Fast inverse square root is used to estimate the reciprocal of the square root For given...
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Fast inverse square root is used to estimate the reciprocal of the square root For given quadratic function √y x² + 6x + 0.21875 = 0, the real roots are around -5.9633 or -0.03668. By using quadratic formula x = -b+√b²-4ac 2a where √b2 the reciprocal of square root of given decimal number y to solve the quadratic function. 4ac can be estimated by inverse square root. Follow the procedure to estimate a) Let b² - 4ac = (35.125)10, covert it into IEEE-754 32-bit floating point format. b) Shift right function SHIFT_RIGHT(y, 1) means that getting a new number with the same number of bits by discarding the least significant bit (LSB) and prepending Os as the new most significant bit (MSB). For example, SHIFT_RIGHT(1011, 1) = 0101, SHIFT_RIGHT (0001, 1) = 0000. Supposing y is the result of question 3a, perform the shift right function SHIFT_RIGHT(y, 1) and convert the corresponding result into hexadecimal number. c) Magic number (5F3759DF) 16 is the approximation of √2¹27 in IEEE-754 32-bit floating point format. Calculate z = (5F3759DF) 16 - Y16, where y16 is the hexadecimal number calculated in 3b. d) Convert z in part c into IEEE-754 32-bit floating point format and convert this number into decimal number. The decimal number is the estimation of the reciprocal of square root of b². - 4ac. Here, the approximation error is around 2.5%. (Optional (inside bracket, no mark), by using √x = x to √x estimate the roots of quadratic equation, approximation error of one of the root will be larger than 10%. Newton's method can be applied to increase the accuracy by Yn+1 = Yn (1.5 — *v²) if y = where - Yn is the estimated value and n is number of iteration.) Fast inverse square root is used to estimate the reciprocal of the square root For given quadratic function √y x² + 6x + 0.21875 = 0, the real roots are around -5.9633 or -0.03668. By using quadratic formula x = -b+√b²-4ac 2a where √b2 the reciprocal of square root of given decimal number y to solve the quadratic function. 4ac can be estimated by inverse square root. Follow the procedure to estimate a) Let b² - 4ac = (35.125)10, covert it into IEEE-754 32-bit floating point format. b) Shift right function SHIFT_RIGHT(y, 1) means that getting a new number with the same number of bits by discarding the least significant bit (LSB) and prepending Os as the new most significant bit (MSB). For example, SHIFT_RIGHT(1011, 1) = 0101, SHIFT_RIGHT (0001, 1) = 0000. Supposing y is the result of question 3a, perform the shift right function SHIFT_RIGHT(y, 1) and convert the corresponding result into hexadecimal number. c) Magic number (5F3759DF) 16 is the approximation of √2¹27 in IEEE-754 32-bit floating point format. Calculate z = (5F3759DF) 16 - Y16, where y16 is the hexadecimal number calculated in 3b. d) Convert z in part c into IEEE-754 32-bit floating point format and convert this number into decimal number. The decimal number is the estimation of the reciprocal of square root of b². - 4ac. Here, the approximation error is around 2.5%. (Optional (inside bracket, no mark), by using √x = x to √x estimate the roots of quadratic equation, approximation error of one of the root will be larger than 10%. Newton's method can be applied to increase the accuracy by Yn+1 = Yn (1.5 — *v²) if y = where - Yn is the estimated value and n is number of iteration.)
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Related Book For
Fundamentals Of Momentum Heat And Mass Transfer
ISBN: 9781118947463
6th Edition
Authors: James Welty, Gregory L. Rorrer, David G. Foster
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