Fast way to get a close power-of-2 number (floating-point)
Function s = get_scale(z) computes the "close power of 2". Since the fraction bits of s
are zero, the inverse of s is just an (inexpensive) integer subtraction: see function inv_of_scale.
On x86 get_scale and inv_of_scale compile to quite efficient assembly with clang.
Compiler clang translates the ternary operators to minsd and maxsd,
see also Peter Cordes' comment.
With gcc, it is slightly more efficient to
translate these functions to x86 intrinsics
code (get_scale_x86 and inv_of_scale_x86), see Godbolt.
Note that C explicitly permits type-punning
through a union, whereas C++ (c++11) has no such permission
Although gcc 8.2 and clang 7.0 do not complain about the union, you can improve
the C++ portabily by using the memcpy trick instead of the
union trick. Such a modification of the code should be trivial.
The code should handle subnormals correctly.
#include<stdio.h>
#include<stdint.h>
#include<immintrin.h>
/* gcc -Wall -m64 -O3 -march=sandybridge dbl_scale.c */
union dbl_int64{
double d;
uint64_t i;
};
double get_scale(double t){
union dbl_int64 x;
union dbl_int64 x_min;
union dbl_int64 x_max;
uint64_t mask_i;
/* 0xFEDCBA9876543210 */
x_min.i = 0x0010000000000000ull;
x_max.i = 0x7FD0000000000000ull;
mask_i = 0x7FF0000000000000ull;
x.d = t;
x.i = x.i & mask_i; /* Set fraction bits to zero, take absolute value */
x.d = (x.d < x_min.d) ? x_min.d : x.d; /* If subnormal: set exponent to 1 */
x.d = (x.d > x_max.d) ? x_max.d : x.d; /* If exponent is very large: set exponent to 7FD, otherwise the inverse is a subnormal */
return x.d;
}
double get_scale_x86(double t){
__m128d x = _mm_set_sd(t);
__m128d x_min = _mm_castsi128_pd(_mm_set1_epi64x(0x0010000000000000ull));
__m128d x_max = _mm_castsi128_pd(_mm_set1_epi64x(0x7FD0000000000000ull));
__m128d mask = _mm_castsi128_pd(_mm_set1_epi64x(0x7FF0000000000000ull));
x = _mm_and_pd(x, mask);
x = _mm_max_sd(x, x_min);
x = _mm_min_sd(x, x_max);
return _mm_cvtsd_f64(x);
}
/* Compute the inverse 1/t of a double t with all zero fraction bits */
/* and exponent between the limits of function get_scale */
/* A single integer subtraction is much less expensive than a */
/* floating point division. */
double inv_of_scale(double t){
union dbl_int64 x;
/* 0xFEDCBA9876543210 */
uint64_t inv_mask = 0x7FE0000000000000ull;
x.d = t;
x.i = inv_mask - x.i;
return x.d;
}
double inv_of_scale_x86(double t){
__m128i inv_mask = _mm_set1_epi64x(0x7FE0000000000000ull);
__m128d x = _mm_set_sd(t);
__m128i x_i = _mm_sub_epi64(inv_mask, _mm_castpd_si128(x));
return _mm_cvtsd_f64(_mm_castsi128_pd(x_i));
}
int main(){
int n = 14;
int i;
/* Several example values, 4.94e-324 is the smallest subnormal */
double y[14] = { 4.94e-324, 1.1e-320, 1.1e-300, 1.1e-5, 0.7, 1.7, 123.1, 1.1e300,
1.79e308, -1.1e-320, -0.7, -1.7, -123.1, -1.1e307};
double z, s, u;
printf("Portable code:\n");
printf(" x pow_of_2 inverse pow2*inv x*inverse \n");
for (i = 0; i < n; i++){
z = y[i];
s = get_scale(z);
u = inv_of_scale(s);
printf("%14e %14e %14e %14e %14e\n", z, s, u, s*u, z*u);
}
printf("\nx86 specific SSE code:\n");
printf(" x pow_of_2 inverse pow2*inv x*inverse \n");
for (i = 0; i < n; i++){
z = y[i];
s = get_scale_x86(z);
u = inv_of_scale_x86(s);
printf("%14e %14e %14e %14e %14e\n", z, s, u, s*u, z*u);
}
return 0;
}
The output looks fine:
Portable code:
x pow_of_2 inverse pow2*inv x*inverse
4.940656e-324 2.225074e-308 4.494233e+307 1.000000e+00 2.220446e-16
1.099790e-320 2.225074e-308 4.494233e+307 1.000000e+00 4.942713e-13
1.100000e-300 7.466109e-301 1.339386e+300 1.000000e+00 1.473324e+00
1.100000e-05 7.629395e-06 1.310720e+05 1.000000e+00 1.441792e+00
7.000000e-01 5.000000e-01 2.000000e+00 1.000000e+00 1.400000e+00
1.700000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.700000e+00
1.231000e+02 6.400000e+01 1.562500e-02 1.000000e+00 1.923437e+00
1.100000e+300 6.696929e+299 1.493222e-300 1.000000e+00 1.642544e+00
1.790000e+308 4.494233e+307 2.225074e-308 1.000000e+00 3.982882e+00
-1.099790e-320 2.225074e-308 4.494233e+307 1.000000e+00 -4.942713e-13
-7.000000e-01 5.000000e-01 2.000000e+00 1.000000e+00 -1.400000e+00
-1.700000e+00 1.000000e+00 1.000000e+00 1.000000e+00 -1.700000e+00
-1.231000e+02 6.400000e+01 1.562500e-02 1.000000e+00 -1.923437e+00
-1.100000e+307 5.617791e+306 1.780059e-307 1.000000e+00 -1.958065e+00
x86 specific SSE code:
x pow_of_2 inverse pow2*inv x*inverse
4.940656e-324 2.225074e-308 4.494233e+307 1.000000e+00 2.220446e-16
1.099790e-320 2.225074e-308 4.494233e+307 1.000000e+00 4.942713e-13
1.100000e-300 7.466109e-301 1.339386e+300 1.000000e+00 1.473324e+00
1.100000e-05 7.629395e-06 1.310720e+05 1.000000e+00 1.441792e+00
7.000000e-01 5.000000e-01 2.000000e+00 1.000000e+00 1.400000e+00
1.700000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.700000e+00
1.231000e+02 6.400000e+01 1.562500e-02 1.000000e+00 1.923437e+00
1.100000e+300 6.696929e+299 1.493222e-300 1.000000e+00 1.642544e+00
1.790000e+308 4.494233e+307 2.225074e-308 1.000000e+00 3.982882e+00
-1.099790e-320 2.225074e-308 4.494233e+307 1.000000e+00 -4.942713e-13
-7.000000e-01 5.000000e-01 2.000000e+00 1.000000e+00 -1.400000e+00
-1.700000e+00 1.000000e+00 1.000000e+00 1.000000e+00 -1.700000e+00
-1.231000e+02 6.400000e+01 1.562500e-02 1.000000e+00 -1.923437e+00
-1.100000e+307 5.617791e+306 1.780059e-307 1.000000e+00 -1.958065e+00
Vectorization
Function get_scale should vectorize with compilers that support auto-vectorization. The following piece of
code vectorizes very well with clang (no need to write SSE/AVX intrinsics code).
/* Test how well get_scale vectorizes: */
void get_scale_vec(double * __restrict__ t, double * __restrict__ x){
int n = 1024;
int i;
for (i = 0; i < n; i++){
x[i] = get_scale(t[i]);
}
}
Unfortunately gcc doesn't find the vmaxpd and vminpd instructions.
Based on wim's answer, here's another solution, which can be faster, as it has one less instruction. The output is a little bit different, but still fulfills the requirements.
The idea is to use bit operations to fix border cases: put a 01 to the lsb of the exponent, no matter of its value. So, exponent:
- 0 becomes 1 (-1023 becomes -1022)
- 2046 becomes 2045 (1023 becomes 1022)
- other exponents modified as well, but just slightly: the number can become two times larger compared to wim's solution (when exponent lsb changes from
00to01), or halved (when 10->01) or 1/4 (when 11->01)
So, this modified routine works (and I think that it's pretty cool that the problem can be solved with only 2 fast asm instructions):
#include<stdio.h>
#include<stdint.h>
#include<immintrin.h>
/* gcc -Wall -m64 -O3 -march=sandybridge dbl_scale.c */
union dbl_int64{
double d;
uint64_t i;
};
double get_scale(double t){
union dbl_int64 x;
uint64_t and_i;
uint64_t or_i;
/* 0xFEDCBA9876543210 */
and_i = 0x7FD0000000000000ull;
or_i = 0x0010000000000000ull;
x.d = t;
x.i = (x.i & and_i)|or_i; /* Set fraction bits to zero, take absolute value */
return x.d;
}
double get_scale_x86(double t){
__m128d x = _mm_set_sd(t);
__m128d x_and = _mm_castsi128_pd(_mm_set1_epi64x(0x7FD0000000000000ull));
__m128d x_or = _mm_castsi128_pd(_mm_set1_epi64x(0x0010000000000000ull));
x = _mm_and_pd(x, x_and);
x = _mm_or_pd(x, x_or);
return _mm_cvtsd_f64(x);
}
/* Compute the inverse 1/t of a double t with all zero fraction bits */
/* and exponent between the limits of function get_scale */
/* A single integer subtraction is much less expensive than a */
/* floating point division. */
double inv_of_scale(double t){
union dbl_int64 x;
/* 0xFEDCBA9876543210 */
uint64_t inv_mask = 0x7FE0000000000000ull;
x.d = t;
x.i = inv_mask - x.i;
return x.d;
}
double inv_of_scale_x86(double t){
__m128i inv_mask = _mm_set1_epi64x(0x7FE0000000000000ull);
__m128d x = _mm_set_sd(t);
__m128i x_i = _mm_sub_epi64(inv_mask, _mm_castpd_si128(x));
return _mm_cvtsd_f64(_mm_castsi128_pd(x_i));
}
int main(){
int n = 14;
int i;
/* Several example values, 4.94e-324 is the smallest subnormal */
double y[14] = { 4.94e-324, 1.1e-320, 1.1e-300, 1.1e-5, 0.7, 1.7, 123.1, 1.1e300,
1.79e308, -1.1e-320, -0.7, -1.7, -123.1, -1.1e307};
double z, s, u;
printf("Portable code:\n");
printf(" x pow_of_2 inverse pow2*inv x*inverse \n");
for (i = 0; i < n; i++){
z = y[i];
s = get_scale(z);
u = inv_of_scale(s);
printf("%14e %14e %14e %14e %14e\n", z, s, u, s*u, z*u);
}
printf("\nx86 specific SSE code:\n");
printf(" x pow_of_2 inverse pow2*inv x*inverse \n");
for (i = 0; i < n; i++){
z = y[i];
s = get_scale_x86(z);
u = inv_of_scale_x86(s);
printf("%14e %14e %14e %14e %14e\n", z, s, u, s*u, z*u);
}
return 0;
}
You can use
double frexp (double x, int* exp);
Returned value is the fractional part of of x and exp is the exponent (minus the offset).
Alternatively, the following code gets the exponent part of a double.
int get_exp(double *d) {
long long *l = (long long *) d;
return ((*l & (0x7ffLL << 52) )>> 52)-1023 ;
}