi Oishi, "Accurate Floating-Point Summation", _Technical Report
    05.12_, Faculty for Information and Communication Sciences, Hamburg
    University of Technology, November 13, 2005.
*/
float ULP(float q)
{
    // SmallestPositive is the smallest positive floating-point number.
    static const float SmallestPositive = FLT_EPSILON * FLT_MIN;

    /*  Scale is .75 ULP, so multiplying it by any significand in [1, 2) yields
        something in [.75 ULP, 1.5 ULP) (even with rounding).
    */
    static const float Scale = 0.75f * FLT_EPSILON;

    q = fabsf(q);

    /*  In fmaf(q, -Scale, q), we subtract q*Scale from q, and q*Scale is
        something more than .5 ULP but less than 1.5 ULP.  That must produce q
        - 1 ULP.  Then we subtract that from q, so we get 1 ULP.

        The significand 1 is of particular interest.  We subtract .75 ULP from
        q, which is midway between the greatest two floating-point numbers less
        than q.  Since we round to even, the lesser one is selected, which is
        less than q by 1 ULP of q, although 2 ULP of itself.
    */
    return fmaxf(SmallestPositive, q - fmaf(q, -Scale, q));
}

下面的返回值在它传递的值之后以浮点数表示的下一个值（将-0和+0视为相同）。

#include <float.h>
#include <math.h>


/*  Return the next floating-point value after the finite value q.

    This was inspired by Algorithm 3.5 in Siegfried M. Rump, Takeshi Ogita, and
    Shin'ichi Oishi, "Accurate Floating-Point Summation", _Technical Report
    05.12_, Faculty for Information and Communication Sciences, Hamburg
    University of Technology, November 13, 2005.
*/
float NextAfterf(float q)
{
    /*  Scale is .625 ULP, so multiplying it by any significand in [1, 2)
        yields something in [.625 ULP, 1.25 ULP].
    */
    static const float Scale = 0.625f * FLT_EPSILON;

    /*  Either of the following may be used, according to preference and
        performance characteristics.  In either case, use a fused multiply-add
        (fmaf) to add to q a number that is in [.625 ULP, 1.25 ULP].  When this
        is rounded to the floating-point format, it must produce the next
        number after q.
    */
#if 0
    // SmallestPositive is the smallest positive floating-point number.
    static const float SmallestPositive = FLT_EPSILON * FLT_MIN;

    if (fabsf(q) < 2*FLT_MIN)
        return q + SmallestPositive;

    return fmaf(fabsf(q), Scale, q);
#else
    return fmaf(fmaxf(fabsf(q), FLT_MIN), Scale, q);
#endif
}

使用库例程，但fmaxf（其参数的最大值）和fabsf（绝对值）很容易替换。 fmaf 应该编译为具有融合乘加的架构上的硬件指令。否则，此用法中的fmaf(a, b, c) 可以替换为(double) a * b + c。 （IEEE-754 binary64 有足够的范围和精度来替换fmaf。double 的其他选择可能没有。）

融合乘法的另一种替代方法是为q * Scale 不正常的情况添加一些测试并单独处理这些情况。对于其他情况，可以使用普通的*和+运算符分别进行乘法和加法。