Archive for October, 2009

Following my earlier article on timing various square-root functions on the x86 Clomid For Sale, , commenter LeeN suggested that it would be useful to also test their impact on a more realistic scenario than square-rooting long arrays of independent numbers. In real gameplay code the most common use for sqrts is in finding the length of a vector or normalizing it, like when you need to perform a distance check between two characters to determine whether they can see/shoot/etc each other. So, I wrote up a group of normalize functions, each using a different sqrt technique, Cheap Clomid no rx, and timed them.

The testbed was, as last time, an array of 2048 single-precision floating point numbers, this time interpreted as a packed list of 682 three-dimensional vectors. This number was chosen so that both it and the output array were sure to fit in the L1 cache; however, because three floats add up to twelve bytes, is Clomid addictive, this means that three out of four vectors were not aligned to a 16-byte boundary, which is significant for the SIMD test case as I had to use the movups unaligned load op. Each timing case consisted of looping over this array of vectors 2048 times, normalizing each and writing the result to memory, Clomid For Sale.

Each normalize function computed the length of the vector 1/√(x2 + y2 + z2), multiplied each component by the reciprocal, and then wrote it back through an output pointer. Get Clomid, The main difference was in how the reciprocal square root was computed:


  • via the x87 FPU, by simply compiling 1.0f/sqrt( x*x + y*y + z*z )

  • via the SSE scalar unit, by compiling 1.0f/sqrt( x*x + y*y + z*z ) with the /arch:SSE2 option set; this causes the compiler to issue a sqrtss followed by an fdivie, it computes the square root and then divides one by it
  • via the SSE scalar unit, by using the estimated reciprocal square root intrinsic and then performing one step of Newton-Raphson iteration

  • via the SSE SIMD unit, working on the whole vector at once

In all cases the results were accurate to 22 bits of precision. The results for 1, doses Clomid work,396,736 vector normalizations were:






















MethodTotal timeTime per vector
Compiler 1.0/sqrt(x)
x87 FPU FSQRT
52.469ms37.6ns
Compiler 1.0/sqrt(x)
SSE scalar sqrtss
27.233ms19.5ns
SSE scalar ops
rsqrtss with one NR step
21.631ms15.5ns
SSE SIMD ops
rsqrtss with one NR step
20.034ms14.3ns

Two things jump out here. First, even when the square root op is surrounded by lots of other math — multiplies, adds, Clomid use, loads, stores — optimizations such as this can make a huge difference. Clomid For Sale, It's not just the cost of the sqrt itself, but also that it's unpipelined, which means it ties up an execution unit and prevents any other work from being done until it's entirely completed.

Second, in this case, SIMD is only a very modest benefit. That's because the input vectors are unaligned, and the two key steps of this operation, online buying Clomid, the dot product and the square root, are scalar in nature. (This is what's meant by "horizontal" SIMD computation — operations between the components of one vector, rather than between the corresponding words of two vectors. Given a vector V ∋ <x, Comprar en línea Clomid, comprar Clomid baratos, y,z>, the sum x + y + z is horizontal, but with two vectors V1 and V2, V3 = <x1+x2, y1+y2, z1+z2> is vertical.) So it really doesn't play to SIMD's strengths at all, Clomid price.

On the other hand, if I were to normalize four vectors at a time, so that four dot products and four rsqrts could be performed in parallel in the four words of a vector register, then the speed advantage of SIMD would be much greater, Clomid For Sale. But, again, my goal wasn't to test performance in tight loops over packed data — it was to figure out the best way to do something like an angle check in the middle of a character's AI, where you usually deal with one vector at a time.

Source code for my testing functions below the jump. Order Clomid from United States pharmacy, Note that each function writes the normalized vector through an out pointer, but also returns the original vector's length. The hand-written intrinsic versions probably aren't totally optimal, but they ought to be good enough to make the point.

[DDET Naive vector normalize, x87 FPU or SSE scalar]
Source

// Normalizes an assumed 3-element vector starting
// at pointer V, and returns the length of the original
// vector, Clomid used for.
inline float NaiveTestNormalize( float * RESTRICT vOut, const float * RESTRICT vIn )
{
const float l = vIn[0]*vIn[0] + vIn[1]*vIn[1] + vIn[2]*vIn[2];
const float rsqt = 1.0f / sqrt(l);
vOut[0] = vIn[0] * rsqt;
vOut[1] = vIn[1] * rsqt;
vOut[2] = vIn[2] * rsqt;
return rsqt * l;
}

Assembly (x87 FPU)


_TEXT SEGMENT
_vOut$ = 8 ; size = 4
_vIn$ = 12 ; size = 4
?TestNormalize@@YAMPIAMPIBM@Z PROC ; TestNormalize, COMDAT

; 396 : const float l = vIn[0]*vIn[0] + vIn[1]*vIn[1] + vIn[2]*vIn[2];

mov eax, DWORD PTR _vIn$[esp-4]
fld DWORD PTR [eax+8]

; 397 : const float rsqt = 1.0f / sqrt(l);
; 398 : vOut[0] = vIn[0] * rsqt;

mov ecx, DWORD PTR _vOut$[esp-4]
fld DWORD PTR [eax+4]
fld DWORD PTR [eax]
fmul ST(0), Order Clomid online overnight delivery no prescription, ST(0)
fld ST(1)
fmulp ST(2), ST(0)
faddp ST(1), ST(0)
fld ST(1)
fmulp ST(2), ST(0)
faddp ST(1), ST(0)
fld ST(0)
fsqrt
fld1
fdivrp ST(1), ST(0)
fld DWORD PTR [eax]
fmul ST(0), ST(1)
fstp DWORD PTR [ecx]

; 399 : vOut[1] = vIn[1] * rsqt;

fld ST(0)
fmul DWORD PTR [eax+4]
fstp DWORD PTR [ecx+4]

; 400 : vOut[2] = vIn[2] * rsqt;

fld ST(0)
fmul DWORD PTR [eax+8]
fstp DWORD PTR [ecx+8]

; 401 : return rsqt * l;

fmulp ST(1), Clomid mg, ST(0)

; 402 : }

ret 0
?TestNormalize@@YAMPIAMPIBM@Z ENDP ; TestNormalize
_TEXT ENDS

Assembly (compiler-issued SSE scalar)


_TEXT SEGMENT
_l$ = -4 ; size = 4
_vOut$ = 8 ; size = 4
_rsqt$ = 12 ; size = 4
_vIn$ = 12 ; size = 4
?TestNormalize@@YAMPIAMPIBM@Z PROC ; TestNormalize, COMDAT

; 392 : {

push ecx

; 393 : const float l = vIn[0]*vIn[0] + vIn[1]*vIn[1] + vIn[2]*vIn[2];

mov eax, DWORD PTR _vIn$[esp]
movss xmm1, DWORD PTR [eax+4]
movss xmm2, DWORD PTR [eax]
movss xmm0, Buying Clomid online over the counter, DWORD PTR [eax+8]

; 394 : const float rsqt = 1.0f / sqrt(l);
; 395 : vOut[0] = vIn[0] * rsqt;

mov eax, DWORD PTR _vOut$[esp]
movaps xmm3, xmm2
mulss xmm3, xmm2
movaps xmm4, xmm1
mulss xmm4, xmm1
addss xmm3, xmm4
movaps xmm4, low dose Clomid, xmm0
mulss xmm4, xmm0
addss xmm3, xmm4
movss DWORD PTR _l$[esp+4], xmm3
sqrtss xmm4, xmm3 ;; slow full-precision square root gets stored in xmm4
movss xmm3, Purchase Clomid online no prescription, DWORD PTR __real@3f800000 ;; store 1.0 in xmm3
divss xmm3, xmm4 ;; divide 1.0 / xmm4 to get the reciprocal square root !?.
movss DWORD PTR _rsqt$[esp], xmm3

; 396 : vOut[1] = vIn[1] * rsqt;
; 397 : vOut[2] = vIn[2] * rsqt;
; 398 : return rsqt * l;

fld DWORD PTR _rsqt$[esp]
mulss xmm2, xmm3
fmul DWORD PTR _l$[esp+4]
mulss xmm1, xmm3
mulss xmm0, xmm3
movss DWORD PTR [eax], where can i order Clomid without prescription, xmm2
movss DWORD PTR [eax+4], xmm1
movss DWORD PTR [eax+8], xmm0

; 399 : }

pop ecx
ret 0
?TestNormalize@@YAMPIAMPIBM@Z ENDP ; TestNormalize
_TEXT ENDS


[/DDET]

[DDET Vector normalize, hand-written SSE scalar by intrinsics]
Source


// SSE scalar reciprocal sqrt using rsqrt op, plus one Newton-Rhaphson iteration
inline __m128 SSERSqrtNR( const __m128 x )
{
__m128 recip = _mm_rsqrt_ss( x ); // "estimate" opcode
const static __m128 three = { 3, Generic Clomid, 3, 3, 3 }; // aligned consts for fast load
const static __m128 half = { 0.5,0.5,0.5,0.5 };
__m128 halfrecip = _mm_mul_ss( half, recip );
__m128 threeminus_xrr = _mm_sub_ss( three, Clomid samples, _mm_mul_ss( x, _mm_mul_ss ( recip, recip ) ) );
return _mm_mul_ss( halfrecip, threeminus_xrr );
}

inline __m128 SSE_ScalarTestNormalizeFast( float * RESTRICT vOut, float * RESTRICT vIn )
{
__m128 x = _mm_load_ss(&vIn[0]);
__m128 y = _mm_load_ss(&vIn[1]);
__m128 z = _mm_load_ss(&vIn[2]);

const __m128 l = // compute x*x + y*y + z*z
_mm_add_ss(
_mm_add_ss( _mm_mul_ss(x, Clomid online cod, x),
_mm_mul_ss(y,y)
),
_mm_mul_ss( z, z )
);

const __m128 rsqt = SSERSqrtNR( l );
_mm_store_ss( &vOut[0] , _mm_mul_ss( rsqt, x ) );
_mm_store_ss( &vOut[1], Clomid no rx, _mm_mul_ss( rsqt, y ) );
_mm_store_ss( &vOut[2] , _mm_mul_ss( rsqt, z ) );

return _mm_mul_ss( l , rsqt );
}

Assembly


_TEXT SEGMENT
_vOut$ = 8 ; size = 4
_vIn$ = 12 ; size = 4
?SSE_ScalarTestNormalizeFast@@YA?AT__m128@@PIAM0@Z PROC ; SSE_ScalarTestNormalizeFast, Clomid over the counter, COMDAT

push ebp
mov ebp, esp
and esp, -16 ; fffffff0H

mov eax, DWORD PTR _vIn$[ebp]
movss xmm0, DWORD PTR [eax]

movss xmm3, DWORD PTR [eax+4]

movaps xmm7, XMMWORD PTR ?three@?1??SSERSqrtNR@@YA?AT__m128@@T2@@Z@4T2@B
movaps xmm2, ordering Clomid online, xmm0
movss xmm0, DWORD PTR [eax+8]

mov eax, DWORD PTR _vOut$[ebp]
movaps xmm4, xmm0
movaps xmm0, xmm2
mulss xmm0, Australia, uk, us, usa, xmm2
movaps xmm1, xmm3
mulss xmm1, xmm3
addss xmm0, xmm1
movaps xmm1, xmm4
mulss xmm1, xmm4
addss xmm0, xmm1
movaps xmm1, Clomid dosage, xmm0
rsqrtss xmm1, xmm1
movaps xmm5, xmm1
mulss xmm1, xmm5
movaps xmm6, xmm0
mulss xmm6, Buy Clomid from mexico, xmm1
movaps xmm1, XMMWORD PTR ?half@?1??SSERSqrtNR@@YA?AT__m128@@T2@@Z@4T2@B
mulss xmm1, xmm5
subss xmm7, xmm6
mulss xmm1, xmm7
movaps xmm5, xmm1
mulss xmm5, xmm2
movss XMMWORD PTR [eax], Clomid cost, xmm5
movaps xmm2, xmm1
mulss xmm2, xmm3

movss XMMWORD PTR [eax+4], xmm2
movaps xmm2, xmm1
mulss xmm2, Clomid forum, xmm4

movss XMMWORD PTR [eax+8], xmm2

mulss xmm0, xmm1

mov esp, ebp
pop ebp
ret 0
?SSE_ScalarTestNormalizeFast@@YA?AT__m128@@PIAM0@Z ENDP ; SSE_ScalarTestNormalizeFast
_TEXT ENDS


[/DDET]

[DDET Vector normalize, hand-written SSE SIMD by intrinsics]
Source


inline __m128 SSE_SIMDTestNormalizeFast( float * RESTRICT vOut, float * RESTRICT vIn )
{
// load as a SIMD vector
const __m128 vec = _mm_loadu_ps(vIn);
// compute a dot product by computing the square, and
// then rotating the vector and adding, Clomid street price, so that the
// dot ends up in the low term (used by the scalar ops)
__m128 dot = _mm_mul_ps( vec, vec );
// rotate x under y and add together
__m128 rotated = _mm_shuffle_ps( dot, dot, _MM_SHUFFLE( 0,3, Clomid overnight, 2,1 ) ); // YZWX ( shuffle macro is high to low word )
dot = _mm_add_ss( dot, rotated ); // x^2 + y^2 in the low word
rotated = _mm_shuffle_ps( rotated, rotated, _MM_SHUFFLE( 0,3,2, purchase Clomid,1 ) ); // ZWXY
dot = _mm_add_ss( dot, rotated ); // x^2 + y^2 + z^2 in the low word

__m128 recipsqrt = SSERSqrtNR( dot ); // contains reciprocal square root in low term
recipsqrt = _mm_shuffle_ps( recipsqrt, recipsqrt, _MM_SHUFFLE( 0, 0, Clomid wiki, 0, 0 ) ); // broadcast low term to all words

// multiply 1/sqrt(dotproduct) against all vector components, and write back
const __m128 normalized = _mm_mul_ps( vec, recipsqrt );
_mm_storeu_ps(vOut, normalized);
return _mm_mul_ss( dot , recipsqrt );
}

Assembly


_TEXT SEGMENT
_vOut$ = 8 ; size = 4
_vIn$ = 12 ; size = 4
?SSE_SIMDTestNormalizeFast@@YA?AT__m128@@PIAM0@Z PROC ; SSE_SIMDTestNormalizeFast, COMDAT

push ebp
mov ebp, order Clomid from mexican pharmacy, esp
and esp, -16 ; fffffff0H

mov eax, DWORD PTR _vIn$[ebp]
movups xmm2, XMMWORD PTR [eax] ;; load the input vector
movaps xmm5, XMMWORD PTR ?three@?1??SSERSqrtNR@@YA?AT__m128@@T2@@Z@4T2@B ;; load the constant "3"
mov ecx, Clomid pharmacy, DWORD PTR _vOut$[ebp]
movaps xmm0, xmm2
mulps xmm0, xmm2
movaps xmm1, xmm0
shufps xmm1, xmm0, 57 ; shuffle to YZWX
addss xmm0, xmm1 ; add Y to low word of xmm0
shufps xmm1, buy Clomid online cod, xmm1, 57 ; shuffle to ZWXY
addss xmm0, xmm1 ; add Z to low word of xmm0

movaps xmm1, xmm0
rsqrtss xmm1, xmm1 ; reciprocal square root estimate
movaps xmm3, Real brand Clomid online, xmm1
mulss xmm1, xmm3
movaps xmm4, xmm0
mulss xmm4, xmm1
movaps xmm1, XMMWORD PTR ?half@?1??SSERSqrtNR@@YA?AT__m128@@T2@@Z@4T2@B
mulss xmm1, xmm3
subss xmm5, xmm4
mulss xmm1, xmm5 ; Newton-Raphson finishes here; 1/sqrt(dot) is in xmm1's low word

shufps xmm1, xmm1, 0 ; broadcast so that xmm1 has 1/sqrt(dot) in all words
movaps xmm3, xmm1
mulps xmm3, xmm2 ; multiply all words of original vector by 1/sqrt(dot)
movups XMMWORD PTR [ecx], xmm3 ; unaligned save to memory

; return dot * 1 / sqrt(dot) == sqrt(dot) == length of vector
mulss xmm0, xmm1

mov esp, ebp
pop ebp
ret 0
?SSE_SIMDTestNormalizeFast@@YA?AT__m128@@PIAM0@Z ENDP ; SSE_SIMDTestNormalizeFast
_TEXT ENDS


[/DDET].

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Retin-A For Sale, The square root is one of those basic mathematical operations that's totally ubiquitous in any game's source code, and yet also has many competing implementations and performance superstitions around it. The compiler offers a sqrt() builtin function, and so do some CPUs, but some programmers insist on writing their own routines in software, Retin-A from canadian pharmacy. And often it's really the reciprocal square root you want, for normalizing a vector, Retin-A reviews, or trigonometry. But I've never had a clear answer for which technique is really fastest, or exactly what accuracy-vs-speed tradeoffs we make with "estimating" intrinsics.

What is the fastest way to compute a square root, where can i find Retin-A online. It would seem that if the CPU has a native square-root opcode, there's no beating the hardware, but is it really true, Retin-A For Sale.

Such questions vex me, so I went and measured all the different means of computing the square root of a scalar single-precision floating point number that I could think of. After Retin-A, I ran trials on my Intel Core 2 and on the Xenon, comparing each technique for both speed and accuracy, and some of the results were surprising.

In this article I'll describe my results for the Intel hardware; next week I'll turn to the Xenon PPC, buy Retin-A no prescription.

Experimental setup


I'll post the whole source code for my tests elsewhere, but basically each of these trials consists of iterating N times over an array of floating point numbers, Retin-A recreational, calling square root upon each of them and writing it to a second output array.

[DDET (see pseudocode)]

 Retin-A For Sale, inline float TestedFunction( float x )
{
return sqrt(x); // one of many implementations..
}
void TimeSquareRoot()
{
float numbersIn[ ARRAYSIZE ]; // ARRAYSIZE chosen so that both arrays
float numbersOut[ ARRAYSIZE ]; // fit in L1 cache
// assume that numbersIn is filled with random positive numbers, and both arrays are
// prefetched to cache...
StartClockCycleCounter();
for ( int i = 0 ; i < NUMITERATIONS ; ++i )
for ( int j = 0 ; j < ARRAYSIZE ; ++j ) // in some cases I unroll this loop
{
numbersOut[j] = TestedFunction( numbersIn[j] );
}
StopClockCycleCounter();
printf( "%.3f millisec for %d floats\n", Retin-A steet value,
ClockCycleCounterInMilliseconds(), ARRAYSIZE * NUMITERATIONS );

// now measure accuracy
float error = 0;
for ( int i = 0 ; i < ARRAYSIZE ; ++i )
{
double knownAccurate = PerfectSquareRoot( numbersIn[i] );
error += fabs( numbersOut[i] - knownAccurate ) / knownAccurate ;
}
error /= ARRAYSIZE ;
printf( "Average error: %.5f%%\n", Retin-A class, error * 100.0f );
}


[/DDET]

In each case I verified that the compiler was not eliding any computations (it really was performing ARRAYSIZE × NUMITERATIONS many square roots), that it was properly inlining the tested function, and that all the arrays fit into L1 cache so that memory latency wasn't affecting the results. I also only tested scalar square root functions — SIMD would clearly be the fastest way of working on large contiguous arrays, Retin-A brand name, but I wanted to measure the different techniques of computing one square root at a time, as is usually necessary in gameplay code. Buy Retin-A without prescription, Because some of the speedup techniques involve trading off accuracy, I compared the resulting numbers against the perfectly-accurate double-precision square root library routine to get an average error for each test run.

And I performed each run multiple times with different data, averaging the final results together, Retin-A For Sale.

x86 results

I ran my tests on a 2.66Ghz Intel Core 2 workstation. An x86 chip actually has two different means of performing scalar floating-point math, buy no prescription Retin-A online. By default, the compiler uses the old x87 FPU, Buy generic Retin-A, which dates back to 1980 with a stack-based instruction set like one of those old RPN calculators. In 1999, Intel introduced SSE, which added a variety of new instructions to the processor, Retin-A treatment. SSE is mostly thought of as a SIMD instruction set — for operating on four 32-bit floats in a single op — but it also includes an entire set of scalar Retin-A For Sale, floating point instructions that operate on only one float at a time. It's faster than the x87 operations and was meant to deprecate the old x87 pathway. However, Effects of Retin-A, both the MSVC and GCC compilers default to exclusively using the x87 for scalar math, so unless you edit the "code generation" project properties panel (MSVC) or provide a cryptic obscure command line option (GCC), you'll be stuck with code that uses the old slow way.

I timed the following techniques for square root:


  1. The compiler's built in sqrt() function (which compiled to the x87 FSQRT opcode)

  2. The SSE "scalar single square root" opcode sqrtss, order Retin-A no prescription, which MSVC emits if you use the _mm_sqrt_ss intrinsic or if you set /arch:SSE2

  3. The "magic number" approximation technique invented by Greg Walsh at Ardent Computer and made famous by John Carmack in the Quake III source code.

  4. Taking the estimated reciprocal square root of a via the SSE opcode rsqrtss, and multiplying it against a to get the square root via the identity x / √x = √x.

  5. Method (4), Retin-A without a prescription, with one additional step of Newton-Raphson iteration to improve accuracy.

  6. Method (5), with the loop at line 13 of the pseudocode above unrolled to process four floats per iteration.


I also tested three ways of getting the reciprocal square root: Carmack's technique, the rsqrtss SSE op via compiler intrinsic, and rsqrtss with one Newton-Raphson step, Retin-A pics.

The results, for 4096 loops over 4096 single-precision floats, Retin-A without prescription, were:


SQUARE ROOT




























MethodTotal timeTime per floatAvg Error
Compiler sqrt(x) /
x87 FPU FSQRT
404.029ms24ns0.0000%
SSE intrinsic ssqrts 200.395ms11.9ns0.0000%
Carmack's Magic Number rsqrt * x 72.682ms4.33ns0.0990%
SSE rsqrtss * x 20.495ms1.22ns0.0094%
SSE rsqrtss * x
with one NR step
53.401ms3.18ns0.0000%
SSE rsqrtss * x
with one NR step, unrolled by four
48.701ms2.90ns0.0000%





RECIPROCAL SQRT
















MethodTotal timeTime per floatAvg Error
Carmack's Magic Number rsqrt 59.378ms3.54ns0.0990%
SSE rsqrtss 14.202ms0.85ns0.0094%
SSE rsqrtss
with one NR step
45.952ms2.74ns0.0000%

Discussion

Looking at these results, it's clear that there's a dramatic difference in performance between different approaches to performing square root; which one you choose really can have a significant impact on framerate and accuracy. My conclusions are:

Don't trust the compiler to do the right thing. The received wisdom on performance in math functions is usually "don't reinvent the wheel; the library and compiler are smart and optimal." We see here that this is completely wrong, and in fact calling the library sqrt(x) causes the compiler to do exactly the worst possible thing, Retin-A For Sale. The compiler's output for y = sqrt(x); is worse by orders of magnitude compared to any other approach tested here, Retin-A alternatives.

The x87 FPU is really very slow. Intel has been trying to deprecate the old x87 FPU instructions for a decade now, but no compiler in the business defaults to using the new, Retin-A gel, ointment, cream, pill, spray, continuous-release, extended-release, faster SSE scalar opcodes in place of emulating a thirty-year-old 8087. In the case of y = sqrt(x) , by default MSVC and GCC emit something like


fld DWORD PTR [ecx]
fsqrt ;; slow x87 flop
fstp DWORD PTR [eax]

But if I set the /arch:SSE2 option flag, telling the compiler "assume this code will run on a machine with SSE2", Retin-A from canada, it will instead emit the following, which is 2x faster. Retin-A dose,

sqrtss xmm0, DWORD PTR [ecx] ;; faster SSE scalar flop
movss DWORD PTR [eax], xmm0

There was a time when not every PC on the market had SSE2, meaning that there was some sense in using the older, Retin-A price, coupon, more backwards-compatible operations, but that time has long since passed. SSE2 was introduced in 2001 with the Pentium 4 Retin-A For Sale, . Retin-A canada, mexico, india, No one is ever going to try to play your game on a machine that doesn't support it. If your customer's PC has DirectX 9, it has SSE2.

You can beat the hardware. The most surprising thing about these results for me was that it is faster to take a reciprocal square root and multiply it, Retin-A no prescription, than it is to use the native sqrt opcode, by an order of magnitude. Retin-A pictures, Even Carmack's trick, which I had assumed was obsolete in an age of deep pipelines and load-hit-stores, proved faster than the native SSE scalar op. Part of this is that the reciprocal sqrt opcode rsqrtss is an estimate, accurate to twelve bits; but it only takes one step of Newton's Method to converge that estimate to an accuracy of 24 bits while still being four times faster than the hardware square root opcode, Retin-A For Sale.

The question that then bothered me was, no prescription Retin-A online, why is SSE's built-in-to-hardware square root opcode slower than synthesizing it out of two other math operations. The first hint came when I tried unrolling the loop so that it performed four ops inside the inner for():


for ( int i = 0 ; i < NUMITERATIONS ; ++i )
for ( int j = 0 ; j < ARRAYSIZE ; j += 4 ) // in some cases I unroll this loop
{
numbersOut[j + 0] = TestedSqrt( numbersIn[j + 0] );
numbersOut[j + 1] = TestedSqrt( numbersIn[j + 1] );
numbersOut[j + 2] = TestedSqrt( numbersIn[j + 2] );
numbersOut[j + 3] = TestedSqrt( numbersIn[j + 3] );
}

// two implementations of

As you can see from the results above, Retin-A maximum dosage, when TestedSqrt was the rsqrtss followed by a multiply and one step of Newton iteration, unrolling the loop this way provided a modest 8.8% improvement in speed. But when I tried the same thing with the "precise square root" op sqrtss, the difference was negligible:


SSE sqrt: 200.395 msec
average error 0.0000%

SSE sqrt, Retin-A from mexico, unrolled four: 196.741 msec
average error 0.0000%

What this suggests is that unrolling the loop this way allowed the four rsqrt paths to be pipelined, so that while an individual rsqrtss might take 6 cycles to execute before its result was ready, What is Retin-A, other work could proceed during that time so that the four square root operations in the loop overlapped. On the other hand, the non-estimated sqrtss op apparently cannot be overlapped; one sqrt must finish before the next can begin. A look at the Intel® 64 and IA-32 Architectures Optimization Reference Manual confirms: sqrtss Retin-A For Sale, is an unpipelined instruction.

Pipelined operations make a big difference. When the CPU hits an unpipelined instruction, Retin-A natural, every other instruction in the pipeline has to stop and wait for it to retire before proceeding, so it's like putting the handbrake on your processor. Retin-A photos, You can identify nonpipelined operations in appendix C of the Optimization Reference Manual as the ones that have a throughput equal to latency and greater than 4 cycles.

In the case of ssqrt, the processor is probably doing the same thing internally that I'm doing in my "fast" function — taking an estimated reciprocal square root, improving it with Newton's method, and then multiplying it by the input parameter, Retin-A long term. Taken all together, this is far too much work to fit into a single execution unit, so the processor stalls until it's all done. But if you break up the work so that each of those steps is its own instruction, then the CPU can pipeline them all, and get a much higher throughput even if the latency is the same.

Pipeline latency and microcoded instructions are a much bigger deal on the 360 and PS3, whose CPUs don't reorder operations to hide bubbles; there the benefit from unrolling is much greater, as you'll see next week.

Conclusion

Not all square root functions are created equal, and writing your own can have very real performance benefits over trusting the compiler to optimize your code for you (at which it fails miserably). In many cases you can trade off some accuracy for a massive increase in speed, but even in those places where you need full accuracy, writing your own function to leverage the rsqrtss op followed by Newton's method can still give you 32 bits of precision at a 4x-8x improvement over what you will get with the built-in sqrtf() function.

And if you have lots of numbers you need to square root, of course SIMD (rsqrtps) will be faster still.

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