source code
// The Computer Language Benchmarks Game
// https://salsa.debian.org/benchmarksgame-team/benchmarksgame/
//
// Contributed by Mark C. Lewis.
// Modified slightly by Chad Whipkey.
// Converted from Java to C++ and added SSE support by Branimir Maksimovic.
// Converted from C++ to C by Alexey Medvedchikov.
// Modified by Jeremy Zerfas.
#include <stdint.h>
#include <stdalign.h>
#include <immintrin.h>
#include <math.h>
#include <stdio.h>
// intptr_t should be the native integer type on most sane systems.
typedef intptr_t intnative_t;
typedef struct{
double position[3], velocity[3], mass;
} body;
#define SOLAR_MASS (4*M_PI*M_PI)
#define DAYS_PER_YEAR 365.24
#define BODIES_COUNT 5
static body solar_Bodies[]={
{ // Sun
.mass=SOLAR_MASS
},
{ // Jupiter
{
4.84143144246472090e+00,
-1.16032004402742839e+00,
-1.03622044471123109e-01
},
{
1.66007664274403694e-03 * DAYS_PER_YEAR,
7.69901118419740425e-03 * DAYS_PER_YEAR,
-6.90460016972063023e-05 * DAYS_PER_YEAR
},
9.54791938424326609e-04 * SOLAR_MASS
},
{ // Saturn
{
8.34336671824457987e+00,
4.12479856412430479e+00,
-4.03523417114321381e-01
},
{
-2.76742510726862411e-03 * DAYS_PER_YEAR,
4.99852801234917238e-03 * DAYS_PER_YEAR,
2.30417297573763929e-05 * DAYS_PER_YEAR
},
2.85885980666130812e-04 * SOLAR_MASS
},
{ // Uranus
{
1.28943695621391310e+01,
-1.51111514016986312e+01,
-2.23307578892655734e-01
},
{
2.96460137564761618e-03 * DAYS_PER_YEAR,
2.37847173959480950e-03 * DAYS_PER_YEAR,
-2.96589568540237556e-05 * DAYS_PER_YEAR
},
4.36624404335156298e-05 * SOLAR_MASS
},
{ // Neptune
{
1.53796971148509165e+01,
-2.59193146099879641e+01,
1.79258772950371181e-01
},
{
2.68067772490389322e-03 * DAYS_PER_YEAR,
1.62824170038242295e-03 * DAYS_PER_YEAR,
-9.51592254519715870e-05 * DAYS_PER_YEAR
},
5.15138902046611451e-05 * SOLAR_MASS
}
};
// Advance all the bodies in the system by one timestep. Calculate the
// interactions between all the bodies, update each body's velocity based on
// those interactions, and update each body's position by the distance it
// travels in a timestep at it's updated velocity.
static void advance(body bodies[]){
// Figure out how many total different interactions there are between each
// body and every other body. Some of the calculations for these
// interactions will be calculated two at a time by using x86 SSE
// instructions and because of that it will also be useful to have a
// ROUNDED_INTERACTIONS_COUNT that is equal to the next highest even number
// which is equal to or greater than INTERACTIONS_COUNT.
#define INTERACTIONS_COUNT (BODIES_COUNT*(BODIES_COUNT-1)/2)
#define ROUNDED_INTERACTIONS_COUNT (INTERACTIONS_COUNT+INTERACTIONS_COUNT%2)
// It's useful to have two arrays to keep track of the position_Deltas
// and magnitudes of force between the bodies for each interaction. For the
// position_Deltas array, instead of using a one dimensional array of
// structures that each contain the X, Y, and Z components for a position
// delta, a two dimensional array is used instead which consists of three
// arrays that each contain all of the X, Y, and Z components for all of the
// position_Deltas. This allows for more efficient loading of this data into
// SSE registers. Both of these arrays are also set to contain
// ROUNDED_INTERACTIONS_COUNT elements to simplify one of the following
// loops and to also keep the second and third arrays in position_Deltas
// aligned properly.
static alignas(__m128d) double
position_Deltas[3][ROUNDED_INTERACTIONS_COUNT],
magnitudes[ROUNDED_INTERACTIONS_COUNT];
// Calculate the position_Deltas between the bodies for each interaction.
for(intnative_t i=0, k=0; i<BODIES_COUNT-1; ++i)
for(intnative_t j=i+1; j<BODIES_COUNT; ++j, ++k)
for(intnative_t m=0; m<3; ++m)
position_Deltas[m][k]=
bodies[i].position[m]-bodies[j].position[m];
// Calculate the magnitudes of force between the bodies for each
// interaction. This loop processes two interactions at a time which is why
// ROUNDED_INTERACTIONS_COUNT/2 iterations are done.
for(intnative_t i=0; i<ROUNDED_INTERACTIONS_COUNT/2; ++i){
// Load position_Deltas of two bodies into position_Delta.
__m128d position_Delta[3];
for(intnative_t m=0; m<3; ++m)
position_Delta[m]=((__m128d *)position_Deltas[m])[i];
const __m128d distance_Squared=
position_Delta[0]*position_Delta[0]+
position_Delta[1]*position_Delta[1]+
position_Delta[2]*position_Delta[2];
// Doing square roots normally using double precision floating point
// math can be quite time consuming so SSE's much faster single
// precision reciprocal square root approximation instruction is used as
// a starting point instead. The precision isn't quite sufficient to get
// acceptable results so two iterations of the Newton–Raphson method are
// done to improve precision further.
__m128d distance_Reciprocal=
_mm_cvtps_pd(_mm_rsqrt_ps(_mm_cvtpd_ps(distance_Squared)));
for(intnative_t j=0; j<2; ++j)
// Normally the last four multiplications in this equation would
// have to be done sequentially but by placing the last
// multiplication in parentheses, a compiler can then schedule that
// multiplication earlier.
distance_Reciprocal=distance_Reciprocal*1.5-
0.5*distance_Squared*distance_Reciprocal*
(distance_Reciprocal*distance_Reciprocal);
// Calculate the magnitudes of force between the bodies. Typically this
// calculation would probably be done by using a division by the cube of
// the distance (or similarly a multiplication by the cube of its
// reciprocal) but for better performance on modern computers it often
// will make sense to do part of the calculation using a division by the
// distance_Squared which was already calculated earlier. Additionally
// this method is probably a little more accurate due to less rounding
// as well.
((__m128d *)magnitudes)[i]=0.01/distance_Squared*distance_Reciprocal;
}
// Use the calculated magnitudes of force to update the velocities for all
// of the bodies.
for(intnative_t i=0, k=0; i<BODIES_COUNT-1; ++i)
for(intnative_t j=i+1; j<BODIES_COUNT; ++j, ++k){
// Precompute the products of the mass and magnitude since it can be
// reused a couple times.
const double
i_mass_magnitude=bodies[i].mass*magnitudes[k],
j_mass_magnitude=bodies[j].mass*magnitudes[k];
for(intnative_t m=0; m<3; ++m){
bodies[i].velocity[m]-=position_Deltas[m][k]*j_mass_magnitude;
bodies[j].velocity[m]+=position_Deltas[m][k]*i_mass_magnitude;
}
}
// Use the updated velocities to update the positions for all of the bodies.
for(intnative_t i=0; i<BODIES_COUNT; ++i)
for(intnative_t m=0; m<3; ++m)
bodies[i].position[m]+=0.01*bodies[i].velocity[m];
}
// Calculate the momentum of each body and conserve momentum of the system by
// adding to the Sun's velocity the appropriate opposite velocity needed in
// order to offset that body's momentum.
static void offset_Momentum(body bodies[]){
for(intnative_t i=0; i<BODIES_COUNT; ++i)
for(intnative_t m=0; m<3; ++m)
bodies[0].velocity[m]-=
bodies[i].velocity[m]*bodies[i].mass/SOLAR_MASS;
}
// Output the total energy of the system.
static void output_Energy(body bodies[]){
double energy=0;
for(intnative_t i=0; i<BODIES_COUNT; ++i){
// Add the kinetic energy for each body.
energy+=0.5*bodies[i].mass*(
bodies[i].velocity[0]*bodies[i].velocity[0]+
bodies[i].velocity[1]*bodies[i].velocity[1]+
bodies[i].velocity[2]*bodies[i].velocity[2]);
// Add the potential energy between this body and every other body.
for(intnative_t j=i+1; j<BODIES_COUNT; ++j){
double position_Delta[3];
for(intnative_t m=0; m<3; ++m)
position_Delta[m]=bodies[i].position[m]-bodies[j].position[m];
energy-=bodies[i].mass*bodies[j].mass/sqrt(
position_Delta[0]*position_Delta[0]+
position_Delta[1]*position_Delta[1]+
position_Delta[2]*position_Delta[2]);
}
}
// Output the total energy of the system.
printf("%.9f\n", energy);
}
int main(int argc, char *argv[]){
offset_Momentum(solar_Bodies);
output_Energy(solar_Bodies);
for(intnative_t n=atoi(argv[1]); n--; advance(solar_Bodies));
output_Energy(solar_Bodies);
}