Accurate Sum and Dot Product with New Instruction for High-Precision Computing on ARMv8 Processor
The accumulation of rounding errors can lead to unreliable results. Therefore, accurate and efficient algorithms are required. A processor from the ARMv8 architecture has introduced new instructions for high-precision computation. We have redesigned and implemented accurate summation and the accurat...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/2/270 |
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| Summary: | The accumulation of rounding errors can lead to unreliable results. Therefore, accurate and efficient algorithms are required. A processor from the ARMv8 architecture has introduced new instructions for high-precision computation. We have redesigned and implemented accurate summation and the accurate dot product. The number of floating-point operations has been reduced from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>7</mn><mi>n</mi><mo>−</mo><mn>5</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>10</mn><mi>n</mi><mo>−</mo><mn>5</mn></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>7</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></semantics></math></inline-formula>, compared with the classic compensated precision algorithms. It has been proven that our accurate summation and dot algorithms’ error bounds are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>γ</mi><mi>n</mi></msub><mi>cond</mi><mo>+</mo><mi>u</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mi>n</mi></msub><msub><mi>γ</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>cond</mi><mo>+</mo><mi>u</mi></mrow></semantics></math></inline-formula>, where ‘cond’ denotes the condition number, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mi>n</mi></msub><mo>=</mo><mi>n</mi><mo>·</mo><mi>u</mi><mo>/</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>n</mi><mo>·</mo><mi>u</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and <i>u</i> denotes the relative rounding error unit. Our accurate summation and dot product achieved a 1.69× speedup and a 1.14× speedup, respectively, on a simulation platform. Numerical experiments also illustrate that, under round-towards-zero mode, our algorithms are as accurate as the classic compensated precision algorithms. |
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| ISSN: | 2227-7390 |