我正在开发一个加密应用程序，使用 Crypto++
作为此应用程序的一个晦涩部分，我需要确定在特定数值范围内可以存在的唯一浮点值的最大数量。

显然在现实中 0 和 1 之间存在无穷多个数字 - 但并非所有数字都可以用唯一的浮点值表示。

我有一个最小浮点值和一个最大浮点值。
我需要确定此范围内可能的浮点值的数量。

这很棘手，因为浮点值间隔得越远，离 0 越远。

例如，0 和 1 之间的可能浮点值的数量与 100,000<> 和 100,001

出于我的目的，我希望计数也包括最小值和最大值。
但是生成独占 计数的算法同样有用，因为我可以根据需要简单地添加 1 或 2。

其他问题:
如果 0 在范围内怎么办？
例如，如果最小值为 -2.0，最大值为正 2.0，我不想对 0 进行两次计数(一次为 0，再次为-0).
另外，如果最小值或最大值为+/-无穷大，会出现什么问题？
(如果最小值或最大值为 NaN，我可能会抛出异常)。

uint32_t RangeValueCount ( float fMin , float fMax )
{
    if ( fMin > fMax )
        swap ( fMin , fMax ) ;  // Ensure fMin <= fMax

    // Calculate the number of possible floating-point values between fMin and fMax.

    return ( *reinterpret_cast < uint32_t* > ( &fMax ) -
             *reinterpret_cast < uint32_t* > ( &fMin ) ) + 1 ;

    // This algorithm is obviously unsafe, assumes IEEE 754
    // How should I account for -0 or infinity?
}

如果这个问题可以解决，我认为该解决方案同样适用于 double 值(可能还有 long double 值，但由于 80 位整数值等，这可能会稍微复杂一些)

最佳答案

这是处理所有有限数字的代码。它需要 IEEE 754 算法。我用更简单、更清晰的代码替换了我以前的版本。这不是距离计算的两种实现，而是有两种将 float 转换为其编码的实现(一种通过复制位，一种通过数学操作)。之后距离的计算就比较简单了(负值要调整，然后距离就简单减法了)。

#include <ctgmath>
#include <cstdint>
#include <cstdlib>
#include <iostream>
#include <limits>


typedef double Float;       //  The floating-point type to use.
typedef std::uint64_t UInt; //  Unsigned integer of same size as Float.


/*  Define a value with only the high bit of a UInt set.  This is also the
    encoding of floating-point -0.
*/
static constexpr UInt HighBit
    = std::numeric_limits<UInt>::max() ^ std::numeric_limits<UInt>::max() >> 1;


//  Return the encoding of a floating-point number by copying its bits.
static UInt EncodingBits(Float x)
{
    UInt result;
    std::memcpy(&result, &x, sizeof result);
    return result;
}


//  Return the encoding of a floating-point number by using math.
static UInt EncodingMath(Float x)
{
    static constexpr int SignificandBits = std::numeric_limits<Float>::digits;
    static constexpr int MinimumExponent = std::numeric_limits<Float>::min_exponent;

    //  Encode the high bit.
    UInt result = std::signbit(x) ? HighBit : 0;

    //  If the value is zero, the remaining bits are zero, so we are done.
    if (x == 0) return result;

    /*  The C library provides a little-known routine to split a floating-point
        number into a significand and an exponent.  Note that this produces a
        normalized significand, not the actual significand encoding.  Notably,
        it brings significands of subnormals up to at least 1/2.  We will
        adjust for that below.  Also, this routine normalizes to [1/2, 1),
        whereas IEEE 754 is usually expressed with [1, 2), but that does not
        bother us.
    */
    int xe;
    Float xf = std::frexp(fabs(x), &xe);

    //  Test whether the number is subnormal.
    if (xe < MinimumExponent)
    {
        /*  For a subnormal value, the exponent encoding is zero, so we only
            have to insert the significand bits.  This scales the significand
            so that its low bit is scaled to the 1 position and then inserts it
            into the encoding.
        */
        result |= (UInt) std::ldexp(xf, xe - MinimumExponent + SignificandBits);
    }
    else
    {
        /*  For a normal value, the significand is encoded without its leading
            bit.  So we subtract .5 to remove that bit and then scale the
            significand so its low bit is scaled to the 1 position.
        */
        result |= (UInt) std::ldexp(xf - .5, SignificandBits);

        /*  The exponent is encoded with a bias of (in C++'s terminology)
            MinimumExponent - 1.  So we subtract that to get the exponent
            encoding and then shift it to the position of the exponent field.
            Then we insert it into the encoding.
        */
        result |= ((UInt) xe - MinimumExponent + 1) << (SignificandBits-1);
    }

    return result;
}


/*  Return the encoding of a floating-point number.  For illustration, we
    get the encoding with two different methods and compare the results.
*/
static UInt Encoding(Float x)
{
    UInt xb = EncodingBits(x);
    UInt xm = EncodingMath(x);

    if (xb != xm)
    {
        std::cerr << "Internal error encoding" << x << ".\n";
        std::cerr << "\tEncodingBits says " << xb << ".\n";
        std::cerr << "\tEncodingMath says " << xm << ".\n";
        std::exit(EXIT_FAILURE);
    }

    return xb;
}


/*  Return the distance from a to b as the number of values representable in
    Float from one to the other.  b must be greater than or equal to a.  0 is
    counted only once.
*/
static UInt Distance(Float a, Float b)
{
    UInt ae = Encoding(a);
    UInt be = Encoding(b);

    /*  For represented values from +0 to infinity, the IEEE 754 binary
        floating-points are in ascending order and are consecutive.  So we can
        simply subtract two encodings to get the number of representable values
        between them (including one endpoint but not the other).

        Unfortunately, the negative numbers are not adjacent and run the other
        direction.  To deal with this, if the number is negative, we transform
        its encoding by subtracting from the encoding of -0.  This gives us a
        consecutive sequence of encodings from the greatest magnitude finite
        negative number to the greatest finite number, in ascending order
        except for wrapping at the maximum UInt value.

        Note that this also maps the encoding of -0 to 0 (the encoding of +0),
        so the two zeroes become one point, so they are counted only once.
    */
    if (HighBit & ae) ae = HighBit - ae;
    if (HighBit & be) be = HighBit - be;

    //  Return the distance between the two transformed encodings.
    return be - ae;
}


static void Try(Float a, Float b)
{
    std::cout << "[" << a << ", " << b << "] contains "
        << Distance(a,b) + 1 << " representable values.\n";
}


int main(void)
{
    if (sizeof(Float) != sizeof(UInt))
    {
        std::cerr << "Error, UInt must be an unsigned integer the same size as Float.\n";
        std::exit(EXIT_FAILURE);
    }

    /*  Prepare some test values:  smallest positive (subnormal) value, largest
        subnormal value, smallest normal value.
    */
    Float S1 = std::numeric_limits<Float>::denorm_min();
    Float N1 = std::numeric_limits<Float>::min();
    Float S2 = N1 - S1;

    //  Test 0 <= a <= b.
    Try( 0,  0);
    Try( 0, S1);
    Try( 0, S2);
    Try( 0, N1);
    Try( 0, 1./3);
    Try(S1, S1);
    Try(S1, S2);
    Try(S1, N1);
    Try(S1, 1./3);
    Try(S2, S2);
    Try(S2, N1);
    Try(S2, 1./3);
    Try(N1, N1);
    Try(N1, 1./3);

    //  Test a <= b <= 0.
    Try(-0., -0.);
    Try(-S1, -0.);
    Try(-S2, -0.);
    Try(-N1, -0.);
    Try(-1./3, -0.);
    Try(-S1, -S1);
    Try(-S2, -S1);
    Try(-N1, -S1);
    Try(-1./3, -S1);
    Try(-S2, -S2);
    Try(-N1, -S2);
    Try(-1./3, -S2);
    Try(-N1, -N1);
    Try(-1./3, -N1);

    //  Test a <= 0 <= b.
    Try(-0., +0.);
    Try(-0., S1);
    Try(-0., S2);
    Try(-0., N1);
    Try(-0., 1./3);
    Try(-S1, +0.);
    Try(-S1, S1);
    Try(-S1, S2);
    Try(-S1, N1);
    Try(-S1, 1./3);
    Try(-S2, +0.);
    Try(-S2, S1);
    Try(-S2, S2);
    Try(-S2, N1);
    Try(-S2, 1./3);
    Try(-N1, +0.);
    Try(-N1, S1);
    Try(-N1, S2);
    Try(-N1, N1);
    Try(-1./3, 1./3);
    Try(-1./3, +0.);
    Try(-1./3, S1);
    Try(-1./3, S2);
    Try(-1./3, N1);
    Try(-1./3, 1./3);

    return 0;
}

关于C++:计算给定范围内可能的浮点值的数量，我们在Stack Overflow上找到一个类似的问题： https://stackoverflow.com/questions/48392390/