(Slides) Efficient Evaluation Methods of Elementary Functions Suitable for SIMD Computation

Efficient Evaluation Methods of Elementary Functions Suitable for SIMD Computation Naoki Shibata Shiga University

Overview
Methods for evaluating following functions using SIMD instructions in double precision
sin, cos, tan
log, exp
asin, acos, atan
Fast
Two times as fast as FPU evaluation
Accurate
Maximum error is within 6 ulps from the true value

Overview (contd.)
Advantages against existing methods
No conditional branches
No gathering/scattering
No table look-ups
x y SIMD operation SIMD operation SIMD operation SIMD operation sin x sin y Apply exactly same series of SIMD operations to x and y , and we get sin x and sin y .

Outline
Overview
Background
Cares needed for SIMD optimization
Related works
Proposed method
Trigonometric functions
Inverse trigonometric functions
Exponential function
Logarithmic function
Evaluation
Accuracy
Speed
Code size
Conclusion

Background
SIMD instructions are now pervasive
SSE in x86 processors
Altivec in Power / PowerPC processors
NEON in ARM processors
Cell Broadband Engine
Many GPU models
Length of SIMD registers is going to be extended
256 bits in Sandy Bridge
512 bits in Larrabee (or Knights Ferry?) GPUs

Background (contd.)
SIMD inst. set does not include instructions for evaluating elmentary functions.
sin, cos, tan, log, exp, asin, acos, atan
Two possibilities
FPU
Elementary functions are available in limited architectures.
Software library
Many are not optimized for SIMD calculation

Cares needed in SIMD optimization
We need special cares for SIMD optimization
Memory access and conditional branches are slow
with modern processor models with long pipeline
Table look-up is slow
Gathering and scattering operations are slow
Some traditionally slow operations are not slow anymore
Division and Square root can now be evaluated using one instruction each
No extra register needed
We need new, specialized algorithms to make efficient uses of SIMD unit

Gathering and scattering operations are slow
We need to compose/decompose each element of vector register
At least one instruction is needed for each element
SIMD ALU may be idle during this operation
Register spills may happen
This requires extra execution of instructions
This may cause memory access

Table look-up is slow
Table look-up is frequently used in traditional implementation for evaluating elementary functions
Both in HW and SW
x y x y Scatter Look-up Look-up x' y' x y Gather Table This part must be repeated for each element

Division and SQRT are not too slow
69 clocks of latency and throughput for 1 execution of SQRTPD instruction[*]
69 clocks of latency and throughput for 1 execution of DIVPD instruction[*]
Memory access latency is 165 clocks
(C2Q 9300, measured by CPU-Z)
The good point is that they do not require extra instruction execution and registers
[*] Intel 64 and IA-32 Architectures Optimization Reference Manual

Challenge
Usually, evaluation of elementary functions is performed using calculation with extra precision
e.g. x86 FPU does evaluation using 80bit calculation, and round the result into 64bit.
With extra-precision calculation, obtaining accurate output is not very hard.
But in the proposed method, 64bit-precision calculation is only available
Thus, error in each step of evaluation leads to the error in the output
So, we cannot tolerate error in each step

Related Works
GNU C Library and FPU emulator in Linux OS utilize conditional branches
Thus, not suitable for SIMD computation
There are many researches on evaluating elementary functions on hardware
Many of them utilize table look-ups
Not suitable for SIMD computation
There are several multiple-precision floating-point computation libraries
The design is very different from our implementation

Outline
Overview
Background
Related works
Proposed method
Evaluation
Accuracy
Speed
Code size
Conclusion

Finds sin x and cos x at the same time
Consists of two steps
Step 1 : Argument is reduced within 0 and  /4
utilizing the symmetry and periodicity
Step 2 : Evaluate sin and cos functions on the reduced argument
tan x can be evaluated by simply calculating (sin x / cos x )

Given argument d , find s and an integer q so that 0 <= s <  /4
Is it as easy as just dividing d with  /4? – No!
Finding q is easy, but s may be inaccurate because of cancellation error
when d is a large number close to a multiple of  /4
Some implementations like FPU on x86 processors exhibits this problem
Reducing argument within 0 and  /4

Cancellation error
Cancellation error happens when we subtract one number from another where
Two numbers are very close
Both numbers have limited precision, and already rounded
Suppose that we are calculating 1 – x , with 5 digits of precision.
True value of x is 0.999985 which is rounded to 0.99999
True value : 1 – 0.999985 = 0.000015
Rounded : 1 – 0.99999 = 0.00001
In this case, the results has only 1 digit of accuracy. Loss of accuracy is caused by cancellation

Solution for this problem
Basic idea : calculate this part with extra precision utilizing properties of IEEE 754
q is assumed to be expressed in 26 bits, which is a half of mantissa
 /4 is split into three parts
so that all of the bits in the lower half of the mantissa are zero,

Solution for this problem (contd.)
Multiplying these numbers with q does not produce an error
because, lower half of these numbers are zero
Subtracting the resulting numbers in descending order from d does not produce cancellation error
Results of each step are all accurate
In this way, we can obtain s accurately.

Simply summing up the terms of the Taylor series does not produce accurate result
Because of accumulation of errors
Using another kind of reduction is common
Utilizing triple angle formula of sine
e.g. First divide x by 3, evaluate Taylor series, and then apply triple angle formula
This speeds up calculation by reducing x
Step 2 in evaluating trigonometric functions

This method is not good for our case.
When sin x is small, difference between two terms are so large and it produces rounding error
So, we use double angle formula of cosine, instead
sin x can be found by the following formula
Step 2 (contd.)

Step 2 (contd.)
But, we have another problem
When x is close to 0 , cos 2 x gets close to 1 , and we have cancellation error
So, we use double angle formula of ( 1 – cos x )
Let f ( x ) be ( 1 - cos x ), and we get
And, these do not produce cancellation or rounding errors

We are going to find atan x , and obtain asin x and acos x from atan x
Again, just evaluating terms of Taylor series produces accumulation of errors
There seems no good way of argument reduction is for inverse trigonometric functions
We propose a new way of argument reduction for atan

New argument reduction for evaluating arc tangent (1/3)
Suppose x = atan d , thus tan x = d
If d <= 1 , we evaluate atan function on the argument of 1 / d instead of d
Then subtract the result from  /2
Now, we can assume that d > 1 and  /4 < x <  /2

We use following formulas
We use (13) to calculate cotangent of argument d
Then we repeatedly use (14) to find cotangent of ( d / 2 n )
which is actually "enlarging" cot d
By (13), enlarging cot d corresponds to reducing tan d
(13) (14)

Suppose that atan 2 =  , tan  = 2
Apply (13) to get
Apply (14) to get
Apply (13) to get
= 0.618 is less than 2, thus we reduced the argument
(13) (14) After argument reduction, we calculate Taylor series of atan

asin and acos functions
We have problem if we simply use the following functions
These produce cancellation errors when | x | is close to 1
This problem can be avoided by small modifications of the formulas

Consists of two steps
Similar to trigonometric functions
Step 1 : Argument is reduced within 0 and log e 2
Step 2 : Further reduce the argument, and evaluate Taylor series

Step 1
We find s and an integer q so that 0 < s <= log e 2
This step is very similar to the first step for trigonometric functions
The same problem arises, and we solve the problem in the same way

Step 2
We can use the following formula to further reduce argument
And the evaluate Taylor series
But, if x is close to 0, we will have rounding errors
Since difference between 1 and other terms are large
So, we find (exp x ) – 1, instead of exp x
And the remaining part is similar to sin and cos

We use the following series, rather than Taylor series
This series converges faster than Taylor series
Just evaluating this series is enough.
This series is well known.

Outline
Overview
Background
Related works
Proposed method
Evaluation
Accuracy
Speed
Code size
Conclusion

Evaluation
Accuracy
We compared the output by proposed method and output by MPFR library
We measured the evaluation accuracy within a few ranges for each function
Speed
We compared the speed of the proposed methods, an FPU and MPFR Library
We used Core i7 920 (2.66GHz), Core 2 Duo E7200 (2.53GHz)

Accuracy Accuracy results for sin, cos and tan Accuracy results for atan Accuracy results for asin and acos

Accuracy (contd.) Accuracy results for exp Accuracy results for log In all cases, error does not exceed 6 ulps

Speed Proposed method is about two times as fast as FPU calculation

Code size
The total code size is very small
Suitable for Cell B.E.
which has only 256K bytes of directly accessible scratch pad memory in each SPE.

Conclusion
We proposed efficient methods for evaluating elementary functions in double precision
It does not include table lookups, scattering from, or gathering into SIMD registers, or conditional branches.
The average and maximum errors were less than 0.67 ulp and 6 ulps, respectively.
The evaluation speed was faster than the FPUs on Intel Core 2 and Core i7 processors.

Thank you
An implementation of the proposed method is now available as public domain software.
Contact
http://freshmeat.net/projects/sleef

(Slides) Efficient Evaluation Methods of Elementary Functions Suitable for SIMD Computation

by Naoki Shibata (柴田直樹)

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