Efficient implementation of log2(__m256d) in AVX2

Question

SVML's __m256d _mm256_log2_pd (__m256d a) is not available on other compilers than Intel, and they say its performance is handicapped on AMD processors. There are some implementations on the internet referred in AVX log intrinsics (_mm256_log_ps) missing in g++-4.8? and SIMD math libraries for SSE and AVX , however they seem to be more SSE than AVX2. There's also Agner Fog's vector library , however it's a large library having much more stuff that just vector log2, so from the implementation in it it's hard to figure out the essential parts for just the vector log2 operation.

So can someone just explain how to implement log2() operation for a vector of 4 double numbers efficiently? I.e. like what __m256d _mm256_log2_pd (__m256d a) does, but available for other compilers and reasonably efficient for both AMD and Intel processors.

EDIT: In my current specific case, the numbers are probabilities between 0 and 1, and logarithm is used for entropy computation: the negation of sum over all i of P[i]*log(P[i]). The range of floating-point exponents for P[i] is large, so the numbers can be close to 0. I'm not sure about accuracy, so would consider any solution starting with 30 bits of mantissa, especially a tuneable solution is preferred.

EDIT2: here is my implementation so far, based on "More efficient series" from https://en.wikipedia.org/wiki/Logarithm#Power_series . How can it be improved? (both performance and accuracy improvements are desired)

namespace {
  const __m256i gDoubleExpMask = _mm256_set1_epi64x(0x7ffULL << 52);
  const __m256i gDoubleExp0 = _mm256_set1_epi64x(1023ULL << 52);
  const __m256i gTo32bitExp = _mm256_set_epi32(0, 0, 0, 0, 6, 4, 2, 0);
  const __m128i gExpNormalizer = _mm_set1_epi32(1023);
  //TODO: some 128-bit variable or two 64-bit variables here?
  const __m256d gCommMul = _mm256_set1_pd(2.0 / 0.693147180559945309417); // 2.0/ln(2)
  const __m256d gCoeff1 = _mm256_set1_pd(1.0 / 3);
  const __m256d gCoeff2 = _mm256_set1_pd(1.0 / 5);
  const __m256d gCoeff3 = _mm256_set1_pd(1.0 / 7);
  const __m256d gCoeff4 = _mm256_set1_pd(1.0 / 9);
  const __m256d gVect1 = _mm256_set1_pd(1.0);
}

__m256d __vectorcall Log2(__m256d x) {
  const __m256i exps64 = _mm256_srli_epi64(_mm256_and_si256(gDoubleExpMask, _mm256_castpd_si256(x)), 52);
  const __m256i exps32_avx = _mm256_permutevar8x32_epi32(exps64, gTo32bitExp);
  const __m128i exps32_sse = _mm256_castsi256_si128(exps32_avx);
  const __m128i normExps = _mm_sub_epi32(exps32_sse, gExpNormalizer);
  const __m256d expsPD = _mm256_cvtepi32_pd(normExps);
  const __m256d y = _mm256_or_pd(_mm256_castsi256_pd(gDoubleExp0),
    _mm256_andnot_pd(_mm256_castsi256_pd(gDoubleExpMask), x));

  // Calculate t=(y-1)/(y+1) and t**2
  const __m256d tNum = _mm256_sub_pd(y, gVect1);
  const __m256d tDen = _mm256_add_pd(y, gVect1);
  const __m256d t = _mm256_div_pd(tNum, tDen);
  const __m256d t2 = _mm256_mul_pd(t, t); // t**2

  const __m256d t3 = _mm256_mul_pd(t, t2); // t**3
  const __m256d terms01 = _mm256_fmadd_pd(gCoeff1, t3, t);
  const __m256d t5 = _mm256_mul_pd(t3, t2); // t**5
  const __m256d terms012 = _mm256_fmadd_pd(gCoeff2, t5, terms01);
  const __m256d t7 = _mm256_mul_pd(t5, t2); // t**7
  const __m256d terms0123 = _mm256_fmadd_pd(gCoeff3, t7, terms012);
  const __m256d t9 = _mm256_mul_pd(t7, t2); // t**9
  const __m256d terms01234 = _mm256_fmadd_pd(gCoeff4, t9, terms0123);

  const __m256d log2_y = _mm256_mul_pd(terms01234, gCommMul);
  const __m256d log2_x = _mm256_add_pd(log2_y, expsPD);

  return log2_x;
}

So far my implementation gives 405 268 490 operations per second, and it seems precise till the 8th digit. The performance is measured with the following function:

#include <chrono>
#include <cmath>
#include <cstdio>
#include <immintrin.h>

// ... Log2() implementation here

const int64_t cnLogs = 100 * 1000 * 1000;

void BenchmarkLog2Vect() {
  __m256d sums = _mm256_setzero_pd();
  auto start = std::chrono::high_resolution_clock::now();
  for (int64_t i = 1; i <= cnLogs; i += 4) {
    const __m256d x = _mm256_set_pd(double(i+3), double(i+2), double(i+1), double(i));
    const __m256d logs = Log2(x);
    sums = _mm256_add_pd(sums, logs);
  }
  auto elapsed = std::chrono::high_resolution_clock::now() - start;
  double nSec = 1e-6 * std::chrono::duration_cast<std::chrono::microseconds>(elapsed).count();
  double sum = sums.m256d_f64[0] + sums.m256d_f64[1] + sums.m256d_f64[2] + sums.m256d_f64[3];
  printf("Vect Log2: %.3lf Ops/sec calculated %.3lf\n", cnLogs / nSec, sum);
}

Comparing to the results of Logarithm in C++ and assembly , the current vector implementation is 4 times faster than std::log2() and 2.5 times faster than std::log().

Specifically, the following approximation formula is used:

Answer 1

Finally here is my best result which on Ryzen 1800X @3.6GHz gives about 0.8 billion of logarithms per second (200 million vectors of 4 logarithms in each) in a single thread, and is accurate till a few last bits in the mantissa. Spoiler: see in the end how to increase performance to 0.87 billion logarithms per second.

Special cases: Negative numbers, negative infinity and NaNs with negative sign bit are treated as if they are very close to 0 (result in some garbage large negative "logarithm" values). Positive infinity and NaNs with positive sign bit result in a logarithm around 1024. If you don't like how special cases are treated, one option is to add code that checks for them and does what suits you better. This will make the computation slower.

namespace {
  // The limit is 19 because we process only high 32 bits of doubles, and out of
  //   20 bits of mantissa there, 1 bit is used for rounding.
  constexpr uint8_t cnLog2TblBits = 10; // 1024 numbers times 8 bytes = 8KB.
  constexpr uint16_t cZeroExp = 1023;
  const __m256i gDoubleNotExp = _mm256_set1_epi64x(~(0x7ffULL << 52));
  const __m256d gDoubleExp0 = _mm256_castsi256_pd(_mm256_set1_epi64x(1023ULL << 52));
  const __m256i cAvxExp2YMask = _mm256_set1_epi64x(
    ~((1ULL << (52-cnLog2TblBits)) - 1) );
  const __m256d cPlusBit = _mm256_castsi256_pd(_mm256_set1_epi64x(
    1ULL << (52 - cnLog2TblBits - 1)));
  const __m256d gCommMul1 = _mm256_set1_pd(2.0 / 0.693147180559945309417); // 2.0/ln(2)
  const __m256i gHigh32Permute = _mm256_set_epi32(0, 0, 0, 0, 7, 5, 3, 1);
  const __m128i cSseMantTblMask = _mm_set1_epi32((1 << cnLog2TblBits) - 1);
  const __m128i gExpNorm0 = _mm_set1_epi32(1023);
  // plus |cnLog2TblBits|th highest mantissa bit
  double gPlusLog2Table[1 << cnLog2TblBits];
} // anonymous namespace

void InitLog2Table() {
  for(uint32_t i=0; i<(1<<cnLog2TblBits); i++) {
    const uint64_t iZp = (uint64_t(cZeroExp) << 52)
      | (uint64_t(i) << (52 - cnLog2TblBits)) | (1ULL << (52 - cnLog2TblBits - 1));
    const double zp = *reinterpret_cast<const double*>(&iZp);
    const double l2zp = std::log2(zp);
    gPlusLog2Table[i] = l2zp;
  }
}

__m256d __vectorcall Log2TblPlus(__m256d x) {
  const __m256d zClearExp = _mm256_and_pd(_mm256_castsi256_pd(gDoubleNotExp), x);
  const __m256d z = _mm256_or_pd(zClearExp, gDoubleExp0);

  const __m128i high32 = _mm256_castsi256_si128(_mm256_permutevar8x32_epi32(
    _mm256_castpd_si256(x), gHigh32Permute));
  // This requires that x is non-negative, because the sign bit is not cleared before
  //   computing the exponent.
  const __m128i exps32 = _mm_srai_epi32(high32, 20);
  const __m128i normExps = _mm_sub_epi32(exps32, gExpNorm0);

  // Compute y as approximately equal to log2(z)
  const __m128i indexes = _mm_and_si128(cSseMantTblMask,
    _mm_srai_epi32(high32, 20 - cnLog2TblBits));
  const __m256d y = _mm256_i32gather_pd(gPlusLog2Table, indexes,
    /*number of bytes per item*/ 8);
  // Compute A as z/exp2(y)
  const __m256d exp2_Y = _mm256_or_pd(
    cPlusBit, _mm256_and_pd(z, _mm256_castsi256_pd(cAvxExp2YMask)));

  // Calculate t=(A-1)/(A+1). Both numerator and denominator would be divided by exp2_Y
  const __m256d tNum = _mm256_sub_pd(z, exp2_Y);
  const __m256d tDen = _mm256_add_pd(z, exp2_Y);

  // Compute the first polynomial term from "More efficient series" of https://en.wikipedia.org/wiki/Logarithm#Power_series
  const __m256d t = _mm256_div_pd(tNum, tDen);

  const __m256d log2_z = _mm256_fmadd_pd(t, gCommMul1, y);

  // Leading integer part for the logarithm
  const __m256d leading = _mm256_cvtepi32_pd(normExps);

  const __m256d log2_x = _mm256_add_pd(log2_z, leading);
  return log2_x;
}

It uses a combination of lookup table approach and a 1st degree polynomial, mostly described on Wikipedia (the link is in the code comments). I can afford to allocate 8KB of L1 cache here (which is a half of 16KB L1 cache available per logical core), because logarithm computation is really the bottleneck for me and there is not much more anything that needs L1 cache.

However, if you need more L1 cache for the other needs, you can decrease the amount of cache used by logarithm algorithm by reducing cnLog2TblBits to e.g. 5 at expense of decreasing the accuracy of logarithm computation.

Or to keep the accuracy high, you can increase the number of polynomial terms by adding:

namespace {
  // ...
  const __m256d gCoeff1 = _mm256_set1_pd(1.0 / 3);
  const __m256d gCoeff2 = _mm256_set1_pd(1.0 / 5);
  const __m256d gCoeff3 = _mm256_set1_pd(1.0 / 7);
  const __m256d gCoeff4 = _mm256_set1_pd(1.0 / 9);
  const __m256d gCoeff5 = _mm256_set1_pd(1.0 / 11);
}

And then changing the tail of Log2TblPlus() after line const __m256d t = _mm256_div_pd(tNum, tDen);:

  const __m256d t2 = _mm256_mul_pd(t, t); // t**2

  const __m256d t3 = _mm256_mul_pd(t, t2); // t**3
  const __m256d terms01 = _mm256_fmadd_pd(gCoeff1, t3, t);
  const __m256d t5 = _mm256_mul_pd(t3, t2); // t**5
  const __m256d terms012 = _mm256_fmadd_pd(gCoeff2, t5, terms01);
  const __m256d t7 = _mm256_mul_pd(t5, t2); // t**7
  const __m256d terms0123 = _mm256_fmadd_pd(gCoeff3, t7, terms012);
  const __m256d t9 = _mm256_mul_pd(t7, t2); // t**9
  const __m256d terms01234 = _mm256_fmadd_pd(gCoeff4, t9, terms0123);
  const __m256d t11 = _mm256_mul_pd(t9, t2); // t**11
  const __m256d terms012345 = _mm256_fmadd_pd(gCoeff5, t11, terms01234);

  const __m256d log2_z = _mm256_fmadd_pd(terms012345, gCommMul1, y);

Then comment // Leading integer part for the logarithm and the rest unchanged follow.

Normally you don't need that many terms, even for a few-bit table, I just provided the coefficients and computations for reference. It's likely that if cnLog2TblBits==5, you won't need anything beyond terms012. But I haven't done such measurements, you need to experiment what suits your needs.

The less polynomial terms you compute, obviously, the faster the computations are.

---

EDIT: this question In what situation would the AVX2 gather instructions be faster than individually loading the data? suggests that you may get a performance improvement if

const __m256d y = _mm256_i32gather_pd(gPlusLog2Table, indexes,
  /*number of bytes per item*/ 8);

is replaced by

const __m256d y = _mm256_set_pd(gPlusLog2Table[indexes.m128i_u32[3]],
  gPlusLog2Table[indexes.m128i_u32[2]],
  gPlusLog2Table[indexes.m128i_u32[1]],
  gPlusLog2Table[indexes.m128i_u32[0]]);

For my implementation it saves about 1.5 cycle, reducing the total cycle count to compute 4 logarithms from 18 to 16.5, thus the performance rises to 0.87 billion logarithms per second. I'm leaving the current implementation as is because it's more idiomatic and shoud be faster once the CPUs start doing gather operations right (with coalescing like GPUs do).

EDIT2: on Ryzen CPU (but not on Intel) you can get a little more speedup (about 0.5 cycle) by replacing

const __m128i high32 = _mm256_castsi256_si128(_mm256_permutevar8x32_epi32(
  _mm256_castpd_si256(x), gHigh32Permute));

with

  const __m128 hiLane = _mm_castpd_ps(_mm256_extractf128_pd(x, 1));
  const __m128 loLane = _mm_castpd_ps(_mm256_castpd256_pd128(x));
  const __m128i high32 = _mm_castps_si128(_mm_shuffle_ps(loLane, hiLane,
    _MM_SHUFFLE(3, 1, 3, 1)));