Math.hh

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#ifndef MATH_HH
#define MATH_HH
 
#include "likely.hh"
#include <cassert>
#include <cmath>
#include <cstdint>
 
#ifdef _MSC_VER
#include <intrin.h>
#pragma intrinsic(_BitScanForward)
#endif
 
// These constants are very common extensions, but not guaranteed to be defined
// by <cmath> when compiling in a strict standards compliant mode. Also e.g.
// visual studio does not provide them.
#ifndef M_E
#define M_E    2.7182818284590452354
#endif
#ifndef M_LN2
#define M_LN2  0.69314718055994530942 // log_e 2
#endif
#ifndef M_LN10
#define M_LN10 2.30258509299404568402 // log_e 10
#endif
#ifndef M_PI
#define M_PI   3.14159265358979323846
#endif
 
namespace Math {
 
template<typename T>
[[nodiscard]] constexpr T log2p1(T x) noexcept
{
        T result = 0;
        while (x) {
                ++result;
                x >>= 1;
        }
        return result;
}
 
template<typename T>
[[nodiscard]] constexpr bool ispow2(T x) noexcept
{
        return x && ((x & (x - 1)) == 0);
}
 
template<typename T>
[[nodiscard]] constexpr T floodRight(T x) noexcept
{
        x |= x >> 1;
        x |= x >> 2;
        x |= x >> 4;
        x |= x >> ((sizeof(x) >= 2) ?  8 : 0); // Written in a weird way to
        x |= x >> ((sizeof(x) >= 4) ? 16 : 0); // suppress compiler warnings.
        x |= x >> ((sizeof(x) >= 8) ? 32 : 0); // Generates equally efficient
        return x;                              // code.
}
 
template<typename T>
[[nodiscard]] constexpr T ceil2(T x) noexcept
{
        // classical implementation:
        //   unsigned res = 1;
        //   while (x > res) res <<= 1;
        //   return res;
 
        // optimized version
        x += (x == 0); // can be removed if argument is never zero
        return floodRight(x - 1) + 1;
}
 
[[nodiscard]] inline int16_t clipIntToShort(int x)
{
        static_assert((-1 >> 1) == -1, "right-shift must preserve sign");
        return likely(int16_t(x) == x) ? x : (0x7FFF - (x >> 31));
}
 
[[nodiscard]] inline uint8_t clipIntToByte(int x)
{
        static_assert((-1 >> 1) == -1, "right-shift must preserve sign");
        return likely(uint8_t(x) == x) ? x : ~(x >> 31);
}
 
[[nodiscard]] constexpr unsigned reverseNBits(unsigned x, unsigned bits)
{
        unsigned ret = 0;
        while (bits--) {
                ret = (ret << 1) | (x & 1);
                x >>= 1;
        }
        return ret;
 
        /* Just for fun I tried the asm version below (the carry-flag trick
         * cannot be described in plain C). It's correct and generates shorter
         * code (both less instructions and less bytes). But it doesn't
         * actually run faster on the machine I tested on, or only a tiny bit
         * (possibly because of dependency chains and processor stalls???).
         * However a big disadvantage of this asm version is that when called
         * with compile-time constant arguments, this version performs exactly
         * the same, while the version above can be further optimized by the
         * compiler (constant-propagation, loop unrolling, ...).
        unsigned ret = 0;
        if (bits) {
                asm (
                "1:     shr     %[VAL]\n"
                "       adc     %[RET],%[RET]\n"
                "       dec     %[BITS]\n"
                "       jne     1b\n"
                        : [VAL]  "+r" (val)
                        , [BITS] "+r" (bits)
                        , [RET]  "+r" (ret)
                );
        }
        return ret;
        */
 
        /* Maarten suggested the following approach with O(lg(N)) time
         * complexity (the version above is O(N)).
         *  - reverse full (32-bit) word: O(lg(N))
         *  - shift right over 32-N bits: O(1)
         * Note: In some lower end CPU the shift-over-N-bits instruction itself
         *       is O(N), in that case this whole algorithm is O(N)
         * Note2: Instead of '32' it's also possible to use a lower power of 2,
         *        as long as it's bigger than or equal to N.
         * This algorithm may or may not be faster than the version above, I
         * didn't try it yet. Also because this routine is _NOT_ performance
         * critical _AT_ALL_ currently.
         */
}
 
[[nodiscard]] constexpr uint8_t reverseByte(uint8_t a)
{
        // Classical implementation (can be extended to 16 and 32 bits)
        //   a = ((a & 0xF0) >> 4) | ((a & 0x0F) << 4);
        //   a = ((a & 0xCC) >> 2) | ((a & 0x33) << 2);
        //   a = ((a & 0xAA) >> 1) | ((a & 0x55) << 1);
        //   return a;
 
        // The versions below are specific to reverse a single byte (can't
        // easily be extended to wider types). Found these tricks on:
        //    http://graphics.stanford.edu/~seander/bithacks.html
#ifdef __x86_64
        // on 64-bit systems this is slightly faster
        return (((a * 0x80200802ULL) & 0x0884422110ULL) * 0x0101010101ULL) >> 32;
#else
        // on 32-bit systems this is faster
        return (((a * 0x0802 & 0x22110) | (a * 0x8020 & 0x88440)) * 0x10101) >> 16;
#endif
}
 
[[nodiscard]] constexpr unsigned countLeadingZeros(uint32_t x)
{
#ifdef __GNUC__
        // actually this only exists starting from gcc-3.4.x
        return __builtin_clz(x); // undefined when x==0
#else
        // gives incorrect result for x==0, but that doesn't matter here
        unsigned lz = 0;
        if (x <= 0x0000ffff) { lz += 16; x <<= 16; }
        if (x <= 0x00ffffff) { lz +=  8; x <<=  8; }
        if (x <= 0x0fffffff) { lz +=  4; x <<=  4; }
        lz += (0x55ac >> ((x >> 27) & 0x1e)) & 0x3;
        return lz;
#endif
}
 
[[nodiscard]] constexpr int countTrailingZeros(uint64_t x)
{
#ifdef __GNUC__
        return __builtin_ctzll(x); // undefined when x==0
#else
        int r = 0;
        if ((x & 0xffffffff) == 0) { r += 32; x >>= 32; }
        if ((x & 0x0000ffff) == 0) { r += 16; x >>= 16; }
        if ((x & 0x000000ff) == 0) { r +=  8; x >>=  8; }
        if ((x & 0x0000000f) == 0) { r +=  4; x >>=  4; }
        if ((x & 0x00000003) == 0) { r +=  2; x >>=  2; }
        if ((x & 0x00000001) == 0) { r +=  1; }
        return r;
#endif
}
 
[[nodiscard]] inline /*constexpr*/ unsigned findFirstSet(uint32_t x)
{
#if defined(__GNUC__)
        return __builtin_ffs(x);
#elif defined(_MSC_VER)
        unsigned long index;
        return _BitScanForward(&index, x) ? index + 1 : 0;
#else
        if (x == 0) return 0;
        int pos = 0;
        if ((x & 0xffff) == 0) { pos += 16; x >>= 16; }
        if ((x & 0x00ff) == 0) { pos +=  8; x >>=  8; }
        if ((x & 0x000f) == 0) { pos +=  4; x >>=  4; }
        if ((x & 0x0003) == 0) { pos +=  2; x >>=  2; }
        if ((x & 0x0001) == 0) { pos +=  1; }
        return pos + 1;
#endif
}
 
// Cubic Hermite Interpolation:
//   Given 4 points: (-1, y[-1]), (0, y[0]), (1, y[1]), (2, y[2])
//   Fit a polynomial:  f(x) = a*x^3 + b*x^2 + c*x + d
//     which passes through the given points at x=0 and x=1
//       f(0) = y[0]
//       f(1) = y[1]
//     and which has specific derivatives at x=0 and x=1
//       f'(0) = (y[1] - y[-1]) / 2
//       f'(1) = (y[2] - y[ 0]) / 2
//   Then evaluate this polynomial at the given x-position (x in [0, 1]).
// For more details see:
//   https://en.wikipedia.org/wiki/Cubic_Hermite_spline
//   https://www.paulinternet.nl/?page=bicubic
[[nodiscard]] constexpr float cubicHermite(const float* y, float x)
{
        assert(0.0f <= x); assert(x <= 1.0f);
        float a = -0.5f*y[-1] + 1.5f*y[0] - 1.5f*y[1] + 0.5f*y[2];
        float b =       y[-1] - 2.5f*y[0] + 2.0f*y[1] - 0.5f*y[2];
        float c = -0.5f*y[-1]             + 0.5f*y[1];
        float d =                    y[0];
        float x2 = x * x;
        float x3 = x * x2;
        return a*x3 + b*x2 + c*x + d;
}
 
} // namespace Math
 
#endif // MATH_HH