precision - 1 - sin x = (1 - sin x) * (1 + sin x) (1 + sin...

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Week 3 - HW submission - Nested multiplication - Loss of precision
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Nested Multiplication p.12 #6: 5x 3 + 2x 2 + x + 10 = ((5x + 2)x + 1)x + 10
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p.12 #8: (c) 6(x+2) 3 + 9(x+2) 7 +3(x+2) 15 - (x+2) 31 (x+2) 3 (6 + 9(x+2) 4 +3(x+2) 12 - (x+2) 28 ) (x+2) 3 (6 + (x+2) 4 (9 + 3(x+2) 8 -(x+2) 24 )) (x+2) 3 (6 + (x+2) 4 (9+(x+2) 8 (3-(x+2) 16 ))) Done! Nested Multiplication
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p.12 #8: (d) -3x 7 + 10x 17 + - 5x 37 + x 127 x 7 (-3 + 10x 10 - 5x 30 + x 120 ) x 7 (-3 + x 10 (10 - 5x 20 + x 110 )) x 7 (-3 + x 10 (10 + x 20 (-5 + x 90 ))) Nested Multiplication
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Loss of precision - Rationalization - Taylor expansion
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Loss of precision Avoid loss of significance around 2: √(x+4) - 2 = (√(x+4) - 2) * √ (x+4) + 2 √(x+4) + 2 = x+4 - 4 = x √(x+4) + 2 √(x+4) + 2 √(x+4) - 2
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Avoid loss of significance around ∏/2
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Unformatted text preview: 1 - sin x = (1 - sin x) * (1 + sin x) (1 + sin x) = (1 - sin 2 x) (1 + sin x) = (cos 2 x ) (1 + sin x) 1 - sin x Loss of precision Loss of precision cos x = 1 - x 2 + x 4- . .. 2! 4! 1 - cos x = 1 - (1 - x 2 + x 4 ) 2! 4! = x 2- x 4 2! 4! Why does this help? If you have x-y, 2 < 1 - y/x < 2 Then p is the minimum bits wrong and q is the maximum bits wrong Subtraction error:-q-p Loss of precision Precision loss example: x - sin(x) around x=0.1 2-10 < 1 - sin 0.1 < 2-9 0.1 Loss of precision 0.00097 < 0.0016 < 0.0019...
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precision - 1 - sin x = (1 - sin x) * (1 + sin x) (1 + sin...

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