How to efficiently perform double/int64 conversion

Question

SSE2 has instructions for converting vectors between single-precision floats and 32-bit integers.

• _mm_cvtps_epi32()
• _mm_cvtepi32_ps()

But there are no equivalents for double-precision and 64-bit integers. In other words, they are missing:

• _mm_cvtpd_epi64()
• _mm_cvtepi64_pd()

It seems that AVX doesn't have them either.

What is the most efficient way to simulate these intrinsics?

Answer 1

This answer is about 64 bit integer to double conversion, without cutting corners. In a previous version of this answer (see paragraph Fast and accurate conversion by splitting ...., below), it was shown that it is quite efficient to split the 64-bit integers in a 32-bit low and a 32-bit high part, convert these parts to double, and compute low + high * 2^32.

The instruction counts of these conversions were:

• int64_to_double_full_range 9 instructions (with mul and add as one fma)
• uint64_to_double_full_range 7 instructions (with mul and add as one fma)

Inspired by Mysticial's updated answer, with better optimized accurate conversions, I further optimized the int64_t to double conversion:

• int64_to_double_fast_precise: 5 instructions.
• uint64_to_double_fast_precise: 5 instructions.

The int64_to_double_fast_precise conversion takes one instruction less than Mysticial's solution. The uint64_to_double_fast_precise code is essentially identical to Mysticial's solution (but with a vpblendd instead of vpblendw). It is included here because of its similarities with the int64_to_double_fast_precise conversion: The instructions are identical, only the constants differ:

#include <stdio.h>
#include <immintrin.h>
#include <stdint.h>

__m256d int64_to_double_fast_precise(const __m256i v)
/* Optimized full range int64_t to double conversion           */
/* Emulate _mm256_cvtepi64_pd()                                */
{
    __m256i magic_i_lo   = _mm256_set1_epi64x(0x4330000000000000);                /* 2^52               encoded as floating-point  */
    __m256i magic_i_hi32 = _mm256_set1_epi64x(0x4530000080000000);                /* 2^84 + 2^63        encoded as floating-point  */
    __m256i magic_i_all  = _mm256_set1_epi64x(0x4530000080100000);                /* 2^84 + 2^63 + 2^52 encoded as floating-point  */
    __m256d magic_d_all  = _mm256_castsi256_pd(magic_i_all);

    __m256i v_lo         = _mm256_blend_epi32(magic_i_lo, v, 0b01010101);         /* Blend the 32 lowest significant bits of v with magic_int_lo                                                   */
    __m256i v_hi         = _mm256_srli_epi64(v, 32);                              /* Extract the 32 most significant bits of v                                                                     */
            v_hi         = _mm256_xor_si256(v_hi, magic_i_hi32);                  /* Flip the msb of v_hi and blend with 0x45300000                                                                */
    __m256d v_hi_dbl     = _mm256_sub_pd(_mm256_castsi256_pd(v_hi), magic_d_all); /* Compute in double precision:                                                                                  */
    __m256d result       = _mm256_add_pd(v_hi_dbl, _mm256_castsi256_pd(v_lo));    /* (v_hi - magic_d_all) + v_lo  Do not assume associativity of floating point addition !!                        */
            return result;                                                        /* With gcc use -O3, then -fno-associative-math is default. Do not use -Ofast, which enables -fassociative-math! */
                                                                                  /* With icc use -fp-model precise                                                                                */
}


__m256d uint64_to_double_fast_precise(const __m256i v)                    
/* Optimized full range uint64_t to double conversion          */
/* This code is essentially identical to Mysticial's solution. */
/* Emulate _mm256_cvtepu64_pd()                                */
{
    __m256i magic_i_lo   = _mm256_set1_epi64x(0x4330000000000000);                /* 2^52        encoded as floating-point  */
    __m256i magic_i_hi32 = _mm256_set1_epi64x(0x4530000000000000);                /* 2^84        encoded as floating-point  */
    __m256i magic_i_all  = _mm256_set1_epi64x(0x4530000000100000);                /* 2^84 + 2^52 encoded as floating-point  */
    __m256d magic_d_all  = _mm256_castsi256_pd(magic_i_all);

    __m256i v_lo         = _mm256_blend_epi32(magic_i_lo, v, 0b01010101);         /* Blend the 32 lowest significant bits of v with magic_int_lo                                                   */
    __m256i v_hi         = _mm256_srli_epi64(v, 32);                              /* Extract the 32 most significant bits of v                                                                     */
            v_hi         = _mm256_xor_si256(v_hi, magic_i_hi32);                  /* Blend v_hi with 0x45300000                                                                                    */
    __m256d v_hi_dbl     = _mm256_sub_pd(_mm256_castsi256_pd(v_hi), magic_d_all); /* Compute in double precision:                                                                                  */
    __m256d result       = _mm256_add_pd(v_hi_dbl, _mm256_castsi256_pd(v_lo));    /* (v_hi - magic_d_all) + v_lo  Do not assume associativity of floating point addition !!                        */
            return result;                                                        /* With gcc use -O3, then -fno-associative-math is default. Do not use -Ofast, which enables -fassociative-math! */
                                                                                  /* With icc use -fp-model precise                                                                                */
}


int main(){
    int i;
    uint64_t j;
    __m256i j_4;
    __m256d v;
    double x[4];
    double x0, x1, a0, a1;

    j = 0ull;
    printf("\nAccurate int64_to_double\n");
    for (i = 0; i < 260; i++){
        j_4= _mm256_set_epi64x(0, 0, -j, j);

        v  = int64_to_double_fast_precise(j_4);
        _mm256_storeu_pd(x,v);
        x0 = x[0];
        x1 = x[1];
        a0 = _mm_cvtsd_f64(_mm_cvtsi64_sd(_mm_setzero_pd(),j));
        a1 = _mm_cvtsd_f64(_mm_cvtsi64_sd(_mm_setzero_pd(),-j));
        printf(" j =%21li   v =%23.1f   v=%23.1f   -v=%23.1f   -v=%23.1f   d=%.1f   d=%.1f\n", j, x0, a0, x1, a1, x0-a0, x1-a1);
        j  = j+(j>>2)-(j>>5)+1ull;
    }

    j = 0ull;
    printf("\nAccurate uint64_to_double\n");
    for (i = 0; i < 260; i++){
        if (i==258){j=-1;}
        if (i==259){j=-2;}
        j_4= _mm256_set_epi64x(0, 0, -j, j);

        v  = uint64_to_double_fast_precise(j_4);
        _mm256_storeu_pd(x,v);
        x0 = x[0];
        x1 = x[1];
        a0 = (double)((uint64_t)j);
        a1 = (double)((uint64_t)-j);
        printf(" j =%21li   v =%23.1f   v=%23.1f   -v=%23.1f   -v=%23.1f   d=%.1f   d=%.1f\n", j, x0, a0, x1, a1, x0-a0, x1-a1);
        j  = j+(j>>2)-(j>>5)+1ull;
    }
    return 0;
}

The conversions may fail if unsafe math optimization options are enabled. With gcc, -O3 is safe, but -Ofast may lead to wrong results, because we may not assume associativity of floating point addition here (the same holds for Mysticial's conversions). With icc use -fp-model precise.

---

Fast and accurate conversion by splitting the 64-bit integers in a 32-bit low and a 32-bit high part.

We assume that both the integer input and the double output are in 256 bit wide AVX registers. Two approaches are considered:

1. int64_to_double_based_on_cvtsi2sd(): as suggested in the comments on the question, use cvtsi2sd 4 times together with some data shuffling. Unfortunately both cvtsi2sd and the data shuffling instructions need execution port 5. This limits the performance of this approach.

2. int64_to_double_full_range(): we can use Mysticial's fast conversion method twice in order to get an accurate conversion for the full 64 bit integer range. The 64-bit integer is split in a 32-bit low and a 32-bit high part ,similarly as in the answers to this question: How to perform uint32/float conversion with SSE? . Each of these pieces is suitable for Mysticial's integer to double conversion. Finally the high part is multiplied by 2^32 and added to the low part. The signed conversion is a little bit more complicted than the unsigned conversion (uint64_to_double_full_range()), because srai_epi64() doesn't exist.

Code:

#include <stdio.h>
#include <immintrin.h>
#include <stdint.h>

/* 
gcc -O3 -Wall -m64 -mfma -mavx2 -march=broadwell cvt_int_64_double.c
./a.out A
time ./a.out B
time ./a.out C
etc.
*/


inline __m256d uint64_to_double256(__m256i x){                  /*  Mysticial's fast uint64_to_double. Works for inputs in the range: [0, 2^52)     */
    x = _mm256_or_si256(x, _mm256_castpd_si256(_mm256_set1_pd(0x0010000000000000)));
    return _mm256_sub_pd(_mm256_castsi256_pd(x), _mm256_set1_pd(0x0010000000000000));
}

inline __m256d int64_to_double256(__m256i x){                   /*  Mysticial's fast int64_to_double. Works for inputs in the range: (-2^51, 2^51)  */
    x = _mm256_add_epi64(x, _mm256_castpd_si256(_mm256_set1_pd(0x0018000000000000)));
    return _mm256_sub_pd(_mm256_castsi256_pd(x), _mm256_set1_pd(0x0018000000000000));
}


__m256d int64_to_double_full_range(const __m256i v)
{
    __m256i msk_lo       =_mm256_set1_epi64x(0xFFFFFFFF);
    __m256d cnst2_32_dbl =_mm256_set1_pd(4294967296.0);                 /* 2^32                                                                    */

    __m256i v_lo         = _mm256_and_si256(v,msk_lo);                  /* extract the 32 lowest significant bits of v                             */
    __m256i v_hi         = _mm256_srli_epi64(v,32);                     /* 32 most significant bits of v. srai_epi64 doesn't exist                 */
    __m256i v_sign       = _mm256_srai_epi32(v,32);                     /* broadcast sign bit to the 32 most significant bits                      */
            v_hi         = _mm256_blend_epi32(v_hi,v_sign,0b10101010);  /* restore the correct sign of v_hi                                        */
    __m256d v_lo_dbl     = int64_to_double256(v_lo);                    /* v_lo is within specified range of int64_to_double                       */ 
    __m256d v_hi_dbl     = int64_to_double256(v_hi);                    /* v_hi is within specified range of int64_to_double                       */ 
            v_hi_dbl     = _mm256_mul_pd(cnst2_32_dbl,v_hi_dbl);        /* _mm256_mul_pd and _mm256_add_pd may compile to a single fma instruction */
    return _mm256_add_pd(v_hi_dbl,v_lo_dbl);                            /* rounding occurs if the integer doesn't exist as a double                */   
}


__m256d int64_to_double_based_on_cvtsi2sd(const __m256i v)
{   __m128d zero         = _mm_setzero_pd();                            /* to avoid uninitialized variables in_mm_cvtsi64_sd                       */
    __m128i v_lo         = _mm256_castsi256_si128(v);
    __m128i v_hi         = _mm256_extracti128_si256(v,1);
    __m128d v_0          = _mm_cvtsi64_sd(zero,_mm_cvtsi128_si64(v_lo));
    __m128d v_2          = _mm_cvtsi64_sd(zero,_mm_cvtsi128_si64(v_hi));
    __m128d v_1          = _mm_cvtsi64_sd(zero,_mm_extract_epi64(v_lo,1));
    __m128d v_3          = _mm_cvtsi64_sd(zero,_mm_extract_epi64(v_hi,1));
    __m128d v_01         = _mm_unpacklo_pd(v_0,v_1);
    __m128d v_23         = _mm_unpacklo_pd(v_2,v_3);
    __m256d v_dbl        = _mm256_castpd128_pd256(v_01);
            v_dbl        = _mm256_insertf128_pd(v_dbl,v_23,1);
    return v_dbl;
}


__m256d uint64_to_double_full_range(const __m256i v)                    
{
    __m256i msk_lo       =_mm256_set1_epi64x(0xFFFFFFFF);
    __m256d cnst2_32_dbl =_mm256_set1_pd(4294967296.0);                 /* 2^32                                                                    */

    __m256i v_lo         = _mm256_and_si256(v,msk_lo);                  /* extract the 32 lowest significant bits of v                             */
    __m256i v_hi         = _mm256_srli_epi64(v,32);                     /* 32 most significant bits of v                                           */
    __m256d v_lo_dbl     = uint64_to_double256(v_lo);                   /* v_lo is within specified range of uint64_to_double                      */ 
    __m256d v_hi_dbl     = uint64_to_double256(v_hi);                   /* v_hi is within specified range of uint64_to_double                      */ 
            v_hi_dbl     = _mm256_mul_pd(cnst2_32_dbl,v_hi_dbl);        
    return _mm256_add_pd(v_hi_dbl,v_lo_dbl);                            /* rounding may occur for inputs >2^52                                     */ 
}



int main(int argc, char **argv){
  int i;
  uint64_t j;
  __m256i j_4, j_inc;
  __m256d v, v_acc;
  double x[4];
  char test = argv[1][0];

  if (test=='A'){               /* test the conversions for several integer values                                       */
    j = 1ull;
    printf("\nint64_to_double_full_range\n");
    for (i = 0; i<30; i++){
      j_4= _mm256_set_epi64x(j-3,j+3,-j,j);
      v  = int64_to_double_full_range(j_4);
      _mm256_storeu_pd(x,v);
      printf("j =%21li    v =%23.1f    -v=%23.1f    v+3=%23.1f    v-3=%23.1f  \n",j,x[0],x[1],x[2],x[3]);
      j  = j*7ull;
    }

    j = 1ull;
    printf("\nint64_to_double_based_on_cvtsi2sd\n");
    for (i = 0; i<30; i++){
      j_4= _mm256_set_epi64x(j-3,j+3,-j,j);
      v  = int64_to_double_based_on_cvtsi2sd(j_4);
      _mm256_storeu_pd(x,v);
      printf("j =%21li    v =%23.1f    -v=%23.1f    v+3=%23.1f    v-3=%23.1f  \n",j,x[0],x[1],x[2],x[3]);
      j  = j*7ull;
    }

    j = 1ull;                       
    printf("\nuint64_to_double_full_range\n");
    for (i = 0; i<30; i++){
      j_4= _mm256_set_epi64x(j-3,j+3,j,j);
      v  = uint64_to_double_full_range(j_4);
      _mm256_storeu_pd(x,v);
      printf("j =%21lu    v =%23.1f   v+3=%23.1f    v-3=%23.1f \n",j,x[0],x[2],x[3]);
      j  = j*7ull;    
    }
  }
  else{
    j_4   = _mm256_set_epi64x(-123,-4004,-312313,-23412731);  
    j_inc = _mm256_set_epi64x(1,1,1,1);  
    v_acc = _mm256_setzero_pd();
    switch(test){

      case 'B' :{                  
        printf("\nLatency int64_to_double_cvtsi2sd()\n");      /* simple test to get a rough idea of the latency of int64_to_double_cvtsi2sd()     */
        for (i = 0; i<1000000000; i++){
          v  =int64_to_double_based_on_cvtsi2sd(j_4);
          j_4= _mm256_castpd_si256(v);                         /* cast without conversion, use output as an input in the next step                 */
        }
        _mm256_storeu_pd(x,v);
      }
      break;

      case 'C' :{                  
        printf("\nLatency int64_to_double_full_range()\n");    /* simple test to get a rough idea of the latency of int64_to_double_full_range()    */
        for (i = 0; i<1000000000; i++){
          v  = int64_to_double_full_range(j_4);
          j_4= _mm256_castpd_si256(v);
        }
        _mm256_storeu_pd(x,v);
      }
      break;

      case 'D' :{                  
        printf("\nThroughput int64_to_double_cvtsi2sd()\n");   /* simple test to get a rough idea of the throughput of int64_to_double_cvtsi2sd()   */
        for (i = 0; i<1000000000; i++){
          j_4   = _mm256_add_epi64(j_4,j_inc);                 /* each step a different input                                                       */
          v     = int64_to_double_based_on_cvtsi2sd(j_4);
          v_acc = _mm256_xor_pd(v,v_acc);                      /* use somehow the results                                                           */
        }
        _mm256_storeu_pd(x,v_acc);
      }
      break;

      case 'E' :{                  
        printf("\nThroughput int64_to_double_full_range()\n"); /* simple test to get a rough idea of the throughput of int64_to_double_full_range() */
        for (i = 0; i<1000000000; i++){
          j_4   = _mm256_add_epi64(j_4,j_inc);  
          v     = int64_to_double_full_range(j_4);
          v_acc = _mm256_xor_pd(v,v_acc);           
        }    
        _mm256_storeu_pd(x,v_acc);
      }
      break;

      default : {}
    }  
    printf("v =%23.1f    -v =%23.1f    v =%23.1f    -v =%23.1f  \n",x[0],x[1],x[2],x[3]);
  }

  return 0;
}

The actual performance of these functions may depend on the surrounding code and the cpu generation.

Timing results for 1e9 conversions (256 bit wide) with simple tests B, C, D, and E in the code above, on an intel skylake i5 6500 system:

Latency experiment int64_to_double_based_on_cvtsi2sd()      (test B)  5.02 sec.
Latency experiment int64_to_double_full_range()             (test C)  3.77 sec.
Throughput experiment int64_to_double_based_on_cvtsi2sd()   (test D)  2.82 sec.
Throughput experiment int64_to_double_full_range()          (test E)  1.07 sec.

The difference in throughput between int64_to_double_full_range() and int64_to_double_based_on_cvtsi2sd() is larger than I expected.

Answer 2

There's no single instruction until AVX512, which added conversion to/from 64-bit integers, signed or unsigned. (Also support for conversion to/from 32-bit unsigned). See intrinsics like _mm512_cvtpd_epi64 and the narrower AVX512VL versions, like _mm256_cvtpd_epi64.

If you only have AVX2 or less, you'll need tricks like below for packed-conversion. (For scalar, x86-64 has scalar int64_t <-> double or float from SSE2, but scalar uint64_t <-> FP requires tricks until AVX512 adds unsigned conversions. Scalar 32-bit unsigned can be done by zero-extending to 64-bit signed.)

---

If you're willing to cut corners, double <-> int64 conversions can be done in only two instructions:

• If you don't care about infinity or NaN.
• For double <-> int64_t, you only care about values in the range [-2^51, 2^51].
• For double <-> uint64_t, you only care about values in the range [0, 2^52).

double -> uint64_t

//  Only works for inputs in the range: [0, 2^52)
__m128i double_to_uint64(__m128d x){
    x = _mm_add_pd(x, _mm_set1_pd(0x0010000000000000));
    return _mm_xor_si128(
        _mm_castpd_si128(x),
        _mm_castpd_si128(_mm_set1_pd(0x0010000000000000))
    );
}

double -> int64_t

//  Only works for inputs in the range: [-2^51, 2^51]
__m128i double_to_int64(__m128d x){
    x = _mm_add_pd(x, _mm_set1_pd(0x0018000000000000));
    return _mm_sub_epi64(
        _mm_castpd_si128(x),
        _mm_castpd_si128(_mm_set1_pd(0x0018000000000000))
    );
}

uint64_t -> double

//  Only works for inputs in the range: [0, 2^52)
__m128d uint64_to_double(__m128i x){
    x = _mm_or_si128(x, _mm_castpd_si128(_mm_set1_pd(0x0010000000000000)));
    return _mm_sub_pd(_mm_castsi128_pd(x), _mm_set1_pd(0x0010000000000000));
}

int64_t -> double

//  Only works for inputs in the range: [-2^51, 2^51]
__m128d int64_to_double(__m128i x){
    x = _mm_add_epi64(x, _mm_castpd_si128(_mm_set1_pd(0x0018000000000000)));
    return _mm_sub_pd(_mm_castsi128_pd(x), _mm_set1_pd(0x0018000000000000));
}

---

Rounding Behavior:

• For the double -> uint64_t conversion, rounding works correctly following the current rounding mode. (which is usually round-to-even)
• For the double -> int64_t conversion, rounding will follow the current rounding mode for all modes except truncation. If the current rounding mode is truncation (round towards zero), it will actually round towards negative infinity.

---

How does it work?

Despite this trick being only 2 instructions, it's not entirely self-explanatory.

The key is to recognize that for double-precision floating-point, values in the range [2^52, 2^53) have the "binary place" just below the lowest bit of the mantissa. In other words, if you zero out the exponent and sign bits, the mantissa becomes precisely the integer representation.

To convert x from double -> uint64_t, you add the magic number M which is the floating-point value of 2^52. This puts x into the "normalized" range of [2^52, 2^53) and conveniently rounds away the fractional part bits.

Now all that's left is to remove the upper 12 bits. This is easily done by masking it out. The fastest way is to recognize that those upper 12 bits are identical to those of M. So rather than introducing an additional mask constant, we can simply subtract or XOR by M. XOR has more throughput.

Converting from uint64_t -> double is simply the reverse of this process. You add back the exponent bits of M. Then un-normalize the number by subtracting M in floating-point.

The signed integer conversions are slightly trickier since you need to deal with the 2's complement sign-extension. I'll leave those as an exercise for the reader.

Related: A fast method to round a double to a 32-bit int explained

---

Full Range int64 -> double:

After many years, I finally had a need for this.

• 5 instructions for uint64_t -> double
• 6 instructions for int64_t -> double

uint64_t -> double

__m128d uint64_to_double_full(__m128i x){
    __m128i xH = _mm_srli_epi64(x, 32);
    xH = _mm_or_si128(xH, _mm_castpd_si128(_mm_set1_pd(19342813113834066795298816.)));          //  2^84
    __m128i xL = _mm_blend_epi16(x, _mm_castpd_si128(_mm_set1_pd(0x0010000000000000)), 0xcc);   //  2^52
    __m128d f = _mm_sub_pd(_mm_castsi128_pd(xH), _mm_set1_pd(19342813118337666422669312.));     //  2^84 + 2^52
    return _mm_add_pd(f, _mm_castsi128_pd(xL));
}

int64_t -> double

__m128d int64_to_double_full(__m128i x){
    __m128i xH = _mm_srai_epi32(x, 16);
    xH = _mm_blend_epi16(xH, _mm_setzero_si128(), 0x33);
    xH = _mm_add_epi64(xH, _mm_castpd_si128(_mm_set1_pd(442721857769029238784.)));              //  3*2^67
    __m128i xL = _mm_blend_epi16(x, _mm_castpd_si128(_mm_set1_pd(0x0010000000000000)), 0x88);   //  2^52
    __m128d f = _mm_sub_pd(_mm_castsi128_pd(xH), _mm_set1_pd(442726361368656609280.));          //  3*2^67 + 2^52
    return _mm_add_pd(f, _mm_castsi128_pd(xL));
}

These work for the entire 64-bit range and are correctly rounded to the current rounding behavior.

These are similar wim's answer below - but with more abusive optimizations. As such, deciphering these will also be left as an exercise to the reader.