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

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Naoki Shibata
Naoki Shibata

The document proposes efficient methods for evaluating elementary functions like sin, cos, tan, log, and exp using SIMD instructions. The methods are twice as fast as floating point unit evaluation and have a maximum error of 6 ulps. They avoid conditional branches, gathering/scattering operations, and table lookups. Trigonometric functions are evaluated in two steps - argument reduction followed by a series evaluation. Inverse trigonometric, exponential and logarithmic functions are also efficiently evaluated in a similar manner suitable for SIMD computation. Evaluation accuracy and speed are evaluated against existing methods and the code size is kept small.

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Efficient Evaluation Methods of Elementary Functions Suitable for SIMD Computation Naoki Shibata Shiga University
Overview ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Overview (contd.) ,[object Object],[object Object],[object Object],[object Object],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 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Background ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Background (contd.) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Cares needed in SIMD optimization ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Gathering and scattering operations are slow ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Table look-up is slow ,[object Object],[object Object],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 ,[object Object],[object Object],[object Object],[object Object],[object Object],[*] Intel 64 and IA-32 Architectures Optimization Reference Manual
Challenge ,[object Object],[object Object],[object Object],[object Object],[object Object],So, we cannot tolerate error in each step
Related Works ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Trigonometric functions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],Reducing argument within 0 and  /4
Cancellation error ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Solution for this problem ,[object Object],[object Object],[object Object],[object Object]
Solution for this problem (contd.) ,[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Step 2 in evaluating trigonometric functions
[object Object],[object Object],[object Object],[object Object],Step 2 (contd.)
Step 2 (contd.) ,[object Object],[object Object],[object Object],[object Object],[object Object]
Inverse trigonometric functions ,[object Object],[object Object],[object Object],[object Object]
New argument reduction for evaluating arc tangent (1/3) ,[object Object],[object Object],[object Object],[object Object]
New argument reduction for evaluating arc tangent (2/3) ,[object Object],[object Object],[object Object],[object Object],[object Object],(13) (14)
New argument reduction for evaluating arc tangent (3/3) ,[object Object],[object Object],[object Object],[object Object],[object Object],(13) (14) After argument reduction, we calculate Taylor series of atan
asin and acos functions ,[object Object],[object Object],[object Object]
Exponential function ,[object Object],[object Object],[object Object],[object Object]
Step 1 ,[object Object],[object Object],[object Object]
Step 2 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Logarithmic function ,[object Object],[object Object],[object Object],[object Object]
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Evaluation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
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 ,[object Object],[object Object],[object Object]
Conclusion ,[object Object],[object Object],[object Object],[object Object]
Thank you ,[object Object],[object Object],http://freshmeat.net/projects/sleef

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

  • 1. Efficient Evaluation Methods of Elementary Functions Suitable for SIMD Computation Naoki Shibata Shiga University
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  • 33. Accuracy Accuracy results for sin, cos and tan Accuracy results for atan Accuracy results for asin and acos
  • 34. Accuracy (contd.) Accuracy results for exp Accuracy results for log In all cases, error does not exceed 6 ulps
  • 35. Speed Proposed method is about two times as fast as FPU calculation
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