Floating Point Performance

CS 301 Lecture, Dr. Lawlor

Floating-point arithmetic is quite fast when everything works properly.  For example, I can add, subtract, and multiply ordinary floats or doubles in about 1ns each:
`   =: 0.433 ns/float   +: 0.845 ns/float   -: 0.847 ns/float   *: 0.848 ns/float`
Other operations are not very fast:
`   /: 77.051 ns/floatsqrt: 24.162 ns/floatcbrt: 75.031 ns/float sin: 16.758 ns/float cos: 21.203 ns/float tan: 17.882 ns/floatasin: 25.842 ns/floatacos: 29.293 ns/floatatan: 17.715 ns/float exp: 33.725 ns/float log: 50.433 ns/float pow: 105.437 ns/float`
Here's the code I used to measure these values.  Note that I'm being very careful in this code to keep the values from exploding to infinity, because (as we see below) such a malfunction can change the speed of the code!
`enum {n_X=1000};double X[n_X];const double k=0.99, k2=1.01, k3=0.000001;int assign_X(void) { for (int i=0;i<n_X-1;i++) X[i]=k;return 0;}int add_X(void) { for (int i=0;i<n_X-1;i++) X[i]=X[i]+k;return 0;}int sub_X(void) { for (int i=0;i<n_X-1;i++) X[i]=X[i]-k;return 0;}int mul_X(void) { for (int i=0;i<n_X-1;i++) X[i]=X[i]*k;return 0;}int div_X(void) { for (int i=0;i<n_X-1;i++) X[i]=X[i]/k2;return 0;}int sqrt_X(void) { for (int i=0;i<n_X-1;i++) X[i]=sqrt(X[i]);return 0;}int cbrt_X(void) { for (int i=0;i<n_X-1;i++) X[i]=cbrt(X[i]);return 0;}int sin_X(void) { for (int i=0;i<n_X-1;i++) X[i]=sin(X[i]);return 0;}int cos_X(void) { for (int i=0;i<n_X-1;i++) X[i]=cos(X[i]);return 0;}int tan_X(void) { for (int i=0;i<n_X-1;i++) X[i]=tan(X[i]);return 0;}int asin_X(void) { for (int i=0;i<n_X-1;i++) X[i]=asin(X[i]);return 0;}int acos_X(void) { for (int i=0;i<n_X-1;i++) X[i]=k3*acos(X[i]);return 0;}int atan_X(void) { for (int i=0;i<n_X-1;i++) X[i]=atan(X[i]);return 0;}int exp_X(void) { for (int i=0;i<n_X-1;i++) X[i]=k3*exp(X[i]);return 0;}int log_X(void) { for (int i=0;i<n_X-1;i++) X[i]=k2+log(X[i]);return 0;}int pow_X(void) { for (int i=0;i<n_X-1;i++) X[i]=pow(X[i],k);return 0;}int foo(void) {	int i;	printf("   =: %.3f ns/float\n",time_function(assign_X)/n_X*1.0e9);	printf("   +: %.3f ns/float\n",time_function(add_X)/n_X*1.0e9);	printf("   -: %.3f ns/float\n",time_function(sub_X)/n_X*1.0e9);	for (i=0;i<n_X;i++) X[i]=1.0; 	printf("   *: %.3f ns/float\n",time_function(mul_X)/n_X*1.0e9);	printf("   /: %.3f ns/float\n",time_function(div_X)/n_X*1.0e9);	printf("sqrt: %.3f ns/float\n",time_function(sqrt_X)/n_X*1.0e9);	printf("cbrt: %.3f ns/float\n",time_function(cbrt_X)/n_X*1.0e9);	printf(" sin: %.3f ns/float\n",time_function(sin_X)/n_X*1.0e9);	printf(" cos: %.3f ns/float\n",time_function(cos_X)/n_X*1.0e9);	printf(" tan: %.3f ns/float\n",time_function(tan_X)/n_X*1.0e9);	printf("asin: %.3f ns/float\n",time_function(asin_X)/n_X*1.0e9);	printf("acos: %.3f ns/float\n",time_function(acos_X)/n_X*1.0e9);	printf("atan: %.3f ns/float\n",time_function(atan_X)/n_X*1.0e9);	for (i=0;i<n_X;i++) X[i]=0.0; 	printf(" exp: %.3f ns/float\n",time_function(exp_X)/n_X*1.0e9);	for (i=0;i<n_X;i++) X[i]=1.0; 	printf(" log: %.3f ns/float\n",time_function(log_X)/n_X*1.0e9);	printf(" pow: %.3f ns/float\n",time_function(pow_X)/n_X*1.0e9);	printf("Final X values: %.4f  %.4f  %.4f\n",X[0],X[n_X/2],X[n_X-2]);	return 0;}`

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Normal (non-Weird) Floats

Recall that a "float" as as defined by IEEE Standard 754 consists of three bitfields:
 Sign Exponent Mantissa (or Fraction) 1 bit--   0 for positive   1 for negative 8 bits--   127 means 20   137 means 210 23 bits-- a binary fraction.

The hardware usually interprets a float as having the value:

value = (-1) sign * 2 (exponent-127) * 1.fraction

Note that the mantissa normally has an implicit leading 1 applied.

Weird: Zeros and Denormals

However, if the "exponent" field is exactly zero, the implicit leading digit is taken to be 0, like this:

value = (-1) sign * 2 (-126) * 0.fraction

Supressing the leading 1 allows you to exactly represent 0: the bit pattern for 0.0 is just exponent==0 and fraction==00000000 (that is, everything zero).  If you set the sign bit to negative, you have "negative zero", a strange curiosity.  Positive and negative zero work the same way in arithmetic operations, and as far as I know there's no reason to prefer one to the other.  The "==" operator claims positive and negative zero are the same!

If the fraction field isn't zero, but the exponent field is, you have a "denormalized number"--these are numbers too small to represent with a leading one.  You always need denormals to represent zero, but denormals (also known as "subnormal" values) also provide a little more range at the very low end--they can store values down to around 1.0e-40 for "float", and 1.0e-310 for "double".

See below for the performance problem with denormals.

Weird: Infinity

If the exponent field is as big as it can get (for "float", 255), this indicates another sort of special number.  If the fraction field is zero, the number is interpreted as positive or negative "infinity".  The hardware will generate "infinity" when dividing by zero, or when another operation exceeds the representable range.
`float z=0.0;float f=1.0/z;std::cout<<f<<"\n";return (int)f;`

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Arithmetic on infinities works just the way you'd expect:infinity plus 1.0 gives infinity, etc. (See tables below).  Positive and negative infinities exist, and work as you'd expect.  Note that while divide-by-integer-zero causes a crash (divide by zero error), divide-by-floating-point-zero just happily returns infinity by default.

Weird: NaN

If you do an operation that doesn't make sense, like:
• 0.0/0.0 (neither zero nor infinity, because we'd want (x/x)==1.0; but not 1.0 either, because we'd want (2*x)/x==2.0...)
• infinity-infinity (might cancel out to anything)
• infinity*0
The machine just gives a special "error" number called a "NaN" (Not-a-Number).  The idea is if you run some complicated program that screws up, you don't want to get a plausible but wrong answer like "4" (like we get with integer overflow!); you want something totally implausible like "nan" to indicate an error happened.   For example, this program prints "nan" and returns -2147483648 (0x80000000):
`float f=sqrt(-1.0);std::cout<<f<<"\n";return (int)f;`

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This is a "NaN", which is represented with a huge exponent and a *nonzero* fraction field.  Positive and negative nans exist, but like zeros both signs seem to work the same.  x86 seems to rewrite the bits of all NaNs to a special pattern it prefers (0x7FC00000 for float, with exponent bits and the leading fraction bit all set to 1).

Performance impact of special values

Machines properly handle ordinary floating-point numbers and zero in hardware at full speed.

However, most modern machines *don't* handle denormals, infinities, or NaNs in hardware--instead when one of these special values occurs, they trap out to software which handles the problem and restarts the computation.  This trapping process takes time, as shown in the following program:
`enum {n_vals=1000};double vals[n_vals];int average_vals(void) {	for (int i=0;i<n_vals-1;i++) 		vals[i]=0.5*(vals[i]+vals[i+1]);	return 0;}int foo(void) {	int i;	for (i=0;i<n_vals;i++) vals[i]=0.0; 	printf(" Zeros: %.3f ns/float\n",time_function(average_vals)/n_vals*1.0e9);	for (i=0;i<n_vals;i++) vals[i]=1.0; 	printf(" Ones: %.3f ns/float\n",time_function(average_vals)/n_vals*1.0e9);	for (i=0;i<n_vals;i++) vals[i]=1.0e-310; 	printf(" Denorm: %.3f ns/float\n",time_function(average_vals)/n_vals*1.0e9);	float x=0.0;	for (i=0;i<n_vals;i++) vals[i]=1.0/x; 	printf(" Inf: %.3f ns/float\n",time_function(average_vals)/n_vals*1.0e9);	for (i=0;i<n_vals;i++) vals[i]=x/x; 	printf(" NaN: %.3f ns/float\n",time_function(average_vals)/n_vals*1.0e9);	return 0;}`
On my P4, this gives 3ns for zeros and ordinary values, 300ns for denormals (a 100x slowdown), and 700ns for infinities and NaNs (a 200x slowdown)!

On my PowerPC 604e, this gives 35ns for zeros, 65ns for denormals (a 2x slowdown), and 35ns for infinities and NaNs (no penalty).

My friends at Illinois and I wrote a paper on this with many more performance details.

Faster Performance via Loop Fusion

Because loading and storing floats is more expensive than performing arithmetic on them, doing several loops through the same data is pretty expensive:
`enum {n=1000};float a[n], b[n];int foo(void) {	for (int i=0;i<n;i++) a[i]=b[i]+3;	for (int i=0;i<n;i++) a[i]+=b[i]*7;	for (int i=0;i<n;i++) a[i]*=4;	return 0;}`

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This takes 2.8ns per float; about one nanosecond float per pass.  We get substantially better performance by "fusing" the loops:
`enum {n=1000};float a[n], b[n];int foo(void) {	for (int i=0;i<n;i++) {		a[i]=b[i]+3;		a[i]+=b[i]*7;		a[i]*=4;	}	return 0;}`

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This takes just 1.54ns per float.  We can do even better with some algebraic transformations:
`enum {n=1000};float a[n], b[n];int foo(void) {	for (int i=0;i<n;i++) {		a[i]=b[i]*32+12;	}	return 0;}`

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Now we're down to 1.0ns per iteration.

Faster Performance via SSE

We start out by making an obvious translation of our C++ code into assembly.  We're using a nasm equ constant for n, and the "times" prefix to make space for the arrays.   Note that we have to store the constants in memory, and reloading the constants every time through the loop makes our version a little slower than the compiler's version above!  This runs at 1.4ns per iteration.
`n equ 1000 ; NASM macromov ecx,0 ; loop counterloopstart:	movss xmm0,[inarr+4*rcx]	movss xmm1,[thirtytwo]	mulss xmm0,xmm1	movss xmm1,[twelve]	addss xmm0,xmm1	movss [outarr+4*rcx],xmm0	add ecx,1	cmp ecx,n	jl loopstartretsection .datainarr: times n dd 1.0outarr: times n dd 0.0twelve: dd 12.0thirtytwo: dd 32.0`

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Loading four floats using "movups" doesn't provide much of a speedup, and we've got to replicate the constant values.  We are down to 1.2ns per iteration:
`n equ 1000 ; NASM macromov ecx,0 ; loop counterloopstart:	movups xmm0,[inarr+4*rcx]	movups xmm1,[thirtytwo]	mulps xmm0,xmm1	movups xmm1,[twelve]	addps xmm0,xmm1	movups [outarr+4*rcx],xmm0	add ecx,4	cmp ecx,n	jl loopstartretsection .datainarr: times n dd 1.0outarr: times n dd 0.0twelve: times 4 dd 12.0thirtytwo: times 4 dd 32.0`

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Switching to an aligned load, "movaps", drops us to 0.36ns per iteration; almost a perfect 4x speedup!
`n equ 1000 ; NASM macromov ecx,0 ; loop counterloopstart:	movaps xmm0,[inarr+4*rcx]	movaps xmm1,[thirtytwo]	mulps xmm0,xmm1	movaps xmm1,[twelve]	addps xmm0,xmm1	movaps [outarr+4*rcx],xmm0	add ecx,4	cmp ecx,n	jl loopstartretsection .dataalign 16inarr: times n dd 1.0outarr: times n dd 0.0twelve: times 4 dd 12.0thirtytwo: times 4 dd 32.0`

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Finally, we can pull the constant loads out of the loop, now down to 0.27ns per float!
`n equ 1000 ; NASM macromov ecx,0 ; loop countermovaps xmm2,[thirtytwo]movaps xmm1,[twelve]loopstart:	movaps xmm0,[inarr+4*rcx]	mulps xmm0,xmm2	addps xmm0,xmm1	movaps [outarr+4*rcx],xmm0	add ecx,4	cmp ecx,n	jl loopstartretsection .dataalign 16inarr: times n dd 1.0outarr: times n dd 0.0twelve: times 4 dd 12.0thirtytwo: times 4 dd 32.0`

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Bonus Speedup: OpenMP

This is really getting far afield from assembly language, but you can use multiple *cores* to speed up some long loops.  For example, this million-float loop is about 4x faster on four cores using an interface called OpenMP (tutorial):
`enum {n=1000000};float a[n], b[n];int foo(void) {#pragma omp parallel for	for (int i=0;i<n;i++) {		a[i]=b[i]*32+12;	}	return 0;}`

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